arXiv subjects

Jan S. Hesthaven

Publications and source records attributed to Jan S. Hesthaven.

At least 37 records · Page 2Linked to original sources

arXiv2022

Rank-adaptive structure-preserving model order reduction of Hamiltonian systems

This work proposes an adaptive structure-preserving model order reduction method for finite-dimensional parametrized Hamiltonian systems modeling non-dissipative phenomena. To overcome the slowly decaying Kolmogorov width typical of transport problems, the full model is approximated on local reduced spaces that are adapted in time using dynamical low-rank approximation techniques. The reduced dynamics is prescribed by approximating the symplectic projection of the Hamiltonian vector field in the tangent space to the local reduced space. This ensures that the canonical symplectic structure of the Hamiltonian dynamics is preserved during the reduction. In addition, accurate approximations with low-rank reduced solutions are obtained by allowing the dimension of the reduced space to change during the time evolution. Whenever the quality of the reduced solution, assessed via an error indicator, is not satisfactory, the reduced basis is augmented in the parameter direction that is worst approximated by the current basis. Extensive numerical tests involving wave interactions, nonlinear transport problems, and the Vlasov equation demonstrate the superior stability properties and considerable runtime speedups of the proposed method as compared to global and traditional reduced basis approaches.

math.NA↗

arXiv2021

FC-based shock-dynamics solver with neural-network localized artificial-viscosity assignment

This paper presents a spectral scheme for the numerical solution of nonlinear conservation laws in non-periodic domains under arbitrary boundary conditions. The approach relies on the use of the Fourier Continuation (FC) method for spectral representation of non-periodic functions in conjunction with smooth localized artificial viscosity assignments produced by means of a Shock-Detecting Neural Network (SDNN). Like previous shock capturing schemes and artificial viscosity techniques, the combined FC-SDNN strategy effectively controls spurious oscillations in the proximity of discontinuities. Thanks to its use of a localized but smooth artificial viscosity term, whose support is restricted to a vicinity of flow-discontinuity points, the algorithm enjoys spectral accuracy and low dissipation away from flow discontinuities, and, in such regions, it produces smooth numerical solutions -- as evidenced by an essential absence of spurious oscillations in level set lines. The FC-SDNN viscosity assignment, which does not require use of problem-dependent algorithmic parameters, induces a significantly lower overall dissipation than other methods, including the Fourier-spectral versions of the previous entropy viscosity method. The character of the proposed algorithm is illustrated with a variety of numerical results for the linear advection, Burgers and Euler equations in one and two-dimensional non-periodic spatial domains.

math.NA↗

arXiv2021

An artificial neural network approach to bifurcating phenomena in computational fluid dynamics

This work deals with the investigation of bifurcating fluid phenomena using a reduced order modelling setting aided by artificial neural networks. We discuss the POD-NN approach dealing with non-smooth solutions set of nonlinear parametrized PDEs. Thus, we study the Navier-Stokes equations describing: (i) the Coanda effect in a channel, and (ii) the lid driven triangular cavity flow, in a physical/geometrical multi-parametrized setting, considering the effects of the domain's configuration on the position of the bifurcation points. Finally, we propose a reduced manifold-based bifurcation diagram for a non-intrusive recovery of the critical points evolution. Exploiting such detection tool, we are able to efficiently obtain information about the pattern flow behaviour, from symmetry breaking profiles to attaching/spreading vortices, even at high Reynolds numbers.

physics.flu-dyn↗

arXiv2021

A hierarchical preconditioner for wave problems in quasilinear complexity

The paper introduces a novel, hierarchical preconditioner based on nested dissection and hierarchical matrix compression. The preconditioner is intended for continuous and discontinuous Galerkin formulations of elliptic problems. We exploit the property that Schur complements arising in such problems can be well approximated by hierarchical matrices. An approximate factorization can be computed matrix-free and in a (quasi-)linear number of operations. The nested dissection is specifically designed to aid the factorization process using hierarchical matrices. We demonstrate the viability of the preconditioner on a range of 2D problems, including the Helmholtz equation and the elastic wave equation. Throughout all tests, including wave phenomena with high wavenumbers, the generalized minimal residual method (GMRES) with the proposed preconditioner converges in a very low number of iterations. We demonstrate that this is due to the hierarchical nature of our approach which makes the high wavenumber limit manageable.

math.NA↗

arXiv2021

Discovery of slow variables in a class of multiscale stochastic systems via neural networks

Finding a reduction of complex, high-dimensional dynamics to its essential, low-dimensional "heart" remains a challenging yet necessary prerequisite for designing efficient numerical approaches. Machine learning methods have the potential to provide a general framework to automatically discover such representations. In this paper, we consider multiscale stochastic systems with local slow-fast time scale separation and propose a new method to encode in an artificial neural network a map that extracts the slow representation from the system. The architecture of the network consists of an encoder-decoder pair that we train in a supervised manner to learn the appropriate low-dimensional embedding in the bottleneck layer. We test the method on a number of examples that illustrate the ability to discover a correct slow representation. Moreover, we provide an error measure to assess the quality of the embedding and demonstrate that pruning the network can pinpoint an essential coordinates of the system to build the slow representation.

math.DS↗

arXiv2021

Multi-fidelity regression using artificial neural networks: efficient approximation of parameter-dependent output quantities

