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Bui-Thanh, Tan

Publications and source records attributed to Bui-Thanh, Tan.

At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Tokamak disruption simulation

UT contribution in this Tokamak Disruption Simulation (TDS) is to develop a parallel high‐order hybridized Discontinuous Galerkin (HDG) methods for large‐s cale MHD simulations. The following are the major goals: 1) Construction of HDG methods for linearized MHD, 2) Rigorous analysis for the HDG formulations for linearized MHD, 3) 2D and 3D simulations to verify the convergent of the HDG methods for linearized MHD, 4) multigrid solvers/preconditioners for HDG formulations, 5) HDG for reconnection problems, 6) Divergence cleaning with HDG, 7) HDG formulations for nonlinear MHD, 8) Picard fixed point HDG approach, 9) IMEX HDG‐DG for nonlinear HDG; 10) parallel large‐scale HDG for linear and nonlinear MHD simulations.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

TNet: A Model-Constrained Tikhonov Network Approach for Inverse Problems

Deep learning (DL), in particular deep neural networks, by default is purely data-driven and in general does not require physics. This is the strength of DL but also one of its key limitations when applied to science and engineering problems in which underlying physical properties—such as stability, conservation, and positivity—and accuracy are required. DL methods in their original forms are often not capable of respecting the underlying mathematical models or achieving desired accuracy even in big-data regimes. On the other hand, many data-driven science and engineering problems, such as inverse problems, typically have limited experimental or observational data, and DL would overfit the data in this case. Leveraging information encoded in the underlying mathematical models, we argue, not only compensates for missing information in low data regimes but also provides opportunities to equip DL methods with the underlying physics, hence promoting better generalization. This paper develops a model-constrained DL approach and its variant TNet—a Tikhonov neural network—which are capable of learning not only information hidden in the training data but also in the underlying mathematical models to solve inverse problems governed by partial differential equations in low data regimes. We provide the constructions and some theoretical results for the proposed approaches for both linear and nonlinear inverse problems. Since TNet is designed to learn inverse solutions with Tikhonov regularization, it is interpretable: in fact it recovers Tikhonov solutions for linear cases while potentially approximating Tikhonov solutions for nonlinear inverse problems. We also prove that data randomization can enhance not only the smoothness of the networks but also their generalizations. Comprehensive numerical results confirm the theoretical findings and show that with even as little as 1 training data sample for one-dimensional (1D) deconvolution, 5 for an inverse 2D heat conductivity problem, 100 for inverse initial conditions for a time-dependent 2D Burgers’s equation, and 50 for inverse initial conditions for 2D Navier–Stokes equations, TNet solutions can be as accurate as Tikhonov solutions while being several orders of magnitude faster. Furthermore, this is possible owing to the model-constrained term, replications, and randomization.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

A unified hp-HDG framework for Friedrichs' PDE systems

Here, this work proposes a unified hp-adaptivity framework for hybridized discontinuous Galerkin (HDG) method for a large class of partial differential equations (PDEs) of Friedrichs' type. In particular, we present unified hp-HDG formulations for abstract one-field and two-field structures and prove their well-posedness. In order to handle non-conforming interfaces we simply take advantage of HDG built-in mortar structures. With split-type mortars and the approximation space of trace, a numerical flux can be derived via Godunov approach and be naturally employed without any additional treatment. As a consequence, the proposed formulations are parameter-free. We perform several numerical experiments for time-independent and linear PDEs including elliptic, hyperbolic, and mixed-type to verify the proposed unified hp-formulations and demonstrate the effectiveness of hp-adaptation. Two adaptivity criteria are considered: one is based on a simple and fast error indicator, while the other is rigorous but more expensive using an adjoint-based error estimate. The numerical results show that these two approaches are comparable in terms of convergence rate even for problems with strong gradients, discontinuities, and singularities.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

An autoencoder compression approach for accelerating large-scale inverse problems

