Showing 1–20 of 20 results for author: Owhadi, H

Aircast-Mars: A Mars Foundation Model for Global Weather Forecasting with HEALPix-Aware Convolutions

Authors: Manmeet Singh, Saptarishi Dhanuka, Naveen Sudharsan, Houman Owhadi, Krista M. Soderlund, Alphan Altinok

Abstract: Foundation models for planetary atmospheres promise fast, lightweight surrogates of expensive general circulation models (GCMs) for mission planning and scientific inquiry. Here we present Aircast-Mars, a deep-learning weather prediction system for Mars trained on the Ensemble Mars Atmosphere Reanalysis System (EMARS) v1.0. We regrid temperature, zonal wind, and meridional wind fields across 28 ve… ▽ More Foundation models for planetary atmospheres promise fast, lightweight surrogates of expensive general circulation models (GCMs) for mission planning and scientific inquiry. Here we present Aircast-Mars, a deep-learning weather prediction system for Mars trained on the Ensemble Mars Atmosphere Reanalysis System (EMARS) v1.0. We regrid temperature, zonal wind, and meridional wind fields across 28 vertical levels onto a hierarchical equal-area isolatitude pixelization (HEALPix) mesh at Nside = 64 (~110 km resolution) and train a HEALPix-aware 2D U-Net inspired by the DLESyM architecture to predict the next hourly atmospheric state. The model employs custom inter-face padding that respects the topology of the 12-face HEALPix sphere and modern ConvNeXt residual blocks with capped Gaussian Error Linear Unit (GELU) activations. While containing 4.3 million trainable parameters, a compact size compared to terrestrial weather foundation models, the network achieves a best validation Mean Squared Error (MSE) of 1.58e-5 in normalized units. Recursive autoregressive rollouts remain stable and physically coherent for 25 hours (one Martian sol), with Root Mean Square Error (RMSE) growing monotonically from ~0.004 at t + 1 h to ~0.031 at t + 25 h without divergence. Compared to a baseline 3D U-Net, the HEALPix-aware architecture reduces validation loss by more than an order of magnitude while using fewer parameters. The model generates a one-hour forecast in approximately 0.5 seconds on a single GPU, offering several orders-of-magnitude speedup over traditional numerical GCMs. These results demonstrate that parsimonious, geometry-respecting neural architectures can capture synoptic-scale Martian atmospheric dynamics and provide a foundation for planetary-scale weather forecasting. △ Less

Submitted 14 June, 2026; originally announced July 2026.

Graphical conditional generative modeling for digital twin modeling

Authors: Zongren Zou, Théo Bourdais, Ricardo Baptista, Houman Owhadi

Abstract: Digital twin modeling, including control and data assimilation under model uncertainty, often faces an open-ended fidelity problem: adding variables, data streams, and time scales can indefinitely increase model complexity, ultimately producing systems that are difficult to maintain, validate, interpret, and use for stress or safety testing. As an alternative, one can seek parsimonious stochastic… ▽ More Digital twin modeling, including control and data assimilation under model uncertainty, often faces an open-ended fidelity problem: adding variables, data streams, and time scales can indefinitely increase model complexity, ultimately producing systems that are difficult to maintain, validate, interpret, and use for stress or safety testing. As an alternative, one can seek parsimonious stochastic surrogate models built only on the variables needed to describe the relevant quantities of interest. We introduce a framework for discovering such variables from observational data by identifying which candidate inputs influence the full conditional law of a target quantity, rather than only its conditional mean. This distinction is essential in stochastic, coarse-grained, or partially observed systems, where dependencies may appear through changes in variability, tail behavior, multimodality, or uncertainty rather than through deterministic functional relationships. The framework couples conditional generative modeling, which learns the conditional distribution of the target given candidate inputs, with Gaussian-process-based analysis of variance (through kernel mode decomposition), which enables iterative pruning of non-influential inputs and interpretable structure discovery. In control settings, the resulting surrogate can be interpreted as a learned Markov decision process: the method identifies not only a transition model, but also the state, action, and memory variables needed to make the learned dynamics effectively Markovian. Across examples involving stochastic dynamical systems, missing variables, PDE control, reinforcement learning, and economic data, the discovered structures yield interpretable stochastic surrogates whose downstream performance is comparable to models trained on the full variable set. △ Less

Submitted 15 June, 2026; originally announced June 2026.

