Showing 1–50 of 132 results for author: Fu, S

Inverse problems for nonlinear Kirchhoff plate equations with multiple unknown parameters

Authors: Song-Ren Fu, Hongyu Liu, Yongyi Yu, Tianyi Zheng

Abstract: This paper provides a comprehensive treatment of inverse boundary value problems for (nonlinear) Kirchhoff plate equations under diverse general settings. We begin by establishing the global well-posedness of the nonlinear forward equations, which not only underpins the subsequent inverse analysis but also holds independent theoretical significance. The inverse problems are then examined for both… ▽ More This paper provides a comprehensive treatment of inverse boundary value problems for (nonlinear) Kirchhoff plate equations under diverse general settings. We begin by establishing the global well-posedness of the nonlinear forward equations, which not only underpins the subsequent inverse analysis but also holds independent theoretical significance. The inverse problems are then examined for both passive and active measurement regimes. With a single passive boundary measurement, we establish the stable recovery of the unknown initial data. In the active regime with infinitely many boundary measurements, our results are twofold. For linear equations featuring generic time-dependent potentials-allowing for spatial unboundedness, we demonstrate the simultaneous recovery of both initial data and coefficients. For nonlinear equations, where both the nonlinearity and initial data are unknown, we develop a novel Runge approximation approach, together with carefully constructed geometric optics solutions and higher-order linearization around nonzero solutions, to prove their simultaneous determination. Furthermore, we introduce a delicate cut-off technique that provides an alternative means of addressing the scenario of vanishing initial data. Notably, the methodologies and results developed herein are readily generalizable to other boundary conditions and plate models, including the classical Euler-Bernoulli equation. △ Less

Submitted 8 August, 2026; originally announced August 2026.

MSC Class: Primary 35R30; Secondary 26A33; 42B37

Convolutive sequences, II: Parametrizations

Authors: Shane Chern, Dennis Eichhorn, Shishuo Fu, James A. Sellers

Abstract: In recent work, the authors defined a sequence $(a_n)_{n\ge 0}$ to be $m$-convolutive exactly if \begin{align*} \sum_{n\ge 0} a_{mn} q^n = \left(\sum_{n\ge 0} a_n q^n\right)^m \end{align*} for a specific positive integer $m$ and provided proofs of the $2$- and $3$-convolutivity of a small number of sequences arising from primitive eta-products. Since the completion of that work, the author… ▽ More In recent work, the authors defined a sequence $(a_n)_{n\ge 0}$ to be $m$-convolutive exactly if \begin{align*} \sum_{n\ge 0} a_{mn} q^n = \left(\sum_{n\ge 0} a_n q^n\right)^m \end{align*} for a specific positive integer $m$ and provided proofs of the $2$- and $3$-convolutivity of a small number of sequences arising from primitive eta-products. Since the completion of that work, the authors have discovered many new instances of convolutive eta-products. The main focus of this work is to unify all but one of these instances in a parametric way. △ Less

Submitted 2 August, 2026; originally announced August 2026.

Comments: 17 pages, 2 tables

MSC Class: 11B83; 05A15; 05A17

Signed Counting on Restricted Partitions and Combinatorial Proofs of Three Identities

Authors: Shishuo Fu, Chenwei Wang

Abstract: Signed enumerations of l-regular partitions by the parity of the number of parts are known to correspond to partitions with congruence conditions. Inspired by the recent combinatorial approaches of Ballantine-Merca and Liu for such identities, we provide combinatorial proofs of three identities for l-regular partitions and their variants. Two of these were originally established analytically by Hi… ▽ More Signed enumerations of l-regular partitions by the parity of the number of parts are known to correspond to partitions with congruence conditions. Inspired by the recent combinatorial approaches of Ballantine-Merca and Liu for such identities, we provide combinatorial proofs of three identities for l-regular partitions and their variants. Two of these were originally established analytically by Hickerson and Robbins, respectively. △ Less

Submitted 19 July, 2026; originally announced July 2026.

Comments: 16pages

MSC Class: 05A17; 05A19; 11P84

When arrow patterns meet classical patterns

Authors: Shishuo Fu, Zhenghe Yang

Abstract: Seeking to bridge the structural divide between a permutation's cycle notation and its one-line notation, Berman and Tenner introduced a novel notion of permutation pattern known as the arrow pattern. Recently, Archer and Laudone initiated a systematic study of arrow pattern avoidance, leaving behind three intriguing conjectures. In this paper, we resolve all three conjectures. First, we enumerate… ▽ More Seeking to bridge the structural divide between a permutation's cycle notation and its one-line notation, Berman and Tenner introduced a novel notion of permutation pattern known as the arrow pattern. Recently, Archer and Laudone initiated a systematic study of arrow pattern avoidance, leaving behind three intriguing conjectures. In this paper, we resolve all three conjectures. First, we enumerate all six subclasses of permutations that simultaneously avoid a classical pattern of length 3 and a fixed arrow pattern of length 3, thereby confirming the first two conjectures. Second, we settle the third conjecture (which involves a different arrow pattern) by providing two independent proofs. These proofs rely on a restriction of Biane's bijection to non-nesting involutions and Krattenthaler's bijection from 321-avoiding permutations to Dyck paths, respectively. △ Less

Submitted 4 July, 2026; originally announced July 2026.

Comments: 17 pages, 3 figures, 4 tables

MSC Class: 05A05; 05A15; 05A19

Edge Multiscale Finite Element Methods

Authors: Shubin Fu, Guanglian Li

Abstract: The objective of this paper is to review recent developments in Edge Multiscale Finite Element Methods (EMsFEM) for partial differential equations with heterogeneous coefficients or highly oscillatory solutions. Using elliptic equations with heterogeneous coefficients as an illustrative example, we present the key ideas of the method. We also analyze the approach while accounting for the discrete… ▽ More The objective of this paper is to review recent developments in Edge Multiscale Finite Element Methods (EMsFEM) for partial differential equations with heterogeneous coefficients or highly oscillatory solutions. Using elliptic equations with heterogeneous coefficients as an illustrative example, we present the key ideas of the method. We also analyze the approach while accounting for the discrete error in the multiscale basis functions. Extensive numerical tests are provided to validate the performance of the method. △ Less

Submitted 4 July, 2026; originally announced July 2026.

Three-Body Earth-Moon Transfers with Different Departure/Arrival Orbital Altitudes: New Phenomenon and Diffusion Model-Augmented Construction

Authors: Shuyue Fu, Wenxuan Zhang, Di Wu, Shengping Gong, Peng Shi

Abstract: Construction of Earth-Moon transfers is the basis of missions to explore the Moon and cislunar space. The traditional grid search method suffers from a relatively low convergence rate and computational efficiency, mainly focusing on the distribution of transfer characteristic parameters. Moreover, when constructing transfers with different departure/arrival orbital altitudes, the process of grid s… ▽ More Construction of Earth-Moon transfers is the basis of missions to explore the Moon and cislunar space. The traditional grid search method suffers from a relatively low convergence rate and computational efficiency, mainly focusing on the distribution of transfer characteristic parameters. Moreover, when constructing transfers with different departure/arrival orbital altitudes, the process of grid search and trajectory correction should be repeated with a low convergence rate and computational efficiency. To address these limitations of the traditional grid search method, this paper is devoted to exploring an effective way to augment the grid search method. Bi-impulsive Earth-Moon transfers from a circular Earth parking orbit to a circular Moon target orbit in the Earth-Moon planar circular restricted three-body problem are considered in this paper. Firstly, the transfers are constructed, and the corresponding solution space is explored in terms of construction parameters, including departure phase angle at the Earth parking orbit, initial-to-circular velocity ratio, and time of flight. An interesting phenomenon about the discontinuous behavior of the time-of-flight distribution with respect to departure phase angle is identified. This phenomenon is further used to train a diffusion model, which aims to augment the traditional grid search method and generate high-quality initial guesses for transfers with different departure/arrival orbital altitudes. The construction results of the proposed method are presented and analyzed. The proposed diffusion model-augmented grid search method improves the convergence rate by 47.34-56.25% and saves the wall-clock time by 39.39-40.52% over the traditional grid search method relatively, while ensuring comparable transfer characteristics. △ Less

Submitted 4 August, 2026; v1 submitted 26 June, 2026; originally announced June 2026.

On mesh patterns of short length: Equidistribution and enumeration

Authors: Qi Fang, Shishuo Fu, Sergey Kitaev, Haijun Li, Xinyu Su, Ziyao Sun

Abstract: The classification and enumeration of short mesh patterns have emerged as two central directions in the area. We make substantial progress on both fronts. We construct an involution and a bijection that establish distributional equivalences for two classes of length-$2$ mesh patterns, thereby resolving a conjecture from 2019 and a recent conjecture. As a consequence, the best known upper bounds fo… ▽ More The classification and enumeration of short mesh patterns have emerged as two central directions in the area. We make substantial progress on both fronts. We construct an involution and a bijection that establish distributional equivalences for two classes of length-$2$ mesh patterns, thereby resolving a conjecture from 2019 and a recent conjecture. As a consequence, the best known upper bounds for the numbers of distribution-equivalence and Wilf-equivalence classes drop to $106$ and $47$, respectively. Combined with the known lower bounds of 105 and 46, conjectured to be exact, these results leave both classifications hinging on a single distribution-equivalence question conjectured in 2019, whose resolution would at once settle the remaining Wilf-equivalence case. We further conjecture that this unresolved equidistribution also holds for involutions, a subclass of all permutations. We also determine the distributions of three additional classes of length-$2$ mesh patterns through a detailed structural analysis. Our work combines bijective techniques with generating-function methods, yielding new insights into the structure and enumeration of short mesh patterns. △ Less

Submitted 12 June, 2026; originally announced June 2026.