Abstract Partial differential equation (PDE)-constrained inverse problems are some of the most challenging and computationally demanding problems in computational science today. Fine meshes required to accurately compute the PDE solution introduce an enormous number of parameters and require large-scale computing resources such as more processors and more memory to solve such systems in a reasonable time. For inverse problems constrained by time-dependent PDEs, the adjoint method often employed to compute gradients and higher order derivatives efficiently requires solving a time-reversed, so-called adjoint PDE that depends on the forward PDE solution at each timestep. This necessitates the storage of a high-dimensional forward solution vector at every timestep. Such a procedure quickly exhausts the available memory resources. Several approaches that trade additional computation for reduced memory footprint have been proposed to mitigate the memory bottleneck, including checkpointing and compression strategies. In this work, we propose a close-to-ideal scalable compression approach using autoencoders to eliminate the need for checkpointing and substantial memory storage, thereby reducing the time-to-solution and memory requirements. We compare our approach with checkpointing and an off-the-shelf compression approach on an earth-scale ill-posed seismic inverse problem. The results verify the expected close-to-ideal speedup for the gradient and Hessian-vector product using the proposed autoencoder compression approach. To highlight the usefulness of the proposed approach, we combine the autoencoder compression with the data-informed active subspace (DIAS) prior showing how the DIAS method can be affordably extended to large-scale problems without the need for checkpointing and large memory.

Mathematics↗
DOE OSTI2023

On unifying randomized methods for inverse problems

This work unifies the analysis of various randomized methods for solving linear and nonlinear inverse problems with Gaussian priors by framing the problem in a stochastic optimization setting. By doing so, we show that many randomized methods are variants of a sample average approximation (SAA). More importantly, we are able to prove a single theoretical result that guarantees the asymptotic convergence for a variety of randomized methods. Additionally, viewing randomized methods as an SAA enables us to prove, for the first time, a single non-asymptotic error result that holds for randomized methods under consideration. Another important consequence of our unified framework is that it allows us to discover new randomization methods. Here, we present various numerical results for linear, nonlinear, algebraic, and PDE-constrained inverse problems that verify the theoretical convergence results and provide a discussion on the apparently different convergence rates and the behavior for various randomized methods.

42 ENGINEERING↗
DOE OSTI2022

Forward and inverse modeling of fault transmissibility in subsurface flows

Characterizing physical properties of faults, such as their transmissibility, is crucial for performing predictive numerical simulation of subsurface flows, such as those encountered in petroleum engineering and remediation of subsurface contamination. Here, this paper provides a complete investigation of the inverse problem for fault transmissibility in subsurface flow models, under appropriate assumptions on fault structure. In particular, the following aspects are considered: 1) fault modeling and well-posedness of the forward problem; 2) finite element (FEM) discretizations of the forward problem and their rigorous a priori convergence analysis; 3) Well-posedness of the Bayesian inverse problem, FEM discretization of the infinite dimensional Bayesian inverse formulation, and its rigorous a priori analysis. Moreover, computation of the maximum a posteriori (MAP) point via fast inexact Newton-conjugate gradient optimization and a Laplace approximation of the Bayesian posterior are also presented. Numerical results illustrate the use of the proposed fault model in forward and inverse problems for subsurface flows in two dimensional domains with multiple faults.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

DIAS: A Data-Informed Active Subspace Regularization Framework for Inverse Problems

This paper presents a regularization framework that aims to improve the fidelity of Tikhonov inverse solutions. At the heart of the framework is the data-informed regularization idea that only data-uninformed parameters need to be regularized, while the data-informed parameters, on which data and forward model are integrated, should remain untouched. We propose to employ the active subspace method to determine the data-informativeness of a parameter. The resulting framework is thus called a data-informed (DI) active subspace (DIAS) regularization. Four proposed DIAS variants are rigorously analyzed, shown to be robust with the regularization parameter and capable of avoiding polluting solution features informed by the data. They are thus well suited for problems with small or reasonably small noise corruptions in the data. Furthermore, the DIAS approaches can effectively reuse any Tikhonov regularization codes/libraries. Though they are readily applicable for nonlinear inverse problems, we focus on linear problems in this paper in order to gain insights into the framework. Various numerical results for linear inverse problems are presented to verify theoretical findings and to demonstrate advantages of the DIAS framework over the Tikhonov, truncated SVD, and the TSVD-based DI approaches.