Structure-Preserving Neural Surrogates with Tractable Uncertainty Quantification

Authors: Handi Zhang, Adrienne M. Propp, Brooks Kinch, Houman Owhadi, Nathaniel Trask

Abstract: Recent advances in scientific machine learning provide a means of near-real-time solution to partial differential equations (PDEs), but lack the theoretical underpinnings of conventional simulators that support contemporary verification and validation. In this work, we construct data-driven reduced-order models that serve as structure-preserving, real-time surrogates. Remarkably, the exterior calc… ▽ More Recent advances in scientific machine learning provide a means of near-real-time solution to partial differential equations (PDEs), but lack the theoretical underpinnings of conventional simulators that support contemporary verification and validation. In this work, we construct data-driven reduced-order models that serve as structure-preserving, real-time surrogates. Remarkably, the exterior calculus that imposes physical conservation structure also exposes topological structure that we use to build a Gaussian process (GP) representation of uncertainty in state-flux relationships, ultimately yielding a Dirichlet-to-Neumann map for quantities of interest with closed-form expressions for posterior uncertainty. We specifically propose structure-preserving $H(\mathrm{div})$--$L^2$ subspaces of conventional Raviart--Thomas and $dgP_0$ elements prescribed by a lightweight transformer. Reduced-order dynamics consistent with this subspace are learned by posing a conservation law in which a GP describes the fluxes between volumes. This work hinges on a novel interface between mixed FEM spaces and GP regression; when training is posed as the optimal recovery problem (ORP), the resulting GP regression can be written as an optimization problem with equality constraints that impose a conservation structure, amenable to a fast Schur-complement training strategy. The trained model can then be solved in real time with closed-form estimators for boundary fluxes driven by prescribed Dirichlet data. The paper includes RKHS posterior error bounds for linear functionals to support uncertainty quantification, as well as numerical experiments demonstrating the accuracy of the posterior distribution as a surrogate for error estimation. △ Less

Submitted 10 June, 2026; originally announced June 2026.

MSC Class: 65N30; 81Q30; 68T07; 60G15; 90C70

Fluids You Can Trust: Property-Preserving Operator Learning for Incompressible Flows

Authors: Ramansh Sharma, Matthew Lowery, Houman Owhadi, Varun Shankar

Abstract: We present a novel property-preserving kernel-based operator learning method for incompressible flows governed by the incompressible Navier--Stokes equations. Traditional numerical solvers incur significant computational costs to respect incompressibility. Operator learning offers efficient surrogate models, but current neural operators fail to exactly enforce physical properties such as incompres… ▽ More We present a novel property-preserving kernel-based operator learning method for incompressible flows governed by the incompressible Navier--Stokes equations. Traditional numerical solvers incur significant computational costs to respect incompressibility. Operator learning offers efficient surrogate models, but current neural operators fail to exactly enforce physical properties such as incompressibility, periodicity, and turbulence. Our kernel method maps input functions to expansion coefficients of output functions in a property-preserving kernel basis, ensuring that predicted velocity fields $\textit{analytically}$ and $\textit{simultaneously}$ preserve the aforementioned physical properties. Our method leverages efficient numerical linear algebra, simple rootfinding, and streaming to allow for training at-scale on desktop GPUs. We also present universal approximation results and both pessimistic and more realistic $\textit{a priori}$ convergence rates for our framework. We evaluate the method on challenging 2D and 3D, laminar and turbulent, incompressible flow problems. Our method achieves up to six orders of magnitude lower relative $\ell_2$ errors upon generalization and trains up to five orders of magnitude faster compared to neural operators, despite our method being trained on desktop GPUs and neural operators being trained on cutting-edge GPU servers. Moreover, while our method enforces incompressibility analytically, neural operators exhibit very large deviations. Our results show that our method provides an accurate and efficient surrogate for incompressible flows. △ Less

Submitted 14 April, 2026; v1 submitted 17 February, 2026; originally announced February 2026.

Bilevel optimization for learning hyperparameters: Application to solving PDEs and inverse problems with Gaussian processes

Authors: Nicholas H. Nelsen, Houman Owhadi, Andrew M. Stuart, Xianjin Yang, Zongren Zou