Comments: 18 pages,4 figures, comments are welcome

MSC Class: 05A05; 05A15; 05A19

An algebraic multiscale preconditioner for large sparse SPD matrices

Authors: Yingjie Zhou, Shubin Fu, Eric Tsz Shun Chung

Abstract: We present a two-grid algebraic multiscale preconditioner for large sparse symmetric positive definite systems arising from elliptic problems with highly heterogeneous coefficients. The coarse space is constructed directly from the system matrix by graph partitioning and local generalized eigenvalue solvers, yielding basis functions that capture the low-energy modes responsible for slow convergenc… ▽ More We present a two-grid algebraic multiscale preconditioner for large sparse symmetric positive definite systems arising from elliptic problems with highly heterogeneous coefficients. The coarse space is constructed directly from the system matrix by graph partitioning and local generalized eigenvalue solvers, yielding basis functions that capture the low-energy modes responsible for slow convergence. The method requires no geometric information, making it suitable for unstructured and matrix-only settings, and its construction is naturally parallelizable. Numerical results for heterogeneous Darcy flow problems show robustness with respect to coefficient contrast and problem size, better performance than standard algebraic multigrid on challenging large-scale cases, and good parallel scalability. △ Less

Submitted 3 June, 2026; originally announced June 2026.

Applying Two-Grid Preconditioner for Subsurface Flow Simulation using Attention-enhanced Hybrid Network to Accelerate Multiscale Discretization in High-contrast Media

Authors: Peiqi Li, Jie Chen, Shubin Fu

Abstract: In this paper, we study the efficient numerical solution of Darcy equations in strongly heterogeneous media with high-contrast permeability and propose a hybrid framework that combines learning with multiscale numerical methods. The learning component is used for the prediction of multiscale basis functions in the mixed generalized multiscale finite element method (mixed GMsFEM), with the goal of… ▽ More In this paper, we study the efficient numerical solution of Darcy equations in strongly heterogeneous media with high-contrast permeability and propose a hybrid framework that combines learning with multiscale numerical methods. The learning component is used for the prediction of multiscale basis functions in the mixed generalized multiscale finite element method (mixed GMsFEM), with the goal of reducing the repeated local computations required in the offline stage. Once these basis functions are predicted, the global system is assembled and the pressure field is computed by a two-grid preconditioned solver. The resulting method accelerates the costly local basis-construction stage while retaining the multiscale discretization and preconditioned iterative structure of the underlying solver. Numerical experiments on two-dimensional heterogeneous Darcy problems show that the proposed framework yields more accurate final pressure reconstruction than several representative learning-based methods and remains stable under strong heterogeneity and high-contrast coefficients. In comparison with the traditional mixed GMsFEM, its main advantage lies in the efficiency of the basis-generation stage, while the quality of the global solve is still ensured by the two-grid preconditioner. These results indicate that accelerating multiscale basis construction through learning, while preserving a mature numerical solver for the global problem, provides a viable approach for high-resolution Darcy-type simulations. △ Less

Submitted 15 April, 2026; originally announced June 2026.

Sharp Exponent of Stable Standing Waves for the Perturbated Hartree Equation

Authors: Guoyi Fu, Shanshan Fu, Xiaoguang Li, Jian Zhang, Shihui Zhu

Abstract: This paper is concerned with the stability of standing waves for the mass-critical Hartree equation with a focusing perturbation by the variational method. The profile decomposition theory is employed to prove the attainability of the cross constrained variational problem, and then the comparison of two cross constrained variational problems is derived. The sharp criteria of blowup, the orbital st… ▽ More This paper is concerned with the stability of standing waves for the mass-critical Hartree equation with a focusing perturbation by the variational method. The profile decomposition theory is employed to prove the attainability of the cross constrained variational problem, and then the comparison of two cross constrained variational problems is derived. The sharp criteria of blowup, the orbital stability, and strong instability of standing waves without any frequency constraint are obtained. This improves the cross constrained variational argument proposed by Zhang (2005). △ Less

Submitted 25 March, 2026; originally announced March 2026.

MSC Class: 35Q55

Multiplicities of eigenvalues and quadratic representations of integers

Authors: Siqi Fu, Andrew Pendleton

Abstract: We study the set $M$ of all multiplicities of non-zero eigenvalues for the Laplace operator on a two-dimensional rectangle or torus. We show that for a rectangle with the side length ratio $r$, $M=\mathbb{N}$, the set of all positive integers, if and only if $r^2$ is rational. For a torus whose generating vectors have a length ratio $r$ and the angle between them $θ$, we show that $M$ is an infini… ▽ More We study the set $M$ of all multiplicities of non-zero eigenvalues for the Laplace operator on a two-dimensional rectangle or torus. We show that for a rectangle with the side length ratio $r$, $M=\mathbb{N}$, the set of all positive integers, if and only if $r^2$ is rational. For a torus whose generating vectors have a length ratio $r$ and the angle between them $θ$, we show that $M$ is an infinite set if and only if both $r\cosθ$ and $r^2$ are rational. In this case, $M=2\mathbb{N}$, $4\mathbb{N}$, or $6\mathbb{N}$, and we obtain a characterization for each of these cases in term of $r\cosθ$ and $r^2$. In the case when at least one of $r\cosθ$ or $r^2$ is irrational, we show that $M=\{2\}$ or $\{2, 4\}$, and obtain a characterization for these cases. We prove these results by studying the number of integral lattice points on dilated ellipses. △ Less

Submitted 2 April, 2026; v1 submitted 15 March, 2026; originally announced March 2026.

Comments: 21 pages, 0 figures. Added a remark and a reference

MSC Class: 11E25; 35P15

Discontinuous Behavior of Time-of-Flight Distribution for Bi-impulsive Earth-Moon Transfers in the Three-Body Model

Authors: Shuyue Fu, Di Wu, Shengping Gong

Abstract: As interest in the Earth-Moon transfers renewed around the world, understanding the solution space of transfer trajectories facilitates the construction of transfers. This paper is devoted to reporting a novel or less-reported phenomenon about the solution space of bi-impulsive Earth-Moon transfers in the Earth-Moon planar circular restricted three-body problem. Differing from the previous works f… ▽ More As interest in the Earth-Moon transfers renewed around the world, understanding the solution space of transfer trajectories facilitates the construction of transfers. This paper is devoted to reporting a novel or less-reported phenomenon about the solution space of bi-impulsive Earth-Moon transfers in the Earth-Moon planar circular restricted three-body problem. Differing from the previous works focusing on the transfer characteristics of the solution space, we focus on the distribution of the construction parameters, i.e., departure phase angle at the Earth parking orbit, initial-to-circular velocity ratio, and time of flight. Firstly, the construction method of bi-impulsive transfers is described, and the solutions satisfying the given constraints are obtained from the grid search method and trajectory correction. Then, the distribution of the obtained solutions is analyzed, and an interesting phenomenon about the discontinuous behavior of the time-of-flight distribution for each departure phase angle is observed and briefly reported. This phenomenon can further provide useful insight into the construction of bi-impulsive transfers, deepening the understanding of the corresponding solution space. △ Less

Submitted 30 October, 2025; originally announced October 2025.

Parity patterns meet Genocchi numbers, I: four labelings and three bijections

Authors: Quan Yuan, Qi Fang, Shishuo Fu, Haijun Li

Abstract: Hetyei introduced in 2019 the homogenized Linial arrangement and showed that its regions are counted by the median Genocchi numbers. In the course of devising a different proof of Hetyei's result, Lazar and Wachs considered another hyperplane arrangement that is associated with certain bipartite graph called Ferrers graph. We bijectively label the regions of this latter arrangement with permutatio… ▽ More Hetyei introduced in 2019 the homogenized Linial arrangement and showed that its regions are counted by the median Genocchi numbers. In the course of devising a different proof of Hetyei's result, Lazar and Wachs considered another hyperplane arrangement that is associated with certain bipartite graph called Ferrers graph. We bijectively label the regions of this latter arrangement with permutations whose ascents are subject to a parity restriction. This labeling not only establishes the equivalence between two enumerative results due to Hetyei and Lazar-Wachs, repectively, but also motivates us to derive and investigate a Seidel-like triangle that interweaves Genocchi numbers of both kinds. Applying similar ideas, we introduce three more variants of permutations with analogous parity restrictions. We provide labelings for regions of the aforementioned arrangement using these three sets of restricted permutations as well. Furthermore, bijections from our first permutation model to two previously known permutation models are established. △ Less

Submitted 15 October, 2025; originally announced October 2025.