Nguyen, Hai↗
DOE OSTI2021

A scalable exponential-DG approach for nonlinear conservation laws: With application to Burger and Euler equations

In this work, we propose an Exponential DG framework for partial differential equations. We decompose 7 governing equations into linear and nonlinear parts to which we apply the discontinuous Galerkin 8 (DG) spatial discretization. In particular, we construct the linear part using Jacobian that effectively 9 capture stiff characteristics in the system. The former is integrated analytically, whereas the latter 10 is approximated. This approach i) is stable with a large Courant number (Cr > 1); ii) supports 11 high-order solutions both in time and space; iii) is computationally favorable compared to IMEX 12 DG methods with no preconditioner; iv) becomes comparable to explicit RKDG methods on uniform 13 mesh and beneficial on non-uniform grid for Euler equations; v) is scalable in a modern massively 14 parallel computing architecture due to its explicit nature of exponential time integrators and com15 pact communication stencil of DG method. Numerical results demonstrate the performance of our 16 proposed methods through various examples. We also discuss the stability and convergence analysis 17 for our exponential DG scheme in the context of Burgers equation.

42 ENGINEERING↗
DOE OSTI2020

Sequential ensemble transform for Bayesian inverse problems

In this work, we present the Sequential Ensemble Transform (SET) method, an approach for generating approximate samples from a Bayesian posterior distribution. The method explores the posterior distribution by solving a sequence of discrete optimal transport problems to produce a series of transport plans which map prior samples to posterior samples. We prove that the sequence of Dirac mixture distributions produced by the SET method converges weakly to the true posterior as the sample size approaches infinity. Furthermore, our numerical results indicate that, when compared to standard Sequential Monte Carlo (SMC) methods, the SET approach is more robust to the choice of Markov mutation kernels and requires less computational efforts to reach a similar accuracy when used to explore complex posterior distributions. Finally, we describe adaptive schemes that allow to completely automate the use of the SET method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2019

Unified Geometric Multigrid Algorithm for Hybridized High-Order Finite Element Methods

In this paper, we consider a standard elliptic partial differential equation and propose a geometric multigrid algorithm based on Dirichlet-to-Neumann (DtN) maps for hybridized high-order finite element methods. The proposed unified approach is applicable to any locally conservative hybridized finite element method including multinumerics with different hybridized methods in different parts of the domain. For these methods, the linear system involves only the unknowns residing on the mesh skeleton, and constructing intergrid transfer operators is therefore not trivial. The key to our geometric multigrid algorithm is the physics-based energy-preserving intergrid transfer operators which depend only on the fine scale DtN maps. Thanks to these operators, we completely avoid upscaling of parameters and no information regarding subgrid physics is explicitly required on coarse meshes. Moreover, our algorithm is agglomeration-based and can straightforwardly handle unstructured meshes. We perform extensive numerical studies with hybridized mixed methods, hybridized discontinuous Galerkin methods, weak Galerkin methods, and hybridized versions of interior penalty discontinuous Galerkin methods on a range of elliptic problems including subsurface flow through highly heterogeneous porous media. We compare the performance of different smoothers and analyze the effect of stabilization parameters on the scalability of the multigrid algorithm.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

IMEX HDG-DG: A coupled implicit hybridized discontinuous Galerkin and explicit discontinuous Galerkin approach for shallow water systems

We propose IMEX HDG-DG schemes for planar and spherical shallow water systems. Of interest is subcritical flow, where the speed of the gravity wave is faster than that of nonlinear advection. In order to simulate these flows efficiently, we split the governing system into a stiff part describing the gravity wave and a non-stiff part associated with nonlinear advection. The former is discretized implicitly with the HDG method while an explicit Runge-Kutta DG discretization is employed for the latter. The proposed IMEX HDG-DG framework: 1) facilitates high-order solutions both in time and space; 2) avoids overly small time-step sizes; 3) requires only one linear system solve per time stage; 4) relative to DG generates smaller and sparser linear systems while promoting further parallelism; and 5) suppresses the fast modes in the system with a large time-step size. Numerical results for various test cases demonstrate that our methods are beneficial for applications where non-stiff terms are accurately treated while stiff terms are less accurately handled.