Abstract: Methods for solving scientific computing and inference problems, such as kernel- and neural network-based approaches for partial differential equations (PDEs), inverse problems, and supervised learning tasks, depend crucially on the choice of hyperparameters. Specifically, the efficacy of such methods, and in particular their accuracy, stability, and generalization properties, strongly depends on… ▽ More Methods for solving scientific computing and inference problems, such as kernel- and neural network-based approaches for partial differential equations (PDEs), inverse problems, and supervised learning tasks, depend crucially on the choice of hyperparameters. Specifically, the efficacy of such methods, and in particular their accuracy, stability, and generalization properties, strongly depends on the choice of hyperparameters. While bilevel optimization offers a principled framework for hyperparameter tuning, its nested optimization structure can be computationally demanding, especially in PDE-constrained contexts. In this paper, we propose an efficient strategy for hyperparameter optimization within the bilevel framework by employing a Gauss-Newton linearization of the inner optimization step. Our approach provides closed-form updates, eliminating the need for repeated costly PDE solves. As a result, each iteration of the outer loop reduces to a single linearized PDE solve, followed by explicit gradient-based hyperparameter updates. We demonstrate the effectiveness of the proposed method through Gaussian process models applied to nonlinear PDEs and to PDE inverse problems. Extensive numerical experiments highlight substantial improvements in accuracy and robustness compared to conventional random hyperparameter initialization. In particular, experiments with additive kernels and neural network-parameterized deep kernels demonstrate the method's scalability and effectiveness for high-dimensional hyperparameter optimization. △ Less

Submitted 7 October, 2025; originally announced October 2025.

Discovery of Probabilistic Dirichlet-to-Neumann Maps on Graphs

Authors: Adrienne M. Propp, Jonas A. Actor, Elise Walker, Houman Owhadi, Nathaniel Trask, Daniel M. Tartakovsky

Abstract: Dirichlet-to-Neumann maps enable the coupling of multiphysics simulations across computational subdomains by ensuring continuity of state variables and fluxes at artificial interfaces. We present a novel method for learning Dirichlet-to-Neumann maps on graphs using Gaussian processes, specifically for problems where the data obey a conservation constraint from an underlying partial differential eq… ▽ More Dirichlet-to-Neumann maps enable the coupling of multiphysics simulations across computational subdomains by ensuring continuity of state variables and fluxes at artificial interfaces. We present a novel method for learning Dirichlet-to-Neumann maps on graphs using Gaussian processes, specifically for problems where the data obey a conservation constraint from an underlying partial differential equation. Our approach combines discrete exterior calculus and nonlinear optimal recovery to infer relationships between vertex and edge values. This framework yields data-driven predictions with uncertainty quantification across the entire graph, even when observations are limited to a subset of vertices and edges. By optimizing over the reproducing kernel Hilbert space norm while applying a maximum likelihood estimation penalty on kernel complexity, our method ensures that the resulting surrogate strictly enforces conservation laws without overfitting. We demonstrate our method on two representative applications: subsurface fracture networks and arterial blood flow. Our results show that the method maintains high accuracy and well-calibrated uncertainty estimates even under severe data scarcity, highlighting its potential for scientific applications where limited data and reliable uncertainty quantification are critical. △ Less

Submitted 23 January, 2026; v1 submitted 2 June, 2025; originally announced June 2025.

MSC Class: 90C70; 60G15; 05C90

Can Specific THz Fields Induce Collective Base-Flipping in DNA? A Stochastic Averaging and Resonant Enhancement Investigation Based on a New Mesoscopic Model

Authors: Wang Sang Koon, Houman Owhadi, Molei Tao, Tomohiro Yanao

Abstract: We study the metastability, internal frequencies, activation mechanism, energy transfer, and the collective base-flipping in a mesoscopic DNA via resonance with specific electric fields. Our new mesoscopic DNA model takes into account not only the issues of helicity and the coupling of an electric field with the base dipole moments, but also includes environmental effects such as fluid viscosity a… ▽ More We study the metastability, internal frequencies, activation mechanism, energy transfer, and the collective base-flipping in a mesoscopic DNA via resonance with specific electric fields. Our new mesoscopic DNA model takes into account not only the issues of helicity and the coupling of an electric field with the base dipole moments, but also includes environmental effects such as fluid viscosity and thermal noise. And all the parameter values are chosen to best represent the typical values for the opening and closing dynamics of a DNA. Our study shows that while the mesocopic DNA is metastable and robust to environmental effects, it is vulnerable to certain frequencies that could be targeted by specific THz fields for triggering its collective base-flipping dynamics and causing large amplitude separation of base pairs. Based on applying Freidlin-Wentzell method of stochastic averaging and the newly developed theory of resonant enhancement to our mesoscopic DNA model, our semi-analytic estimates show that the required fields should be THz fields with frequencies around 0.28 THz and with amplitudes in the order of 450 kV/cm. These estimates compare well with the experimental data of Titova et al., which have demonstrated that they could affect the function of DNA in human skin tissues by THz pulses with frequencies around 0.5 THz and with a peak electric field at 220 kV/cm. Moreover, our estimates also conform to a number of other experimental results which appeared in the last couple years. △ Less

Submitted 18 March, 2024; originally announced June 2024.