Comments: 35 pages, 4 tables, and 4 figures

MSC Class: 05A05; 05A15; 05A19; 52C35

An involution for trivariate symmetries of vincular patterns

Authors: Joanna N. Chen, Shishuo Fu, Jiang Zeng

Abstract: We provide a bijective proof of the equidistribution of two pairs of vincular patterns in permutations, thereby resolving a recent open problem of Bitonti, Deb, and Sokal (arXiv:2412.10214). Since the bijection is involutive, we also confirm their conjecture on the equidistribution of triple vincular patterns. Somewhat unexpectedly, we show that this involution is closed on the set of Baxter permu… ▽ More We provide a bijective proof of the equidistribution of two pairs of vincular patterns in permutations, thereby resolving a recent open problem of Bitonti, Deb, and Sokal (arXiv:2412.10214). Since the bijection is involutive, we also confirm their conjecture on the equidistribution of triple vincular patterns. Somewhat unexpectedly, we show that this involution is closed on the set of Baxter permutations, thereby implying another trivariate symmetries of vincular patterns. The proof of this second result requires a variant of a characterization of Baxter permutations in terms of restricted Laguerre histories, first given by Viennot using the Françon-Viennot bijection. △ Less

Submitted 16 September, 2025; originally announced September 2025.

Comments: 19 pages, 3 figures

Signed counting of partition matrices

Authors: Shane Chern, Shishuo Fu

Abstract: We prove that the signed counting (with respect to the parity of the ``$\operatorname{inv}$'' statistic) of partition matrices equals the cardinality of a subclass of inversion sequences. In the course of establishing this result, we introduce an interesting class of partition matrices called improper partition matrices. We further show that a subset of improper partition matrices is equinumerous… ▽ More We prove that the signed counting (with respect to the parity of the ``$\operatorname{inv}$'' statistic) of partition matrices equals the cardinality of a subclass of inversion sequences. In the course of establishing this result, we introduce an interesting class of partition matrices called improper partition matrices. We further show that a subset of improper partition matrices is equinumerous with the set of Motzkin paths. Such an equidistribution is established both analytically and bijectively. △ Less

Submitted 22 April, 2026; v1 submitted 28 August, 2025; originally announced August 2025.

Comments: 28 pages. Most of Section 4 has been rewritten, with more questions raised in Outlook

MSC Class: 05A05; 05A15; 05A19

Families of Transfers from circular low Earth orbit to Distant Prograde Orbit around the Moon

Authors: Shuyue Fu, Di Wu, Yihan Peng, Peng Shi, Shengping Gong

Abstract: Distant prograde orbits around the Moon exhibit remarkable potential for practical applications such as cislunar surveillance activities and low-energy transfers due to their instability. Previous works on transfers from circular low Earth orbit to distant prograde orbits mainly focused on construction methods based on dynamical structures, lacking a comprehensive analysis of the solution space of… ▽ More Distant prograde orbits around the Moon exhibit remarkable potential for practical applications such as cislunar surveillance activities and low-energy transfers due to their instability. Previous works on transfers from circular low Earth orbit to distant prograde orbits mainly focused on construction methods based on dynamical structures, lacking a comprehensive analysis of the solution space of this transfer scenario. This paper investigates the solution space and identifies families of transfers from a 167 km circular low Earth orbit to a 1:1 distant prograde orbit. In particular, grid search and trajectory continuation are performed to construct these transfer trajectories. Initial guesses of the transfers are selected in the 1:1 distant prograde orbit through a backward propagation strategy and are then corrected to satisfy specified constraints. Based on the obtained solutions, a linear predictor is derived to predict more feasible solutions and a predictor-corrector continuation method is used to extend the solution space. Twelve transfer families are identified, most of which are new or previously underexplored. The distributions of construction parameters and transfer characteristics of these twelve families are analyzed and discussed, showing which families are applicable to which types of specific practical missions. Comparison between the obtained solution and solution developed by previous works is further performed to imply the effects of the selection of dynamical model on transfer construction. △ Less

Submitted 3 August, 2025; originally announced August 2025.

Convolutive sequences, I: Through the lens of integer partition functions

Authors: Shane Chern, Dennis Eichhorn, Shishuo Fu, James A. Sellers

Abstract: Motivated by the convolutive behavior of the counting function for partitions with designated summands in which all parts are odd, we consider coefficient sequences $(a_n)_{n\ge 0}$ of primitive eta-products that satisfy the generic convolutive property \begin{align*} \sum_{n\ge 0} a_{mn} q^n = \left(\sum_{n\ge 0} a_n q^n\right)^m \end{align*} for a specific positive integer $m$. Given the… ▽ More Motivated by the convolutive behavior of the counting function for partitions with designated summands in which all parts are odd, we consider coefficient sequences $(a_n)_{n\ge 0}$ of primitive eta-products that satisfy the generic convolutive property \begin{align*} \sum_{n\ge 0} a_{mn} q^n = \left(\sum_{n\ge 0} a_n q^n\right)^m \end{align*} for a specific positive integer $m$. Given the results of an exhaustive search of the Online Encyclopedia of Integer Sequences for such sequences for $m$ up to $6$, we first focus on the case where $m=2$ with our attention mainly paid to the combinatorics of two $2$-convolutive sequences, featuring bijective proofs for both. For other $2$-convolutive sequences discovered in the OEIS, we apply generating function manipulations to show their convolutivity. We also give two examples of $3$-convolutive sequences. Finally, we discuss other convolutive series that are not eta-products. △ Less

Submitted 3 December, 2025; v1 submitted 15 July, 2025; originally announced July 2025.

Comments: 23 pages, accepted by Experimental Mathematics

MSC Class: 05A17; 11P81

On fourteen equidistribution conjectures of Lv and Zhang and monotone mesh patterns with corner shadings

Authors: Qi Fang, Shishuo Fu, Sergey Kitaev, Haijun Li

Abstract: Three complementation-like involutions are constructed on permutations to prove, and in some cases generalize, all remaining fourteen joint symmetric equidistribution conjectures of Lv and Zhang. Further enumerative results are obtained for several classes of (mesh) pattern-avoiding permutations, where the shadings of all involved mesh patterns are restricted to an opposing pair of corners. Three complementation-like involutions are constructed on permutations to prove, and in some cases generalize, all remaining fourteen joint symmetric equidistribution conjectures of Lv and Zhang. Further enumerative results are obtained for several classes of (mesh) pattern-avoiding permutations, where the shadings of all involved mesh patterns are restricted to an opposing pair of corners. △ Less

Submitted 7 July, 2025; originally announced July 2025.

A residual driven multiscale method for Darcy's flow in perforated domains

Authors: Wei Xie, Shubin Fu, Yin Yang, Yunqing Huang

Abstract: In this paper, we present a residual-driven multiscale method for simulating Darcy flow in perforated domains, where complex geometries and highly heterogeneous permeability make direct simulations computationally expensive. To address this, we introduce a velocity elimination technique that reformulates the mixed velocity-pressure system into a pressure-only formulation, significantly reducing co… ▽ More In this paper, we present a residual-driven multiscale method for simulating Darcy flow in perforated domains, where complex geometries and highly heterogeneous permeability make direct simulations computationally expensive. To address this, we introduce a velocity elimination technique that reformulates the mixed velocity-pressure system into a pressure-only formulation, significantly reducing complexity by focusing on the dominant pressure variable. Our method is developed within the Generalized Multiscale Finite Element Method (GMsFEM) framework. For each coarse block, we construct offline basis functions from local spectral problems that capture key geometric and physical features. Online basis functions are then adaptively enriched using residuals, allowing the method to incorporate global effects such as source terms and boundary conditions, thereby improving accuracy. We provide detailed error analysis demonstrating how the offline and online spaces contribute to the accuracy and efficiency of the solution. Numerical experiments confirm the method's effectiveness, showing substantial reductions in computational cost while maintaining high accuracy, particularly through adaptive online enrichment. These results highlight the method's potential for efficient and accurate simulation of Darcy flow in complex, heterogeneous perforated domains. △ Less

Submitted 9 March, 2026; v1 submitted 29 June, 2025; originally announced June 2025.

MatExPre: A matrix exponential preconditioner for the high-frequency Helmholtz equation

Authors: Shubin Fu, Qing Huo Liu, Qiwei Zhan, Eric T. Chung, Changqing Ye

Abstract: In this article, we present a new preconditioner, MatExPre, for the high-frequency Helmholtz equation by leveraging the properties of matrix exponentials. Our approach begins by reformulating the Helmholtz equation into a Schrödinger-like equation and constructing a time-domain solver based on a fixed-point iteration. We then establish a rigorous connection between the time-domain solver and matri… ▽ More In this article, we present a new preconditioner, MatExPre, for the high-frequency Helmholtz equation by leveraging the properties of matrix exponentials. Our approach begins by reformulating the Helmholtz equation into a Schrödinger-like equation and constructing a time-domain solver based on a fixed-point iteration. We then establish a rigorous connection between the time-domain solver and matrix exponential integrators, which enables us to derive algebraic preconditioners that rely solely on sparse matrix-vector products. Spectral analysis and a detailed numerical implementation strategy, including performance improvements achieved through complex shifting, are discussed. Finally, numerical experiments on 2D and large-scale 3D homogeneous and inhomogeneous models, including benchmark seismic examples, substantiate the effectiveness and scalability of the proposed methods. △ Less

Submitted 4 June, 2025; originally announced June 2025.