42 ENGINEERING↗
DOE OSTI2019

Scalable Matrix-Free Adaptive Product-Convolution Approximation for Locally Translation-Invariant Operators

We present an adaptive grid matrix-free operator approximation scheme based on a "product-convolution" interpolation of convolution operators. This scheme is appropriate for operators that are locally translation-invariant, even if these operators are high rank or full rank. Such operators arise in Schur complement methods for solving partial differential equations (PDEs), as Hessians in PDE-constrained optimization and inverse problems, as integral operators, as covariance operators, and as Dirichlet-to-Neumann maps. Constructing the approximation requires computing the impulse responses of the operator to point sources centered on nodes in an adaptively refined grid of sample points. A randomized a posteriori error estimator drives the adaptivity. Once constructed, the approximation can be efficiently applied to vectors using the fast Fourier transform. The approximation can be efficiently converted to hierarchical matrix (H-matrix) format, then inverted or factorized using scalable H-matrix arithmetic. The quality of the approximation degrades gracefully as fewer sample points are used, allowing cheap lower quality approximations to be used as preconditioners. Furthermore, this yields an automated method to construct preconditioners for locally translation-invariant Schur complements. We directly address issues related to boundaries and prove that our scheme eliminates boundary artifacts. We test the scheme on a spatially varying blurring kernel, on the nonlocal component of an interface Schur complement for the Poisson operator, and on the data misfit Hessian for an advection dominated advection-diffusion inverse problem. Numerical results show that the scheme outperforms existing methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Active Subspace Methods for Data-Intensive Inverse Problems (Final Report)

Fostered by advances in computing, simulation and data science have emerged as the third and fourth paradigms enabling scientific discovery. Many mathematical challenges arise when integrating these paradigms to produce new knowledge. One of the most important challenges is the inverse or calibration problem: find the inputs of a simulation such that its outputs agree with a given set of measurements. As both the cost of the complex simulations and the volume of measurement data sets increase, intuition-driven trial-and-error methods for calibration quickly become untenable; modern scientists need more computationally efficient and mathematically rigorous methods. When measurements and simulated predictions contain uncertainty, the Bayesian perspective on statistical inference provides a popular mathematical framework for developing computational methods to solve inverse problems. Markov chain Monte Carlo (MCMC) methods are a mainstay of this framework; they characterize a probability distribution on the space of model inputs that is conditioned on the given measurements. However, despite recent advances, MCMC methods remain impractical for expensive simulations with more than a handful of inputs. At its heart, MCMC searches the space of inputs for corresponding outputs that match measurements, and so it is afflicted with the curse of dimensionality. Additionally, MCMC methods become less efficient as the volume of the measurement data increases, particularly if the measurements are redundant. To make MCMC—and Bayesian methods, in general—amenable to today’s computer models with millions of inputs and terabytes of measurement data, one must devise methods to reduce the dimension of both the simulation inputs and the measurements. For many models used in practice, the outputs vary primarily on a low-dimensional manifold of the nominally high-dimensional input space. Recently developed active subspace methods discover this low-dimensional manifold and exploit it to enable analyses typically suited for low-dimensional settings (e.g., statistical inversion of complex systems). We hypothesize that active subspace methods can provide the necessary dimension reduction, enabling existing Bayesian methods to find solutions for a large class of otherwise intractable statistical inverse problems. We will test the hypothesis with a series of mathematical analyses and computational experiments that will proceed along three research veins: (i) integrating active subspace methods into methods for statistical inversion, (ii) addressing current challenges in active subspace methods, and (iii) incorporating methods for nonlinear dimensionality reduction into active subspace methods. We will apply the methods we develop to real inverse problems in chemical kinetics and turbulent flame modeling. Our team combines expertise in methods for inverse problems, dimension reduction, and reduced order modeling. With a history of active collaboration and thoughtful engagement, we bring enthusiasm and fresh perspectives to a well studied subject. We are excited by the potential opportunity to work together to extend and apply active subspace methods to data-intensive inverse problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

An improved iterative HDG approach for partial differential equations

Here we propose and analyze an iterative high-order hybridized discontinuous Galerkin (iHDG) discretization for linear partial differential equations. We improve our previous work [45] in several directions: 1) the improved iHDG approach converges in a finite number of iterations for the scalar transport equation; 2) it is unconditionally convergent for both the linearized shallow water system and the convection–diffusion equation; 3) it has improved stability and convergence rates; 4) we uncover a relationship between the number of iterations and time stepsize, solution order, meshsize and the equation parameters. This allows us to choose the time stepsize such that the number of iterations is approximately independent of the solution order and the meshsize; and 5) we provide both strong and weak scalings of the improved iHDG approach up to 16,384 cores. A connection between iHDG and time integration methods such as parareal and implicit/explicit methods are discussed. Extensive numerical results for linear (and nonlinear) PDEs are presented to verify the theoretical findings.

97 MATHEMATICS AND COMPUTING↗