Comments: 37 pages, 8 figures

Kolmogorov n-Widths for Multitask Physics-Informed Machine Learning (PIML) Methods: Towards Robust Metrics

Authors: Michael Penwarden, Houman Owhadi, Robert M. Kirby

Abstract: Physics-informed machine learning (PIML) as a means of solving partial differential equations (PDE) has garnered much attention in the Computational Science and Engineering (CS&E) world. This topic encompasses a broad array of methods and models aimed at solving a single or a collection of PDE problems, called multitask learning. PIML is characterized by the incorporation of physical laws into the… ▽ More Physics-informed machine learning (PIML) as a means of solving partial differential equations (PDE) has garnered much attention in the Computational Science and Engineering (CS&E) world. This topic encompasses a broad array of methods and models aimed at solving a single or a collection of PDE problems, called multitask learning. PIML is characterized by the incorporation of physical laws into the training process of machine learning models in lieu of large data when solving PDE problems. Despite the overall success of this collection of methods, it remains incredibly difficult to analyze, benchmark, and generally compare one approach to another. Using Kolmogorov n-widths as a measure of effectiveness of approximating functions, we judiciously apply this metric in the comparison of various multitask PIML architectures. We compute lower accuracy bounds and analyze the model's learned basis functions on various PDE problems. This is the first objective metric for comparing multitask PIML architectures and helps remove uncertainty in model validation from selective sampling and overfitting. We also identify avenues of improvement for model architectures, such as the choice of activation function, which can drastically affect model generalization to "worst-case" scenarios, which is not observed when reporting task-specific errors. We also incorporate this metric into the optimization process through regularization, which improves the models' generalizability over the multitask PDE problem. △ Less

Submitted 4 September, 2024; v1 submitted 16 February, 2024; originally announced February 2024.

Gaussian Process Hydrodynamics

Authors: Houman Owhadi

Abstract: We present a Gaussian Process (GP) approach (Gaussian Process Hydrodynamics, GPH) for approximating the solution of the Euler and Navier-Stokes equations. As in Smoothed Particle Hydrodynamics (SPH), GPH is a Lagrangian particle-based approach involving the tracking of a finite number of particles transported by the flow. However, these particles do not represent mollified particles of matter but… ▽ More We present a Gaussian Process (GP) approach (Gaussian Process Hydrodynamics, GPH) for approximating the solution of the Euler and Navier-Stokes equations. As in Smoothed Particle Hydrodynamics (SPH), GPH is a Lagrangian particle-based approach involving the tracking of a finite number of particles transported by the flow. However, these particles do not represent mollified particles of matter but carry discrete/partial information about the continuous flow. Closure is achieved by placing a divergence-free GP prior $ξ$ on the velocity field and conditioning on vorticity at particle locations. Known physics (e.g., the Richardson cascade and velocity-increments power laws) is incorporated into the GP prior through physics-informed additive kernels. This approach allows us to coarse-grain turbulence in a statistical manner rather than a deterministic one. By enforcing incompressibility and fluid/structure boundary conditions through the selection of the kernel, GPH requires much fewer particles than SPH. Since GPH has a natural probabilistic interpretation, numerical results come with uncertainty estimates enabling their incorporation into a UQ pipeline and the adding/removing of particles (quantas of information) in an adapted manner. The proposed approach is amenable to analysis, it inherits the complexity of state-of-the-art solvers for dense kernel matrices, and it leads to a natural definition of turbulence as information loss. Numerical experiments support the importance of selecting physics-informed kernels and illustrate the major impact of such kernels on accuracy and stability. Since the proposed approach has a Bayesian interpretation, it naturally enables data assimilation and making predictions and estimations based on mixing simulation data with experimental data. △ Less

Submitted 28 January, 2023; v1 submitted 21 September, 2022; originally announced September 2022.