MSC Class: 65M22; 65F08

A refined view of a curious identity for partitions into odd parts with designated summands

Authors: Shishuo Fu, James Sellers

Abstract: In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called partitions with designated summands. These are constructed by taking unrestricted integer partitions and designating exactly one of each occurrence of a part. In the same work, they also considered the restricted partitions with designated summands wherein all parts must be odd, and they denoted the corresp… ▽ More In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called partitions with designated summands. These are constructed by taking unrestricted integer partitions and designating exactly one of each occurrence of a part. In the same work, they also considered the restricted partitions with designated summands wherein all parts must be odd, and they denoted the corresponding function by $\mathrm{PDO}(n)$. △ Less

Submitted 27 May, 2025; originally announced May 2025.

Comments: Appearing at Discrete Mathematics, 16 pages

MSC Class: 11P84; 05A17; 05A15; 33B10

Design and Continuation of Nonlinear Teardrop Hovering Formation along the Near Rectilinear Halo Orbit

Authors: Shuyue Fu, Yihan Peng, Shengping Gong, Peng Shi

Abstract: This short communication is devoted to the design and continuation of a teardrop hovering formation along the Near Rectilinear Halo orbit and provides further insights into future on-orbit services in the cislunar space. First, we extend the concept of the teardrop hovering formation to scenarios along the Near Rectilinear Halo orbit in the Earth-Moon circular restricted three-body problem. Then,… ▽ More This short communication is devoted to the design and continuation of a teardrop hovering formation along the Near Rectilinear Halo orbit and provides further insights into future on-orbit services in the cislunar space. First, we extend the concept of the teardrop hovering formation to scenarios along the Near Rectilinear Halo orbit in the Earth-Moon circular restricted three-body problem. Then, we develop two methods for designing these formations based on the nonlinear model for relative motion. The first method addresses the design of the teardrop hovering formations with relatively short revisit distances, while the second method continues hovering trajectories from short to longer revisit distances. In particular, new continuation method is developed to meet the design requirements of this new scenario. Simulation results verify the effectiveness of the proposed methods, and a near-natural teardrop hovering formation is achieved by considering the dynamical properties near the NRHO. Comparisons between design results obtained using linear and nonlinear models further strengthen the necessity of using the nonlinear model. △ Less

Submitted 16 April, 2025; originally announced April 2025.

Journal ref: Aerospace Science and Technology, 2025

Symmetric statistics on rational Dyck paths

Authors: Lilan Dai, Shishuo Fu, Dun Qiu

Abstract: Rational Dyck paths are the rational generalization of classical Dyck paths. They play an important role in Catalan combinatorics, and have multiple applications in algebra and geometry. Two statistics over rational Dyck paths called run and ratio-run are introduced. They both have symmetric joint distributions with the return statistic. We give combinatorial proofs and algebraic proofs of the sym… ▽ More Rational Dyck paths are the rational generalization of classical Dyck paths. They play an important role in Catalan combinatorics, and have multiple applications in algebra and geometry. Two statistics over rational Dyck paths called run and ratio-run are introduced. They both have symmetric joint distributions with the return statistic. We give combinatorial proofs and algebraic proofs of the symmetries, generalizing a result of Li and Lin. △ Less

Submitted 23 February, 2026; v1 submitted 25 March, 2025; originally announced March 2025.

Comments: 17 pages, 5 figures

MSC Class: 05A05; 05A15; 05A19

Analytical Strategies and Winning Conditions for Elliptic-Orbit Target-Attacker-Defender Game

Authors: Shuyue Fu, Shengping Gong, Di Wu, Peng Shi

Abstract: This paper proposes an analytical framework for the orbital Target-Attacker-Defender game with a non-maneuvering target along elliptic orbits. Focusing on the linear quadratic game, we derive an analytical solution to the matrix Riccati equation, which yields analytical Nash-equilibrium strategies for the game. Based on the analytical strategies, we derive the analytical form of the necessary and… ▽ More This paper proposes an analytical framework for the orbital Target-Attacker-Defender game with a non-maneuvering target along elliptic orbits. Focusing on the linear quadratic game, we derive an analytical solution to the matrix Riccati equation, which yields analytical Nash-equilibrium strategies for the game. Based on the analytical strategies, we derive the analytical form of the necessary and sufficient winning conditions for the attacker. The simulation results show good consistency between the analytical and numerical methods, exhibiting 0.004$\%$ relative error in the cost function. The analytical method achieves over 99.9$\%$ reduction in CPU time compared to the conventional numerical method, strengthening the advantage of developing the analytical strategies. Furthermore, we verify the proposed winning conditions and investigate the effects of eccentricity on the game outcomes. Our analysis reveals that for games with hovering initial states, the initial position of the defender should be constrained inside a mathematically definable set to ensure that the attacker wins the game. This constrained set further permits geometric interpretation through our proposed method. This work establishes the analytical framework for orbital Target-Attacker-Defender games, providing fundamental insights into the solution analysis of the game. △ Less

Submitted 28 March, 2025; v1 submitted 18 March, 2025; originally announced March 2025.

Comments: Correction on Eq. (78) for this paper and Eq. (55) for the article published in Aerospace Science and Technology (doi:10.1016/j.ast.2025.109946)

Journal ref: Journal of Guidance, Control, and Dynamics, 2025

A novel multipoint stress control volume method for linear elasticity on quadrilateral grids

Authors: Shubin Fu, Lina Zhao

Abstract: In this paper, we develop a novel control volume method that is locally conservative and locking-free for linear elasticity problem on quadrilateral grids. The symmetry of stress is weakly imposed through the introduction of a Lagrange multiplier. As such, the method involves three unknowns: stress, displacement and rotation. To ensure the well-posedness of the scheme, a pair of carefully defined… ▽ More In this paper, we develop a novel control volume method that is locally conservative and locking-free for linear elasticity problem on quadrilateral grids. The symmetry of stress is weakly imposed through the introduction of a Lagrange multiplier. As such, the method involves three unknowns: stress, displacement and rotation. To ensure the well-posedness of the scheme, a pair of carefully defined finite element spaces is used for the stress, displacement and rotation such that the inf-sup condition holds. An appealing feature of the method is that piecewise constant functions are used for the approximations of stress, displacement and rotation, which greatly simplifies the implementation. In particular, the stress space is defined delicately such that the stress bilinear form is localized around each vertex, which allows for the local elimination of the stress, resulting in a cell-centered system. By choosing different definitions of the space for rotation, we develop two variants of the method. In particular, the first method uses a constant function for rotation over the interaction region, which allows for further elimination and results in a cell-centered system involving displacement only. A rigorous error analysis is performed for the proposed scheme. We show the optimal convergence for $L^2$-error of the stress and rotation. Moreover, we can also prove the superconvergence for $L^2$-error of displacement. Extensive numerical simulations indicate that our method is efficient and accurate, and can handle problems with discontinuous coefficients. △ Less

Submitted 13 April, 2025; v1 submitted 3 March, 2025; originally announced March 2025.

Comments: 26 pages, 6 figures

Consecutive and quasi-consecutive patterns: $\mathrm{des}$-Wilf classifications and generating functions

Authors: Yan Wang, Qi Fang, Shishuo Fu, Sergey Kitaev, Haijun Li

Abstract: Motivated by a correlation between the distribution of descents over permutations that avoid a consecutive pattern and those avoiding the respective quasi-consecutive pattern, as established in this paper, we obtain a complete $\des$-Wilf classification for quasi-consecutive patterns of length up to 4. For equivalence classes containing more than one pattern, we construct various descent-preservin… ▽ More Motivated by a correlation between the distribution of descents over permutations that avoid a consecutive pattern and those avoiding the respective quasi-consecutive pattern, as established in this paper, we obtain a complete $\des$-Wilf classification for quasi-consecutive patterns of length up to 4. For equivalence classes containing more than one pattern, we construct various descent-preserving bijections to establish the equivalences, which lead to the provision of proper versions of two incomplete bijective arguments previously published in the literature. Additionally, for two singleton classes, we derive explicit bivariate generating functions using the generalized run theorem. △ Less

Submitted 14 February, 2025; originally announced February 2025.

Comments: 20 pages

MSC Class: 05A05; 05A15; 05A19

An adaptive two-grid preconditioner and linearly implicit scheme for shale gas transport in fractured porous media

Authors: Maria Vasilyeva, Ben S. Southworth, Shubin Fu

Abstract: We consider a nonlinear mixed-dimensional model for simulating gas transport in shale formation. The mathematical model consists of a coupled system of nonlinear equations, where flow within fractures is represented using a lower-dimensional representation. For the numerical solution of the coupled transport problem, we construct an unstructured mesh that resolves lower dimensional fractures on th… ▽ More We consider a nonlinear mixed-dimensional model for simulating gas transport in shale formation. The mathematical model consists of a coupled system of nonlinear equations, where flow within fractures is represented using a lower-dimensional representation. For the numerical solution of the coupled transport problem, we construct an unstructured mesh that resolves lower dimensional fractures on the grid level and use the finite element approximation to build a discrete system. To construct an efficient scheme for the resulting nonlinear problem, we use an explicit-implicit method for time integration, where we carefully choose an additive partition of the nonlinear operators to separate the stiff linear component and integrate it implicitly to ensure the stability of the time integration. Next, we invert the linear partition of the operator by constructing an efficient two-grid preconditioner for shale gas transport in fractured porous media. We use a local pointwise smoother on the fine grid and carefully design an adaptive multiscale space for coarse grid approximation based on local generalized eigenvalue problems. We utilize an adaptive thresholding to automatically identify local dominant modes which correspond to the very small eigenvalues in local domains. We remark that such spatial features are automatically captured through our local spectral problems, and connect these to fracture information in the global formulation of the problem. Approximation properties of the local spectral space with convergence of the proposed two-grid algorithm are given. Numerical results are presented for two fracture distributions with 30 and 160 fractures, demonstrating iterative convergence independent of the contrast of fracture and porous matrix permeability. △ Less

Submitted 17 August, 2025; v1 submitted 26 November, 2024; originally announced November 2024.

