Showing papers in "SIAM Journal on Scientific Computing in 2020"
TL;DR: The accuracy and effectiveness of PI-GANs in solving SDEs for up to 30 dimensions is demonstrated, but in principle, PI-gans could tackle very high dimensional problems given more sensor data with low-polynomial growth in computational cost.
Abstract: We developed a new class of physics-informed generative adversarial networks (PI-GANs) to solve forward, inverse, and mixed stochastic problems in a unified manner based on a limited number of scat...
181 citations
TL;DR: Two new Physics-Informed Neural Networks (PINNs) are proposed for solving time-dependent SPDEs, namely the NN-DO/BO methods, which incorporate the DO/BO constraints into the loss function with an implicit form instead of generating explicit expressions for the temporal derivatives of the Do/BO modes.
Abstract: One of the open problems in scientific computing is the long-time integration of nonlinear stochastic partial differential equations (SPDEs), especially with arbitrary initial data. We address this...
159 citations
TL;DR: The framework of inner product norm preserving relaxation Runge-Kutta methods is extended to general convex quantities and is proved analytically and demonstrated in several numerical examples, including applications to high-order entropy-conservative and entropy-stable semi-discretizations on unstructured grids for the compressible Euler and Navier-Stokes equations.
Abstract: The framework of inner product norm preserving relaxation Runge--Kutta methods [D. I. Ketcheson, SIAM J. Numer. Anal., 57 (2019), pp. 2850--2870] is extended to general convex quantities. Conservat...
89 citations
TL;DR: Numerical results on benchmark examples with interacting wave-type structures and time-varying transport speeds and on a model combustor of a single-element rocket engine demonstrate the wide applicability of the approach and the significant runtime speedups compared to full models and traditional reduced models.
Abstract: This work presents a model reduction approach for problems with coherent structures that propagate over time, such as convection-dominated flows and wave-type phenomena. Traditional model reduction...
83 citations
TL;DR: This novel auxiliary variable method based on exponential form of nonlinear free energy potential is more effective and applicable than the traditional SAV method which is very popular to construct energy stable schemes.
Abstract: In this paper, we consider an exponential scalar auxiliary variable (E-SAV) approach to obtain energy stable schemes for a class of phase field models. This novel auxiliary variable method based on...
75 citations
TL;DR: This work presents a systematic approach to developing arbitrarily high-order, unconditionally energy stable numerical schemes for thermodynamically consistent gradient flow models that satisfy energy dissipation requirements.
Abstract: We present a systematic approach to developing arbitrarily high-order, unconditionally energy stable numerical schemes for thermodynamically consistent gradient flow models that satisfy energy diss...
62 citations
TL;DR: The proposed PhaseDNN is able to convert high frequency learning to low frequency one, allowing a uniform learning to wideband functions and will then be applied to find the solution of high frequency wave equations in inhomogeneous media through both differential and integral equation formulations with least square residual loss functions.
Abstract: In this paper, we propose a phase shift deep neural network (PhaseDNN), which provides a uniform wideband convergence in approximating high frequency functions and solutions of wave equations. The ...
57 citations
TL;DR: In this article, the authors propose a gradients for multivariate functions encountered in high-dimensional uncertainty quantification problems often vary most strongly along a few dominant directions in the input parameter space.
Abstract: Multivariate functions encountered in high-dimensional uncertainty quantification problems often vary most strongly along a few dominant directions in the input parameter space. We propose a gradie...
54 citations
TL;DR: The registration procedure is applied, in combination with a linear compression method, to devise low-dimensional representations of solution manifolds with slowly-decaying Kolmogorov $N$-widths; the application to problems in parameterized geometries is considered.
Abstract: We propose a general---i.e., independent of the underlying equation---registration method for parameterized model order reduction. Given the spatial domain $\Omega \subset \mathbb{R}^d$ and the man...
52 citations
TL;DR: This work presents several essential improvements to the powerful scalar auxiliary variable (SAV) approach, by using the introduced scalar variable to control both the nonlinear and the explicit linear models.
Abstract: We present several essential improvements to the powerful scalar auxiliary variable (SAV) approach. Firstly, by using the introduced scalar variable to control both the nonlinear and the explicit l...
52 citations
TL;DR: This work investigates the stability of (discrete) empirical interpolation for nonlinear model reduction and state field approximation from measurements and presents a deterministic sampling strategy that aims to achieve lower approximation errors with fewer points than randomized sampling by taking information about the low-dimensional spaces into account.
Abstract: This work investigates the stability of (discrete) empirical interpolation for nonlinear model reduction and state field approximation from measurements. Empirical interpolation derives approximati...
TL;DR: Numerical results demonstrate that the low-dimensional models learned with the proposed approach match reduced models from traditional model reduction up to numerical errors in practice and are predictive even in situations where models fitted to trajectories without re-projection are inaccurate and unstable.
Abstract: This work introduces a method for learning low-dimensional models from data of high-dimensional black-box dynamical systems. The novelty is that the learned models are exactly the reduced models th...
TL;DR: This paper presents a simple randomized extended average block Kaczmarz algorithm that is suitable for solving large-scale linear systems and has shown good results in both linear and non-linear cases.
Abstract: Randomized iterative algorithms have recently been proposed to solve large-scale linear systems. In this paper, we present a simple randomized extended average block Kaczmarz algorithm that exponen...
TL;DR: Computing units that carry out a fused multiply-add (FMA) operation with matrix arguments, referred to as tensor units by some vendors, have great potential for use in scientific computing.
Abstract: Computing units that carry out a fused multiply-add (FMA) operation with matrix arguments, referred to as tensor units by some vendors, have great potential for use in scientific computing. However...