Comments: 26 pages. See https://www.youtube.com/user/HoumanOwhadi for animations

MSC Class: 35Q30; 76D05; 60G15; 65M75; 65N75; 65N35; 47B34; 41A15; 34B15

Journal ref: Applied Mathematics and Mechanics, 2023

Uncertainty Quantification of the 4th kind; optimal posterior accuracy-uncertainty tradeoff with the minimum enclosing ball

Authors: Hamed Hamze Bajgiran, Pau Batlle Franch, Houman Owhadi, Mostafa Samir, Clint Scovel, Mahdy Shirdel, Michael Stanley, Peyman Tavallali

Abstract: There are essentially three kinds of approaches to Uncertainty Quantification (UQ): (A) robust optimization, (B) Bayesian, (C) decision theory. Although (A) is robust, it is unfavorable with respect to accuracy and data assimilation. (B) requires a prior, it is generally brittle and posterior estimations can be slow. Although (C) leads to the identification of an optimal prior, its approximation s… ▽ More There are essentially three kinds of approaches to Uncertainty Quantification (UQ): (A) robust optimization, (B) Bayesian, (C) decision theory. Although (A) is robust, it is unfavorable with respect to accuracy and data assimilation. (B) requires a prior, it is generally brittle and posterior estimations can be slow. Although (C) leads to the identification of an optimal prior, its approximation suffers from the curse of dimensionality and the notion of risk is one that is averaged with respect to the distribution of the data. We introduce a 4th kind which is a hybrid between (A), (B), (C), and hypothesis testing. It can be summarized as, after observing a sample $x$, (1) defining a likelihood region through the relative likelihood and (2) playing a minmax game in that region to define optimal estimators and their risk. The resulting method has several desirable properties (a) an optimal prior is identified after measuring the data, and the notion of risk is a posterior one, (b) the determination of the optimal estimate and its risk can be reduced to computing the minimum enclosing ball of the image of the likelihood region under the quantity of interest map (which is fast and not subject to the curse of dimensionality). The method is characterized by a parameter in $ [0,1]$ acting as an assumed lower bound on the rarity of the observed data (the relative likelihood). When that parameter is near $1$, the method produces a posterior distribution concentrated around a maximum likelihood estimate with tight but low confidence UQ estimates. When that parameter is near $0$, the method produces a maximal risk posterior distribution with high confidence UQ estimates. In addition to navigating the accuracy-uncertainty tradeoff, the proposed method addresses the brittleness of Bayesian inference by navigating the robustness-accuracy tradeoff associated with data assimilation. △ Less

Submitted 13 September, 2022; v1 submitted 24 August, 2021; originally announced August 2021.

Comments: 49 pages. To appear in the Journal of Computational Physics

MSC Class: 62C20; 62F03; 62F35; 62F25; 68T37

Data-driven geophysical forecasting: Simple, low-cost, and accurate baselines with kernel methods

Authors: Boumediene Hamzi, Romit Maulik, Houman Owhadi

Abstract: Modeling geophysical processes as low-dimensional dynamical systems and regressing their vector field from data is a promising approach for learning emulators of such systems. We show that when the kernel of these emulators is also learned from data (using kernel flows, a variant of cross-validation), then the resulting data-driven models are not only faster than equation-based models but are easi… ▽ More Modeling geophysical processes as low-dimensional dynamical systems and regressing their vector field from data is a promising approach for learning emulators of such systems. We show that when the kernel of these emulators is also learned from data (using kernel flows, a variant of cross-validation), then the resulting data-driven models are not only faster than equation-based models but are easier to train than neural networks such as the long short-term memory neural network. In addition, they are also more accurate and predictive than the latter. When trained on geophysical observational data, for example, the weekly averaged global sea-surface temperature, considerable gains are also observed by the proposed technique in comparison to classical partial differential equation-based models in terms of forecast computational cost and accuracy. When trained on publicly available re-analysis data for the daily temperature of the North-American continent, we see significant improvements over classical baselines such as climatology and persistence-based forecast techniques. Although our experiments concern specific examples, the proposed approach is general, and our results support the viability of kernel methods (with learned kernels) for interpretable and computationally efficient geophysical forecasting for a large diversity of processes. △ Less

Submitted 9 August, 2021; v1 submitted 13 February, 2021; originally announced March 2021.