Analytical Pursuit-Evasion Game Strategy in Arbitrary Keplerian Reference Orbits

Authors: Shuyue Fu, Shengping Gong, Peng Shi

Abstract: This paper develops an analytical strategy for solving the linear quadratic pursuit-evasion game in arbitrary Keplerian reference orbits. The motion of the pursuer and evader is described using the controlled Tschauner-Hempel equations, and the optimal game strategies of the pursuer and evader are presented by the solution of the differential Riccati equation.The analytical solution of the differe… ▽ More This paper develops an analytical strategy for solving the linear quadratic pursuit-evasion game in arbitrary Keplerian reference orbits. The motion of the pursuer and evader is described using the controlled Tschauner-Hempel equations, and the optimal game strategies of the pursuer and evader are presented by the solution of the differential Riccati equation.The analytical solution of the differential Riccati equation is presented for elliptic, parabolic, and hyperbolic reference orbits, thereby enabling an analytical pursuit-evasion game strategy. Then, the procedure to solve the pursuit-evasion game using this analytical strategy is proposed. Simulations of pursuit-evasion game in elliptic, parabolic, and hyperbolic reference orbits validate the effectiveness of the developed analytical strategy. Results indicates that the analytical strategy saves the CPU time by more than 99.8$\%$ compared to the numerical one, highlighting the efficiency of the developed strategy. The developed analytical strategy is also applicable to pursuit-evasion game scenarios considering orbital disturbances. Compared to the conventional strategy, which succeed in only two out of six test scenarios, the developed strategy achieves success in all six cases, particularly demonstrating its effectiveness in high-eccentricity cases. △ Less

Submitted 18 December, 2024; v1 submitted 24 November, 2024; originally announced November 2024.

Journal ref: Aerospace Science and Technology, 2025

On Temporal Decay of Compressible Hookean Viscoelastic Fluids with Relatively Large Elasticity Coefficient

Authors: Shengbin Fu, Wenting Huang, Fei Jiang

Abstract: Recently, Jiang--Jiang (J. Differential Equations 282, 2021) showed the existence of unique strong solutions in spatial periodic domain (denoted by $\mathbb{T}^3$), whenever the elasticity coefficient is larger than the initial velocity perturbation of the rest state. Motivated by Jiang--Jiang's result, we revisit the Cauchy problem of the compressible viscoelastic fluids in Lagrangian coordinates… ▽ More Recently, Jiang--Jiang (J. Differential Equations 282, 2021) showed the existence of unique strong solutions in spatial periodic domain (denoted by $\mathbb{T}^3$), whenever the elasticity coefficient is larger than the initial velocity perturbation of the rest state. Motivated by Jiang--Jiang's result, we revisit the Cauchy problem of the compressible viscoelastic fluids in Lagrangian coordinates. Employing an energy method with temporal weights and an additional asymptotic stability condition of initial density in Lagrangian coordinates, we extend the Jiang--Jiang's result with exponential decay-in-time in $\mathbb{T}^3$ to the one with algebraic decay-in-time in the whole space $\mathbb{R}^3$. Thanks to the algebraic decay of solutions established by the energy method with temporal weights, we can further use the spectral analysis to improve the temporal decay rate of solutions. In particular, we find that the $k$-th order spatial derivatives of both the density and deformation perturbations converge to zero in $L^2(\mathbb{R}^3)$ at a rate of $(1+t)^{-\frac{3}{4}-\frac{k+1}{2}}$, which is faster than the decay rate $(1 +t)^{-\frac{3}{4}-\frac{k}{2}}$ obtained by Hu--Wu (SIAM J. Math. Anal. 45, 2013) for $k=0$ and $ 1$. In addition, it's well-known that the decay rate $(1+t)^{-\frac{3}{4}-\frac{k}{2}}$ of the density perturbation is optimal in the compressible Navier--Stokes equations (A.~Matsumura, T.~Nishida, Proc. Jpn. Acad. Ser-A. 55, 1979). Therefore, our faster temporal decay rates indicate that the elasticity accelerates the decay of the density perturbation after the rest state of a compressible viscoelastic fluid being perturbed. △ Less

Submitted 25 November, 2024; v1 submitted 22 November, 2024; originally announced November 2024.

The Calderón problem for third order nonlocal wave equations with time-dependent nonlinearities and potentials

Authors: Song-Ren Fu, Yongyi Yu, Philipp Zimmermann

Abstract: In this article, we study the Calderón problem for nonlocal generalizations of the semilinear Moore--Gibson--Thompson (MGT) equation and the Jordan--Moore--Gibson--Thompson (JMGT) equation of Westervelt-type. These partial differential equations are third order wave equations that appear in nonlinear acoustics, describe the propagation of high-intensity sound waves and exhibit finite speed of prop… ▽ More In this article, we study the Calderón problem for nonlocal generalizations of the semilinear Moore--Gibson--Thompson (MGT) equation and the Jordan--Moore--Gibson--Thompson (JMGT) equation of Westervelt-type. These partial differential equations are third order wave equations that appear in nonlinear acoustics, describe the propagation of high-intensity sound waves and exhibit finite speed of propagation. For semilinear MGT equations with nonlinearity $g$ and potential $q$, we show the following uniqueness properties of the Dirichlet to Neumann (DN) map $Λ_{q,g}$: (i) If $g$ is a polynomial-type nonlinearity whose $m$-th order derivative is bounded, then $Λ_{q,g}$ uniquely determines $q$ and $(\partial^{\ell}_τg(x,t,0))_{2\leq \ell \leq m}$. (ii) If $g$ is a polyhomogeneous nonlinearity of finite order $L$, then $Λ_{q,g}$ uniquely determines $q$ and $g$. The uniqueness proof for polynomial-type nonlinearities is based on a higher order linearization scheme, while the proof for polyhomogeneous nonlinearities only uses a first order linearization. Finally, we demonstrate that a first linearization suffices to uniquely determine Westervelt-type nonlinearities from the related DN maps. We also remark that all the unknowns, which we wish to recover from the DN data, are allowed to depend on time. △ Less

Submitted 22 January, 2026; v1 submitted 13 November, 2024; originally announced November 2024.

MSC Class: Primary 35R30; secondary 26A33; 42B37

Edge multiscale finite element methods for semilinear parabolic problems with heterogeneous coefficients

Authors: Leonardo A. Poveda, Shubin Fu, Guanglian Li, Eric Chung

Abstract: We develop a new spatial semidiscrete multiscale method based upon the edge multiscale methods to solve semilinear parabolic problems with heterogeneous coefficients and smooth initial data. This method allows for a cheap spatial discretization, which fails to resolve the spatial heterogeneity but maintains satisfactory accuracy independent of the heterogeneity. This is achieved by simultaneously… ▽ More We develop a new spatial semidiscrete multiscale method based upon the edge multiscale methods to solve semilinear parabolic problems with heterogeneous coefficients and smooth initial data. This method allows for a cheap spatial discretization, which fails to resolve the spatial heterogeneity but maintains satisfactory accuracy independent of the heterogeneity. This is achieved by simultaneously constructing a steady-state multiscale ansatz space with certain approximation properties for the evolving solution and the initial data. The approximation properties of the multiscale ansatz space are derived using local-global splitting. A fully discrete scheme is analyzed using a first-order explicit exponential Euler scheme. We derive the error estimates in the $L^{2}$-norm and energy norm under the regularity assumptions for the semilinear term. The convergence rates depend on the coarse grid size and the level parameter. Finally, extensive numerical experiments are carried out to validate the efficiency of the proposed method. △ Less

Submitted 15 December, 2025; v1 submitted 28 October, 2024; originally announced October 2024.

Sequences of odd length in strict partitions II: the $2$-measure and refinements of Euler's theorem

Authors: Shishuo Fu, Haijun Li

Abstract: The number of sequences of odd length in strict partitions (denoted as $\mathrm{sol}$), which plays a pivotal role in the first paper of this series, is investigated in different contexts, both new and old. Namely, we first note a direct link between $\mathrm{sol}$ and the $2$-measure of strict partitions when the partition length is given. This notion of $2$-measure of a partition was introduced… ▽ More The number of sequences of odd length in strict partitions (denoted as $\mathrm{sol}$), which plays a pivotal role in the first paper of this series, is investigated in different contexts, both new and old. Namely, we first note a direct link between $\mathrm{sol}$ and the $2$-measure of strict partitions when the partition length is given. This notion of $2$-measure of a partition was introduced quite recently by Andrews, Bhattacharjee, and Dastidar. We establish a $q$-series identity in three ways, one of them features a Franklin-type involuion. Secondly, still with this new partition statistic $\mathrm{sol}$ in mind, we revisit Euler's partition theorem through the lens of Sylvester-Bessenrodt. Two new bivariate refinements of Euler's theorem are established, which involve notions such as MacMahon's 2-modular Ferrers diagram, the Durfee side of partitions, and certain alternating index of partitions that we believe is introduced here for the first time. △ Less

Submitted 22 October, 2024; originally announced October 2024.