TL;DR: Adaptive second-order Crank-Nicolson time-stepping methods using the recent scalar auxiliary variable (SAV) approach are developed for the time-fractional molecular beam epitaxial models with Capu... as discussed by the authors.
Abstract: Adaptive second-order Crank--Nicolson time-stepping methods using the recent scalar auxiliary variable (SAV) approach are developed for the time-fractional molecular beam epitaxial models with Capu...
TL;DR: This paper obtains optimal error estimates of rational spectral approximation in the fractional Sobolev spaces, and analyses the optimal convergence of the proposed Galerkin scheme to show that the rational method outperforms the Hermite function approach.
Abstract: Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decaying slowly and subject to certain power law. Their numerical solutions are underexplor...
TL;DR: A new class of high-order maximum principle preserving numerical methods is proposed for solving parabolic equations, with application to the semilinear Allen--Cahn equation.
Abstract: A new class of high-order maximum principle preserving numerical methods is proposed for solving parabolic equations, with application to the semilinear Allen--Cahn equation. The proposed method co...
TL;DR: A numerical framework for recovering unknown nonautonomous dynamical systems with time-dependent inputs is presented and the difficulty presented by the nonaut autonomous nature of the system is circumvented.
Abstract: We present a numerical framework for recovering unknown nonautonomous dynamical systems with time-dependent inputs. To circumvent the difficulty presented by the nonautonomous nature of the system,...
TL;DR: In this article, a simple discretization scheme for the hypersingular integral representa- tion of the fractional Laplace operator and solver for the corresponding fractional Lplacian problem is presented.
Abstract: We present a simple discretization scheme for the hypersingular integral representa- tion of the fractional Laplace operator and solver for the corresponding fractional Laplacian problem. Through s...
TL;DR: In this paper, the authors focus on the efficient numerization of PDE-constrained optimization problems with uncertain inputs, such as random forces or material parameters, and propose a model incorporating uncertain inputs.
Abstract: Models incorporating uncertain inputs, such as random forces or material parameters, have been of increasing interest in PDE-constrained optimization. In this paper, we focus on the efficient numer...
TL;DR: Dynamic mode decomposition (DMD), which belongs to the family of singular-value decompositions (SVDs), is a popular tool of data-driven regression and while multiple numerical tests demonstrated the p...
Abstract: Dynamic mode decomposition (DMD), which belongs to the family of singular-value decompositions (SVDs), is a popular tool of data-driven regression. While multiple numerical tests demonstrated the p...
TL;DR: In this paper, the authors consider the problem of nonsmooth convex optimization with linear equality constraints, where the objective function is only accessible through its proximal operator, and show that the problem is NP-hard.
Abstract: We consider the problem of nonsmooth convex optimization with linear equality constraints, where the objective function is only accessible through its proximal operator. This problem arises in many...
TL;DR: In this paper, a hyper-reduction step is needed to reduce the computational complexity of reduced order models for nonlinear dynamical systems, which can achieve a considerable speed-up.
Abstract: Several reduced order models have been successfully developed for nonlinear dynamical systems. To achieve a considerable speed-up, a hyper-reduction step is needed to reduce the computational compl...
TL;DR: The Peng-Robinson equation of state (EoS) has become one of the most extensively applied equations of state in chemical engineering and the petroleum industry due to its excellent accuracy in p...
Abstract: The Peng--Robinson equation of state (PR-EoS) has become one of the most extensively applied equations of state in chemical engineering and the petroleum industry due to its excellent accuracy in p...
TL;DR: A pressure robust weak Galerkin finite element scheme for Stokes equations on polygonal mesh is developed and the major idea for achieving a pressure-independent energy-error estimate is achieved.
Abstract: In this paper, we develop a pressure robust weak Galerkin finite element scheme for Stokes equations on polygonal mesh. The major idea for achieving a pressure-independent energy-error estimate is ...
TL;DR: A numerical approach for approximating unknown Hamiltonian systems using observation data is presented, which is structure-preserving, in the sense that it enforces conservation of the reconstructed Hamiltonian by directly approximating the underlyingunknown Hamiltonian.
Abstract: We present a numerical approach for approximating unknown Hamiltonian systems using observational data. A distinct feature of the proposed method is that it is structure-preserving, in the sense th...
TL;DR: A continuous data assimilation algorithm proposed by Azouani, Olson, and Titi (AOT) in the context of an unknown viscosity is studied and the large-time error between the true solution of t is determined.
Abstract: We study a continuous data assimilation algorithm proposed by Azouani, Olson, and Titi (AOT) in the context of an unknown viscosity. We determine the large-time error between the true solution of t...
TL;DR: This paper studies a radial basis function that serves as a basis for interpolation and approximation techniques for functions with discontinuities in many applications, such as medical imaging.
Abstract: Accurate interpolation and approximation techniques for functions with discontinuities are key tools in many applications, such as medical imaging. In this paper, we study a radial basis function t...
TL;DR: Several efficient numerical schemes are developed which preserve exactly the global constraints for constrained gradient flows, based on the scalar auxiliary variable (SAV) approach.
Abstract: We develop several efficient numerical schemes which preserve exactly the global constraints for constrained gradient flows. Our schemes are based on the scalar auxiliary variable (SAV) approach co...
TL;DR: In this article, a robust numerical path tracking algorithm for polynomial homotopy continuation is proposed, which is robust in the sense that it is designed to prevent path jumping.
Abstract: We propose a new algorithm for numerical path tracking in polynomial homotopy continuation. The algorithm is “robust” in the sense that it is designed to prevent path jumping, and in many cases it ...