Control of a Model of DNA Division via Parametric Resonance

Authors: Wang Sang Koon, Houman Owhadi, Molei Tao, Tomohiro Yanao

Abstract: We study the internal resonance, energy transfer, activation mechanism, and control of a model of DNA division via parametric resonance. While the system is robust to noise, this study shows that it is sensitive to specific fine scale modes and frequencies that could be targeted by low intensity electro-magnetic fields for triggering and controlling the division. The DNA model is a chain of pendul… ▽ More We study the internal resonance, energy transfer, activation mechanism, and control of a model of DNA division via parametric resonance. While the system is robust to noise, this study shows that it is sensitive to specific fine scale modes and frequencies that could be targeted by low intensity electro-magnetic fields for triggering and controlling the division. The DNA model is a chain of pendula in a Morse potential. While the (possibly parametrically excited) system has a large number of degrees of freedom and a large number of intrinsic time scales, global and slow variables can be identified by (i) first reducing its dynamic to two modes exchanging energy between each other and (ii) averaging the dynamic of the reduced system with respect to the phase of the fastest mode. Surprisingly the global and slow dynamic of the system remains Hamiltonian (despite the parametric excitation) and the study of its associated effective potential shows how parametric excitation can turn the unstable open state into a stable one. Numerical experiments support the accuracy of the time-averaged reduced Hamiltonian in capturing the global and slow dynamic of the full system. △ Less

Submitted 16 November, 2012; originally announced November 2012.

Comments: Submitted on October 4, 2012

MSC Class: 37N35

Space-time FLAVORS: finite difference, multisymlectic, and pseudospectral integrators for multiscale PDEs

Authors: Molei Tao, Houman Owhadi, Jerrold E. Marsden

Abstract: We present a new class of integrators for stiff PDEs. These integrators are generalizations of FLow AVeraging integratORS (FLAVORS) for stiff ODEs and SDEs introduced in [Tao, Owhadi and Marsden 2010] with the following properties: (i) Multiscale: they are based on flow averaging and have a computational cost determined by mesoscopic steps in space and time instead of microscopic steps in space an… ▽ More We present a new class of integrators for stiff PDEs. These integrators are generalizations of FLow AVeraging integratORS (FLAVORS) for stiff ODEs and SDEs introduced in [Tao, Owhadi and Marsden 2010] with the following properties: (i) Multiscale: they are based on flow averaging and have a computational cost determined by mesoscopic steps in space and time instead of microscopic steps in space and time; (ii) Versatile: the method is based on averaging the flows of the given PDEs (which may have hidden slow and fast processes). This bypasses the need for identifying explicitly (or numerically) the slow variables or reduced effective PDEs; (iii) Nonintrusive: A pre-existing numerical scheme resolving the microscopic time scale can be used as a black box and easily turned into one of the integrators in this paper by turning the large coefficients on over a microscopic timescale and off during a mesoscopic timescale; (iv) Convergent over two scales: strongly over slow processes and in the sense of measures over fast ones; (v) Structure-preserving: for stiff Hamiltonian PDEs (possibly on manifolds), they can be made to be multi-symplectic, symmetry-preserving (symmetries are group actions that leave the system invariant) in all variables and variational. △ Less

Submitted 1 April, 2011; originally announced April 2011.

Variational and linearly-implicit integrators, with applications

Authors: Molei Tao, Houman Owhadi

Abstract: We show that symplectic and linearly-implicit integrators proposed by [Zhang and Skeel, 1997] are variational linearizations of Newmark methods. When used in conjunction with penalty methods (i.e., methods that replace constraints by stiff potentials), these integrators permit coarse time-stepping of holonomically constrained mechanical systems and bypass the resolution of nonlinear systems. Altho… ▽ More We show that symplectic and linearly-implicit integrators proposed by [Zhang and Skeel, 1997] are variational linearizations of Newmark methods. When used in conjunction with penalty methods (i.e., methods that replace constraints by stiff potentials), these integrators permit coarse time-stepping of holonomically constrained mechanical systems and bypass the resolution of nonlinear systems. Although penalty methods are widely employed, an explicit link to Lagrange multiplier approaches appears to be lacking; such a link is now provided (in the context of two-scale flow convergence [Tao, Owhadi and Marsden, 2010]). The variational formulation also allows efficient simulations of mechanical systems on Lie groups. △ Less

Submitted 4 December, 2014; v1 submitted 23 March, 2011; originally announced March 2011.

Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast

Authors: Houman Owhadi, Lei Zhang

Abstract: We construct finite-dimensional approximations of solution spaces of divergence form operators with $L^\infty$-coefficients. Our method does not rely on concepts of ergodicity or scale-separation, but on the property that the solution space of these operators is compactly embedded in $H^1$ if source terms are in the unit ball of $L^2$ instead of the unit ball of $H^{-1}$. Approximation spaces are… ▽ More We construct finite-dimensional approximations of solution spaces of divergence form operators with $L^\infty$-coefficients. Our method does not rely on concepts of ergodicity or scale-separation, but on the property that the solution space of these operators is compactly embedded in $H^1$ if source terms are in the unit ball of $L^2$ instead of the unit ball of $H^{-1}$. Approximation spaces are generated by solving elliptic PDEs on localized sub-domains with source terms corresponding to approximation bases for $H^2$. The $H^1$-error estimates show that $\mathcal{O}(h^{-d})$-dimensional spaces with basis elements localized to sub-domains of diameter $\mathcal{O}(h^α\ln \frac{1}{h})$ (with $α\in [1/2,1)$) result in an $\mathcal{O}(h^{2-2α})$ accuracy for elliptic, parabolic and hyperbolic problems. For high-contrast media, the accuracy of the method is preserved provided that localized sub-domains contain buffer zones of width $\mathcal{O}(h^α\ln \frac{1}{h})$ where the contrast of the medium remains bounded. The proposed method can naturally be generalized to vectorial equations (such as elasto-dynamics). △ Less

Submitted 4 August, 2011; v1 submitted 3 November, 2010; originally announced November 2010.

Comments: Accepted for publication in SIAM MMS

MSC Class: 34E13; 35B27

Optimal Uncertainty Quantification

Authors: Houman Owhadi, Clint Scovel, Timothy John Sullivan, Mike McKerns, Michael Ortiz

Abstract: We propose a rigorous framework for Uncertainty Quantification (UQ) in which the UQ objectives and the assumptions/information set are brought to the forefront. This framework, which we call \emph{Optimal Uncertainty Quantification} (OUQ), is based on the observation that, given a set of assumptions and information about the problem, there exist optimal bounds on uncertainties: these are obtained… ▽ More We propose a rigorous framework for Uncertainty Quantification (UQ) in which the UQ objectives and the assumptions/information set are brought to the forefront. This framework, which we call \emph{Optimal Uncertainty Quantification} (OUQ), is based on the observation that, given a set of assumptions and information about the problem, there exist optimal bounds on uncertainties: these are obtained as values of well-defined optimization problems corresponding to extremizing probabilities of failure, or of deviations, subject to the constraints imposed by the scenarios compatible with the assumptions and information. In particular, this framework does not implicitly impose inappropriate assumptions, nor does it repudiate relevant information. Although OUQ optimization problems are extremely large, we show that under general conditions they have finite-dimensional reductions. As an application, we develop \emph{Optimal Concentration Inequalities} (OCI) of Hoeffding and McDiarmid type. Surprisingly, these results show that uncertainties in input parameters, which propagate to output uncertainties in the classical sensitivity analysis paradigm, may fail to do so if the transfer functions (or probability distributions) are imperfectly known. We show how, for hierarchical structures, this phenomenon may lead to the non-propagation of uncertainties or information across scales. In addition, a general algorithmic framework is developed for OUQ and is tested on the Caltech surrogate model for hypervelocity impact and on the seismic safety assessment of truss structures, suggesting the feasibility of the framework for important complex systems. The introduction of this paper provides both an overview of the paper and a self-contained mini-tutorial about basic concepts and issues of UQ. △ Less

Submitted 23 May, 2012; v1 submitted 2 September, 2010; originally announced September 2010.

Comments: 90 pages. Accepted for publication in SIAM Review (Expository Research Papers). See SIAM Review for higher quality figures

Journal ref: SIAM Rev. 55(2):271--345, 2013

Temperature and Friction Accelerated Sampling of Boltzmann-Gibbs Distribution

Authors: Molei Tao, Houman Owhadi, Jerrold E. Marsden

Abstract: This paper is concerned with tuning friction and temperature in Langevin dynamics for fast sampling from the canonical ensemble. We show that near-optimal acceleration is achieved by choosing friction so that the local quadratic approximation of the Hamiltonian is a critical damped oscillator. The system is also over-heated and cooled down to its final temperature. The performances of different co… ▽ More This paper is concerned with tuning friction and temperature in Langevin dynamics for fast sampling from the canonical ensemble. We show that near-optimal acceleration is achieved by choosing friction so that the local quadratic approximation of the Hamiltonian is a critical damped oscillator. The system is also over-heated and cooled down to its final temperature. The performances of different cooling schedules are analyzed as functions of total simulation time. △ Less

Submitted 6 July, 2010; originally announced July 2010.