Comments: 17 pages

MSC Class: 11P84; 05A17; 05A15

Sequences of odd length in strict partitions I: the combinatorics of double sum Rogers-Ramanujan type identities

Authors: Shishuo Fu, Haijun Li

Abstract: Strict partitions are enumerated with respect to the weight, the number of parts, and the number of sequences of odd length. We write this trivariate generating function as a double sum $q$-series. Equipped with such a combinatorial set-up, we investigate a handful of double sum identities appeared in recent works of Cao-Wang, Wang-Wang, Wei-Yu-Ruan, Andrews-Uncu, Chern, and Wang, finding partitio… ▽ More Strict partitions are enumerated with respect to the weight, the number of parts, and the number of sequences of odd length. We write this trivariate generating function as a double sum $q$-series. Equipped with such a combinatorial set-up, we investigate a handful of double sum identities appeared in recent works of Cao-Wang, Wang-Wang, Wei-Yu-Ruan, Andrews-Uncu, Chern, and Wang, finding partition theoretical interpretations to all of these identities, and in most cases supplying Franklin-type involutive proofs. This approach dates back more than a century to P. A. MacMahon's interpretations of the celebrated Rogers-Ramanujan identities, and has been further developed by Kurşungöz in the last decade. △ Less

Submitted 13 October, 2024; originally announced October 2024.

Comments: 24 pages

MSC Class: 11P84; 05A19; 05A15

A robust solver for large-scale heat transfer topology optimization

Authors: Yingjie Zhou, Changqing Ye, Yucheng Liu, Shubin Fu, Eric T. Chung

Abstract: This paper presents a large-scale parallel solver, specifically designed to tackle the challenges of solving high-dimensional and high-contrast linear systems in heat transfer topology optimization. The solver incorporates an interpolation technique to accelerate convergence in high-resolution domains, along with a multiscale multigrid preconditioner to handle complex coefficient fields with signi… ▽ More This paper presents a large-scale parallel solver, specifically designed to tackle the challenges of solving high-dimensional and high-contrast linear systems in heat transfer topology optimization. The solver incorporates an interpolation technique to accelerate convergence in high-resolution domains, along with a multiscale multigrid preconditioner to handle complex coefficient fields with significant contrast. All modules of the optimization solver are implemented on a high performance computing cluster by the PETSc numerical library. Through a series of numerical investigations, we demonstrate the effectiveness of our approach in enhancing convergence and robustness during the optimization process, particularly in high-contrast scenarios with resolutions up to $1024^3$. Our performance results indicate that the proposed preconditioner achieves over $2\times$ speedup against the default algebraic multigrid in PETSc for high-contrast cases. △ Less

Submitted 13 January, 2025; v1 submitted 9 October, 2024; originally announced October 2024.

MSC Class: 65F10; 65N55; 65Y05; 80M50

Learning a generalized multiscale prolongation operator

Authors: Yucheng Liu, Shubin Fu, Yingjie Zhou, Changqing Ye, Eric T. Chung

Abstract: In this research, we address Darcy flow problems with random permeability using iterative solvers, enhanced by a two-grid preconditioner based on a generalized multiscale prolongation operator, which has been demonstrated to be stable for high contrast profiles. To circumvent the need for repeatedly solving spectral problems with varying coefficients, we harness deep learning techniques to expedit… ▽ More In this research, we address Darcy flow problems with random permeability using iterative solvers, enhanced by a two-grid preconditioner based on a generalized multiscale prolongation operator, which has been demonstrated to be stable for high contrast profiles. To circumvent the need for repeatedly solving spectral problems with varying coefficients, we harness deep learning techniques to expedite the construction of the generalized multiscale prolongation operator. Considering linear transformations on multiscale basis have no impact on the performance of the preconditioner, we devise a loss function by the coefficient-based distance between subspaces instead of the plain $l^2$-norm of the difference of the corresponding multiscale bases. We discover that leveraging the inherent symmetry in the local spectral problem can effectively accelerate the neural network training process. In scenarios where training data are limited, we utilize the Karhunen-Loève expansion to augment the dataset. Extensive numerical experiments with various types of random coefficient models are exhibited, showing that the proposed method can significantly reduce the time required to generate the prolongation operator while maintaining the original efficiency of the two-grid preconditioner. Notably, the neural network demonstrates strong generalization capabilities, as evidenced by its satisfactory performance on unseen random permeability fields. △ Less

Submitted 13 January, 2025; v1 submitted 9 October, 2024; originally announced October 2024.

MSC Class: 65F08; 65N55; 68T07

On Edge Multiscale Space based Hybrid Schwarz Preconditioner for Helmholtz Problems with Large Wavenumbers

Authors: Shubin Fu, Shihua Gong, Guanglian Li, Yueqi Wang

Abstract: In this work, we develop a novel hybrid Schwarz method, termed as edge multiscale space based hybrid Schwarz (EMs-HS), for solving the Helmholtz problem with large wavenumbers. The problem is discretized using $H^1$-conforming nodal finite element methods on meshes of size $h$ decreasing faster than $k^{-1}$ such that the discretization error remains bounded as the wavenumber $k$ increases. EMs-HS… ▽ More In this work, we develop a novel hybrid Schwarz method, termed as edge multiscale space based hybrid Schwarz (EMs-HS), for solving the Helmholtz problem with large wavenumbers. The problem is discretized using $H^1$-conforming nodal finite element methods on meshes of size $h$ decreasing faster than $k^{-1}$ such that the discretization error remains bounded as the wavenumber $k$ increases. EMs-HS consists of a one-level Schwarz preconditioner (RAS-imp) and a coarse solver in a multiplicative way. The RAS-imp preconditioner solves local problems on overlapping subdomains with impedance boundary conditions in parallel, and combines the local solutions using partition of unity. The coarse space is an edge multiscale space proposed in [13]. The key idea is to first establish a local splitting of the solution over each subdomain by a local bubble part and local Helmholtz harmonic extension part, and then to derive a global splitting by means of the partition of unity. This facilitates representing the solution as the sum of a global bubble part and a global Helmholtz harmonic extension part. We prove that the EMs-HS preconditioner leads to a convergent fixed-point iteration uniformly for large wavenumbers, by rigorously analyzing the approximation properties of the coarse space to the global Helmholtz harmonic extension part and to the solution of the adjoint problem. Distinctly, the theoretical convergence analysis are valid in two extreme cases: using minimal overlapping size among subdomains (of order $h$), or using coarse spaces of optimal dimension (of magnitude $k^d$, where $d$ is the spatial dimension). We provide extensive numerical results on the sharpness of the theoretical findings and also demonstrate the method on challenging heterogeneous models. △ Less

Submitted 15 August, 2024; originally announced August 2024.

Comments: 24 pages

MSC Class: 65F08; 65F10; 65N55

A fast cosine transformation accelerated method for predicting effective thermal conductivity

Authors: Changqing Ye, Shubin Fu, Eric T. Chung

Abstract: Predicting effective thermal conductivity by solving a Partial Differential Equation (PDE) defined on a high-resolution Representative Volume Element (RVE) is a computationally intensive task. In this paper, we tackle the task by proposing an efficient and implementation-friendly computational method that can fully leverage the computing power offered by hardware accelerators, namely, graphical pr… ▽ More Predicting effective thermal conductivity by solving a Partial Differential Equation (PDE) defined on a high-resolution Representative Volume Element (RVE) is a computationally intensive task. In this paper, we tackle the task by proposing an efficient and implementation-friendly computational method that can fully leverage the computing power offered by hardware accelerators, namely, graphical processing units (GPUs). We first employ the Two-Point Flux-Approximation scheme to discretize the PDE and then utilize the preconditioned conjugate gradient method to solve the resulting algebraic linear system. The construction of the preconditioner originates from FFT-based homogenization methods, and an engineered linear programming technique is utilized to determine the homogeneous reference parameters. The fundamental observation presented in this paper is that the preconditioner system can be effectively solved using multiple Fast Cosine Transformations (FCT) and parallel tridiagonal matrix solvers. Regarding the fact that default multiple FCTs are unavailable on the CUDA platform, we detail how to derive FCTs from FFTs with nearly optimal memory usage. Numerical experiments including the stability comparison with standard preconditioners are conducted for 3D RVEs. Our performance reports indicate that the proposed method can achieve a $5$-fold acceleration on the GPU platform over the pure CPU platform and solve the problems with $512^3$ degrees of freedom and reasonable contrast ratios in less than $30$ seconds. △ Less

Submitted 2 April, 2024; originally announced April 2024.

A robust two-level overlapping preconditioner for Darcy flow in high-contrast media

Authors: Changqing Ye, Shubin Fu, Eric T. Chung, Jizu Huang

Abstract: In this article, a two-level overlapping domain decomposition preconditioner is developed for solving linear algebraic systems obtained from simulating Darcy flow in high-contrast media. Our preconditioner starts at a mixed finite element method for discretizing the partial differential equation by Darcy's law with the no-flux boundary condition and is then followed by a velocity elimination techn… ▽ More In this article, a two-level overlapping domain decomposition preconditioner is developed for solving linear algebraic systems obtained from simulating Darcy flow in high-contrast media. Our preconditioner starts at a mixed finite element method for discretizing the partial differential equation by Darcy's law with the no-flux boundary condition and is then followed by a velocity elimination technique to yield a linear algebraic system with only unknowns of pressure. Then, our main objective is to design a robust and efficient domain decomposition preconditioner for this system, which is accomplished by engineering a multiscale coarse space that is capable of characterizing high-contrast features of the permeability field. A generalized eigenvalue problem is solved in each non-overlapping coarse element in a communication-free manner to form the global solver, which is accompanied by local solvers originated from additive Schwarz methods but with a non-Galerkin discretization to derive the two-level preconditioner. We provide a rigorous analysis that indicates that the condition number of the preconditioned system could be bounded above with several assumptions. Extensive numerical experiments with various types of three-dimensional high-contrast models are exhibited. In particular, we study the robustness against the contrast of the media as well as the influences of numbers of eigenfunctions, oversampling sizes, and subdomain partitions on the efficiency of the proposed preconditioner. Besides, strong and weak scalability performances are also examined. △ Less

Submitted 28 March, 2024; originally announced March 2024.

An efficient multiscale multigrid preconditioner for Darcy flow in high-contrast media

Authors: Changqing Ye, Shubin Fu, Eric T. Chung, Jizu Huang

Abstract: In this paper, we develop a multigrid preconditioner to solve Darcy flow in highly heterogeneous porous media. The key component of the preconditioner is to construct a sequence of nested subspaces $W_{\mathcal{L}}\subset W_{\mathcal{L}-1}\subset\cdots\subset W_1=W_h$. An appropriate spectral problem is defined in the space of $W_{i-1}$, then the eigenfunctions of the spectral problems are utilize… ▽ More In this paper, we develop a multigrid preconditioner to solve Darcy flow in highly heterogeneous porous media. The key component of the preconditioner is to construct a sequence of nested subspaces $W_{\mathcal{L}}\subset W_{\mathcal{L}-1}\subset\cdots\subset W_1=W_h$. An appropriate spectral problem is defined in the space of $W_{i-1}$, then the eigenfunctions of the spectral problems are utilized to form $W_i$. The preconditioner is applied to solve a positive semidefinite linear system which results from discretizing the Darcy flow equation with the lowest order Raviart-Thomas spaces and adopting a trapezoidal quadrature rule. Theoretical analysis and numerical investigations of this preconditioner will be presented. In particular, we will consider several typical highly heterogeneous permeability fields whose resolutions are up to $1024^3$ and examine the computational performance of the preconditioner in several aspects, such as strong scalability, weak scalability, and robustness against the contrast of the media. We also demonstrate an application of this preconditioner for solving a two-phase flow benchmark problem. △ Less

Submitted 28 March, 2024; originally announced March 2024.

A group action on cyclic compositions and $γ$-positivity

Authors: Shishuo Fu, Jie Yang

Abstract: Let $w_{n,k,m}$ be the number of Dyck paths of semilength $n$ with $k$ occurrences of $UD$ and $m$ occurrences of $UUD$. We establish in two ways a new interpretation of the numbers $w_{n,k,m}$ in terms of plane trees and internal nodes. The first way builds on a new characterization of plane trees that involves cyclic compositions. The second proof utilizes a known interpretation of $w_{n,k,m}$ i… ▽ More Let $w_{n,k,m}$ be the number of Dyck paths of semilength $n$ with $k$ occurrences of $UD$ and $m$ occurrences of $UUD$. We establish in two ways a new interpretation of the numbers $w_{n,k,m}$ in terms of plane trees and internal nodes. The first way builds on a new characterization of plane trees that involves cyclic compositions. The second proof utilizes a known interpretation of $w_{n,k,m}$ in terms of plane trees and leaves, and a recent involution on plane trees constructed by Li, Lin, and Zhao. Moreover, a group action on the set of cyclic compositions (or equivalently, $2$-dominant compositions) is introduced, which amounts to give a combinatorial proof of the $γ$-positivity of the Narayana polynomial, as well as the $γ$-positivity of the polynomial $W_{2k+1,k}(t):=\sum_{1\le m\le k}w_{2k+1,k,m}t^m$ previously obtained by Bóna et al, with apparently new combinatorial interpretations of their $γ$-coefficients. △ Less

Submitted 1 March, 2024; originally announced March 2024.

Comments: 19 pages, 3 figures

MSC Class: 05A05; 05A10; 05A15; 05C05

Wavelet-based Edge Multiscale Finite Element Methods for Singularly Perturbed Convection-Diffusion Equations

Authors: Shubin Fu, Eric Chung, Guanglian Li

Abstract: We propose a novel efficient and robust Wavelet-based Edge Multiscale Finite Element Method (WEMsFEM) motivated by \cite{MR3980476,GL18} to solve the singularly perturbed convection-diffusion equations. The main idea is to first establish a local splitting of the solution over a local region by a local bubble part and local Harmonic extension part, and then derive a global splitting by means of Pa… ▽ More We propose a novel efficient and robust Wavelet-based Edge Multiscale Finite Element Method (WEMsFEM) motivated by \cite{MR3980476,GL18} to solve the singularly perturbed convection-diffusion equations. The main idea is to first establish a local splitting of the solution over a local region by a local bubble part and local Harmonic extension part, and then derive a global splitting by means of Partition of Unity. This facilitates a representation of the solution as a summation of a global bubble part and a global Harmonic extension part, where the first part can be computed locally in parallel. To approximate the second part, we construct an edge multiscale ansatz space locally with hierarchical bases as the local boundary data that has a guaranteed approximation rate \noteLg{both inside and outside of the layers}. The key innovation of this proposed WEMsFEM lies in a provable convergence rate with little restriction on the mesh size. Its convergence rate with respect to the computational degree of freedom is rigorously analyzed, which is verified by extensive 2-d and 3-d numerical tests. △ Less

Submitted 8 November, 2024; v1 submitted 21 September, 2023; originally announced September 2023.

Inverse problem of recovering a time-dependent nonlinearity appearing in third-order nonlinear acoustic equations

Authors: Song-Ren Fu, Peng-Fei Yao, Yongyi Yu

Abstract: In this paper, we consider the inverse problem of recovering a time-dependent nonlinearity for a third order nonlinear acoustic equation, which is known as the Jordan-Moore-Gibson-Thompson equation (J-M-G-T equation for short). This third order in time equation arises, for example, from the wave propagation in viscous thermally relaxing fluids. The well-posedness of the nonlinear equation is obtai… ▽ More In this paper, we consider the inverse problem of recovering a time-dependent nonlinearity for a third order nonlinear acoustic equation, which is known as the Jordan-Moore-Gibson-Thompson equation (J-M-G-T equation for short). This third order in time equation arises, for example, from the wave propagation in viscous thermally relaxing fluids. The well-posedness of the nonlinear equation is obtained for the small initial and boundary data. By the higher order linearization to the nonlinear equation, and construction of complex geometric optics (CGO for short) solutions for the linearized equation, we derive the uniqueness of recovering the nonlinearity. △ Less

Submitted 5 October, 2023; v1 submitted 21 August, 2023; originally announced August 2023.

Partitions with parts separated by parity: conjugation, congruences and the mock theta functions

Authors: Shishuo Fu, Dazhao Tang

Abstract: Noting a curious link between Andrews' even-odd crank and the Stanley rank, we adopt a combinatorial approach building on the map of conjugation and continue the study of integer partitions with parts separated by parity. Our motivation is twofold. First off, we derive results for certain restricted partitions with even parts below odd parts. These include a Franklin-type involution proving a para… ▽ More Noting a curious link between Andrews' even-odd crank and the Stanley rank, we adopt a combinatorial approach building on the map of conjugation and continue the study of integer partitions with parts separated by parity. Our motivation is twofold. First off, we derive results for certain restricted partitions with even parts below odd parts. These include a Franklin-type involution proving a parametrized identity that generalizes Andrews' bivariate generating function, and two families of Andrews--Beck type congruences. Secondly, we introduce several new subsets of partitions that are stable (i.e., invariant under conjugation) and explore their connections with three third order mock theta functions $ω(q)$, $ν(q)$, and $ψ^{(3)}(q)$, introduced by Ramanujan and Watson. △ Less

Submitted 23 June, 2023; originally announced June 2023.