Comments: 15 pages, 6 figures

From efficient symplectic exponentiation of matrices to symplectic integration of high-dimensional Hamiltonian systems with slowly varying quadratic stiff potentials

Authors: Molei Tao, Houman Owhadi, Jerrold E. Marsden

Abstract: We present a multiscale integrator for Hamiltonian systems with slowly varying quadratic stiff potentials that uses coarse timesteps (analogous to what the impulse method uses for constant quadratic stiff potentials). This method is based on the highly-non-trivial introduction of two efficient symplectic schemes for exponentiations of matrices that only require O(n) matrix multiplications operatio… ▽ More We present a multiscale integrator for Hamiltonian systems with slowly varying quadratic stiff potentials that uses coarse timesteps (analogous to what the impulse method uses for constant quadratic stiff potentials). This method is based on the highly-non-trivial introduction of two efficient symplectic schemes for exponentiations of matrices that only require O(n) matrix multiplications operations at each coarse time step for a preset small number n. The proposed integrator is shown to be (i) uniformly convergent on positions; (ii) symplectic in both slow and fast variables; (iii) well adapted to high dimensional systems. Our framework also provides a general method for iteratively exponentiating a slowly varying sequence of (possibly high dimensional) matrices in an efficient way. △ Less

Submitted 13 April, 2011; v1 submitted 23 June, 2010; originally announced June 2010.

Structure preserving Stochastic Impulse Methods for stiff Langevin systems with a uniform global error of order 1 or 1/2 on position

Authors: Molei Tao, Houman Owhadi, Jerrold E. Marsden

Abstract: Impulse methods are generalized to a family of integrators for Langevin systems with quadratic stiff potentials and arbitrary soft potentials. Uniform error bounds (independent from stiff parameters) are obtained on integrated positions allowing for coarse integration steps. The resulting integrators are explicit and structure preserving (quasi-symplectic for Langevin systems). Impulse methods are generalized to a family of integrators for Langevin systems with quadratic stiff potentials and arbitrary soft potentials. Uniform error bounds (independent from stiff parameters) are obtained on integrated positions allowing for coarse integration steps. The resulting integrators are explicit and structure preserving (quasi-symplectic for Langevin systems). △ Less

Submitted 23 June, 2010; originally announced June 2010.

Non-intrusive and structure preserving multiscale integration of stiff ODEs, SDEs and Hamiltonian systems with hidden slow dynamics via flow averaging

Authors: Molei Tao, Houman Owhadi, Jerrold E. Marsden

Abstract: We introduce a new class of integrators for stiff ODEs as well as SDEs. These integrators are (i) {\it Multiscale}: they are based on flow averaging and so do not fully resolve the fast variables and have a computational cost determined by slow variables (ii) {\it Versatile}: the method is based on averaging the flows of the given dynamical system (which may have hidden slow and fast processes)… ▽ More We introduce a new class of integrators for stiff ODEs as well as SDEs. These integrators are (i) {\it Multiscale}: they are based on flow averaging and so do not fully resolve the fast variables and have a computational cost determined by slow variables (ii) {\it Versatile}: the method is based on averaging the flows of the given dynamical system (which may have hidden slow and fast processes) instead of averaging the instantaneous drift of assumed separated slow and fast processes. This bypasses the need for identifying explicitly (or numerically) the slow or fast variables (iii) {\it Nonintrusive}: A pre-existing numerical scheme resolving the microscopic time scale can be used as a black box and easily turned into one of the integrators in this paper by turning the large coefficients on over a microscopic timescale and off during a mesoscopic timescale (iv) {\it Convergent over two scales}: strongly over slow processes and in the sense of measures over fast ones. We introduce the related notion of two-scale flow convergence and analyze the convergence of these integrators under the induced topology (v) {\it Structure preserving}: for stiff Hamiltonian systems (possibly on manifolds), they can be made to be symplectic, time-reversible, and symmetry preserving (symmetries are group actions that leave the system invariant) in all variables. They are explicit and applicable to arbitrary stiff potentials (that need not be quadratic). Their application to the Fermi-Pasta-Ulam problems shows accuracy and stability over four orders of magnitude of time scales. For stiff Langevin equations, they are symmetry preserving, time-reversible and Boltzmann-Gibbs reversible, quasi-symplectic on all variables and conformally symplectic with isotropic friction. △ Less

Submitted 13 April, 2010; v1 submitted 9 August, 2009; originally announced August 2009.

Comments: 69 pages, 21 figures

MSC Class: 65P10; 70K70; 65C30; 65B99

Journal ref: Multiscale Model. Simul. Vol. 8, Issue 4, pp. 1269-1324 (2010)