Comments: 20 pages, 6 figures

MSC Class: 11P81; 11P83; 05A17

Journal ref: Proceedings of the Royal Society of Edinburgh: Section A Mathematics 155 (2025) 954-974

Convergence of the CEM-GMsFEM for compressible flow in highly heterogeneous media

Authors: Leonardo A. Poveda, Shubin Fu, Eric T. Chung, Lina Zhao

Abstract: This paper presents and analyses a Constraint Energy Minimization Generalized Multiscale Finite Element Method (CEM-GMsFEM) for solving single-phase non-linear compressible flows in highly heterogeneous media. The construction of CEM-GMsFEM hinges on two crucial steps: First, the auxiliary space is constructed by solving local spectral problems, where the basis functions corresponding to small eig… ▽ More This paper presents and analyses a Constraint Energy Minimization Generalized Multiscale Finite Element Method (CEM-GMsFEM) for solving single-phase non-linear compressible flows in highly heterogeneous media. The construction of CEM-GMsFEM hinges on two crucial steps: First, the auxiliary space is constructed by solving local spectral problems, where the basis functions corresponding to small eigenvalues are captured. Then the basis functions are obtained by solving local energy minimization problems over the oversampling domains using the auxiliary space. The basis functions have exponential decay outside the corresponding local oversampling regions. The convergence of the proposed method is provided, and we show that this convergence only depends on the coarse grid size and is independent of the heterogeneities. An online enrichment guided by \emph{a posteriori} error estimator is developed to enhance computational efficiency. Several numerical experiments on a three-dimensional case to confirm the theoretical findings are presented, illustrating the performance of the method and giving efficient and accurate numerical. △ Less

Submitted 30 March, 2023; originally announced March 2023.

MSC Class: 65M12; 65M60; 65M22

Inverse problem of recovering the time-dependent damping and nonlinear terms for wave equations

Authors: Song-Ren Fu

Abstract: In this paper, we consider the inverse boundary problems of recovering the time-dependent nonlinearity and damping term for a semilinear wave equation on a Riemannian manifold. The Carleman estimate and the construction of Gaussian beams together with the higher order linearization are respectively used to derive the uniqueness results of recovering the coefficients. In this paper, we consider the inverse boundary problems of recovering the time-dependent nonlinearity and damping term for a semilinear wave equation on a Riemannian manifold. The Carleman estimate and the construction of Gaussian beams together with the higher order linearization are respectively used to derive the uniqueness results of recovering the coefficients. △ Less

Submitted 6 December, 2022; v1 submitted 4 December, 2022; originally announced December 2022.

Uncertainty Quantification of Nonlinear Lagrangian Data Assimilation Using Linear Stochastic Forecast Models

Authors: Nan Chen, Shubin Fu

Abstract: Lagrangian data assimilation exploits the trajectories of moving tracers as observations to recover the underlying flow field. One major challenge in Lagrangian data assimilation is the intrinsic nonlinearity that impedes using exact Bayesian formulae for the state estimation of high-dimensional systems. In this paper, an analytically tractable mathematical framework for continuous-in-time Lagrang… ▽ More Lagrangian data assimilation exploits the trajectories of moving tracers as observations to recover the underlying flow field. One major challenge in Lagrangian data assimilation is the intrinsic nonlinearity that impedes using exact Bayesian formulae for the state estimation of high-dimensional systems. In this paper, an analytically tractable mathematical framework for continuous-in-time Lagrangian data assimilation is developed. It preserves the nonlinearity in the observational processes while approximating the forecast model of the underlying flow field using linear stochastic models (LSMs). A critical feature of the framework is that closed analytic formulae are available for solving the posterior distribution, which facilitates mathematical analysis and numerical simulations. First, an efficient iterative algorithm is developed in light of the analytically tractable statistics. It accurately estimates the parameters in the LSMs using only a small number of the observed tracer trajectories. Next, the framework facilitates the development of several computationally efficient approximate filters and the quantification of the associated uncertainties. A cheap approximate filter with a diagonal posterior covariance derived from the asymptotic analysis of the posterior estimate is shown to be skillful in recovering incompressible flows. It is also demonstrated that randomly selecting a small number of tracers at each time step as observations can reduce the computational cost while retaining the data assimilation accuracy. Finally, based on a prototype model in geophysics, the framework with LSMs is shown to be skillful in filtering nonlinear turbulent flow fields with strong non-Gaussian features. △ Less

Submitted 9 May, 2023; v1 submitted 28 October, 2022; originally announced October 2022.

Burstein's permutation conjecture, Hong and Li's inversion sequence conjecture, and restricted Eulerian distributions

Authors: Shane Chern, Shishuo Fu, Zhicong Lin

Abstract: Recently, Hong and Li launched a systematic study of length-four pattern avoidance in inversion sequences, and in particular, they conjectured that the number of $0021$-avoiding inversion sequences can be enumerated by the OEIS entry A218225. Meanwhile, Burstein suggested that the same sequence might also count three sets of pattern restricted permutations. The objective of this paper is not only… ▽ More Recently, Hong and Li launched a systematic study of length-four pattern avoidance in inversion sequences, and in particular, they conjectured that the number of $0021$-avoiding inversion sequences can be enumerated by the OEIS entry A218225. Meanwhile, Burstein suggested that the same sequence might also count three sets of pattern restricted permutations. The objective of this paper is not only a confirmation of Hong and Li's conjecture and Burstein's first conjecture, but also two more delicate generating function identities with the $\mathsf{ides}$ statistic concerned in the restricted permutation case, and the $\mathsf{asc}$ statistic concerned in the restricted inversion sequence case, which yield a new equidistribution result. △ Less

Submitted 24 September, 2022; originally announced September 2022.

Comments: 19 pages, comments are welcome

MSC Class: 05A05; 05A15

An involution on restricted Laguerre histories and its applications

Authors: Joanna N. Chen, Shishuo Fu

Abstract: Laguerre histories (restricted or not) are certain weighted Motzkin paths with two types of level steps. They are, on one hand, in natural bijection with the set of permutations, and on the other hand, yield combinatorial interpretations for the moments of Laguerre polynomials via Flajolet's combinatorial theory of continued fractions. In this paper, we first introduce a reflection-like involution… ▽ More Laguerre histories (restricted or not) are certain weighted Motzkin paths with two types of level steps. They are, on one hand, in natural bijection with the set of permutations, and on the other hand, yield combinatorial interpretations for the moments of Laguerre polynomials via Flajolet's combinatorial theory of continued fractions. In this paper, we first introduce a reflection-like involution on restricted Laguerre histories. Then, we demonstrate its power by composing this involution with three bijections due to Fran\ccon-Viennot, Foata-Zeilberger, and Yan-Zhou-Lin, respectively. A host of equidistribution results involving various (multiset-valued) permutation statistics follow from these applications. As byproducts, seven apparently new Mahonian statistics present themselves; new interpretations of known Mahonian statistics are discovered as well. Finally, in our effort to show the interconnections between these Mahonian statistics, we are naturally led to a new link between the variant Yan-Zhou-Lin bijection and the Kreweras complement. △ Less

Submitted 24 August, 2022; originally announced August 2022.

Comments: 7 figures and 3 tables

MSC Class: 05A05; 05A19

A discontinuous Galerkin based multiscale method for heterogeneous elastic wave equations

Authors: Zhongqian Wang, Shubin Fu, Zishang Li, Eric Chung

Abstract: In this paper, we develop a local multiscale model reduction strategy for the elastic wave equation in strongly heterogeneous media, which is achieved by solving the problem in a coarse mesh with multiscale basis functions. We use the interior penalty discontinuous Galerkin (IPDG) to couple the multiscale basis functions that contain important heterogeneous media information. The construction of e… ▽ More In this paper, we develop a local multiscale model reduction strategy for the elastic wave equation in strongly heterogeneous media, which is achieved by solving the problem in a coarse mesh with multiscale basis functions. We use the interior penalty discontinuous Galerkin (IPDG) to couple the multiscale basis functions that contain important heterogeneous media information. The construction of efficient multiscale basis functions starts with extracting dominant modes of carefully defined spectral problems to represent important media feature, which is followed by solving a constraint energy minimization problems. Then a Petrov-Galerkin projection and systematization onto the coarse grid is applied. As a result, an explicit and energy conserving scheme is obtained for fast online simulation. The method exhibits both coarse-mesh and spectral convergence as long as one appropriately chose the oversampling size. We rigorously analyze the stability and convergence of the proposed method. Numerical results are provided to show the performance of the multiscale method and confirm the theoretical results. △ Less

Submitted 10 July, 2022; originally announced July 2022.

Local multiscale model reduction using discontinuous Galerkin coupling for elasticity problems

Authors: Zhongqian Wang, Shubin Fu, Eric Chung

Abstract: In this paper, we consider the constrained energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with discontinuous Galerkin (DG) coupling for the linear elasticity equations in highly heterogeneous and high contrast media. We will introduce the construction of a DG version of the CEM-GMsFEM, such as auxiliary basis functions and offline basis functions. The DG version of the… ▽ More In this paper, we consider the constrained energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with discontinuous Galerkin (DG) coupling for the linear elasticity equations in highly heterogeneous and high contrast media. We will introduce the construction of a DG version of the CEM-GMsFEM, such as auxiliary basis functions and offline basis functions. The DG version of the method offers some advantages such as flexibility in coarse grid construction and sparsity of resulting discrete systems. Moreover, to our best knowledge, this is the first time where the proof of the convergence of the CEM-GMsFEM in the DG form is given. Some numerical examples will be presented to illustrate the performance of the method. △ Less

Submitted 10 October, 2022; v1 submitted 16 April, 2022; originally announced April 2022.