Highly accurate numerical or physical experiments are often time-consuming or expensive to obtain. When time or budget restrictions prohibit the generation of additional data, the amount of available samples may be too limited to provide satisfactory model results. Multi-fidelity methods deal with such problems by incorporating information from other sources, which are ideally well-correlated with the high-fidelity data, but can be obtained at a lower cost. By leveraging correlations between different data sets, multi-fidelity methods often yield superior generalization when compared to models based solely on a small amount of high-fidelity data. In this work, we present the use of artificial neural networks applied to multi-fidelity regression problems. By elaborating a few existing approaches, we propose new neural network architectures for multi-fidelity regression. The introduced models are compared against a traditional multi-fidelity scheme, co-kriging. A collection of artificial benchmarks are presented to measure the performance of the analyzed models. The results show that cross-validation in combination with Bayesian optimization consistently leads to neural network models that outperform the co-kriging scheme. Additionally, we show an application of multi-fidelity regression to an engineering problem. The propagation of a pressure wave into an acoustic horn with parametrized shape and frequency is considered, and the index of reflection intensity is approximated using the multi-fidelity models. A finite element model and a reduced basis model are adopted as the high- and low-fidelity, respectively. It is shown that the multi-fidelity neural network returns outputs that achieve a comparable accuracy to those from the expensive, full-order model, using only very few full-order evaluations combined with a larger amount of inaccurate but cheap evaluations of a reduced order model.

math.NA↗

arXiv2019

Estimation of groundwater storage from seismic data using deep learning

Convolutional neural networks can provide a potential framework to characterize groundwater storage from seismic data. Estimation of key components such as the amount of groundwater stored in an aquifer and delineate water-table level, from active-source seismic data are investigated in this study. The data to train, validate, and test the neural networks are obtained by solving wave propagation in a coupled poroviscoelastic-elastic media. A discontinuous Galerkin method is applied to model wave propagation whereas a deep convolutional neural network is used for the parameter estimation problem. In the numerical experiment, the primary unknowns are the amount of stored groundwater and water-table level, and are estimated, while the remaining parameters, assumed to be of less of interest, are marginalized in the convolutional neural networks-based solution. Results, obtained through synthetic data, illustrate the potential of deep learning methods to extract additional aquifer information from seismic data, which otherwise would be impossible based on a set of reflection seismic sections or velocity tomograms.

physics.comp-ph↗

arXiv2019

Constraint-Aware Neural Networks for Riemann Problems

Neural networks are increasingly used in complex (data-driven) simulations as surrogates or for accelerating the computation of classical surrogates. In many applications physical constraints, such as mass or energy conservation, must be satisfied to obtain reliable results. However, standard machine learning algorithms are generally not tailored to respect such constraints. We propose two different strategies to generate constraint-aware neural networks. We test their performance in the context of front-capturing schemes for strongly nonlinear wave motion in compressible fluid flow. Precisely, in this context so-called Riemann problems have to be solved as surrogates. Their solution describes the local dynamics of the captured wave front in numerical simulations. Three model problems are considered: a cubic flux model problem, an isothermal two-phase flow model, and the Euler equations. We demonstrate that a decrease in the constraint deviation correlates with low discretization errors for all model problems, in addition to the structural advantage of fulfilling the constraint.

physics.comp-ph↗

arXiv2018

Symplectic Model-Reduction with a Weighted Inner Product

In the recent years, considerable attention has been paid to preserving structures and invariants in reduced basis methods, in order to enhance the stability and robustness of the reduced system. In the context of Hamiltonian systems, symplectic model reduction seeks to construct a reduced system that preserves the symplectic symmetry of Hamiltonian systems. However, symplectic methods are based on the standard Euclidean inner products and are not suitable for problems equipped with a more general inner product. In this paper, we generalize symplectic model reduction to allow for the norms and inner products that are most appropriate to the problem while preserving the symplectic symmetry of the Hamiltonian systems. To construct a reduced basis and accelerate the evaluation of nonlinear terms, a greedy generation of a symplectic basis is proposed. Furthermore, it is shown that the greedy approach yields a norm-bounded reduced basis. The accuracy and the stability of this model reduction technique are illustrated through the development of reduced models for a vibrating elastic beam and the sine-Gordon equation.

math.NA↗

arXiv2018

Deep convolutional neural networks for estimating porous material parameters with ultrasound tomography

We study the feasibility of data based machine learning applied to ultrasound tomography to estimate water-saturated porous material parameters. In this work, the data to train the neural networks is simulated by solving wave propagation in coupled poroviscoelastic-viscoelastic-acoustic media. As the forward model, we consider a high-order discontinuous Galerkin method while deep convolutional neural networks are used to solve the parameter estimation problem. In the numerical experiment, we estimate the material porosity and tortuosity while the remaining parameters which are of less interest are successfully marginalized in the neural networks-based inversion. Computational examples confirms the feasibility and accuracy of this approach.

physics.comp-ph↗

arXiv2017

Structure-Preserving Model-Reduction of Dissipative Hamiltonian Systems

Reduced basis methods are popular for approximately solving large and complex systems of differential equations. However, conventional reduced basis methods do not generally preserve conservation laws and symmetries of the full order model. Here, we present an approach for reduced model construction, that preserves the symplectic symmetry of dissipative Hamiltonian systems. The method constructs a closed reduced Hamiltonian system by coupling the full model with a canonical heat bath. This allows the reduced system to be integrated with a symplectic integrator, resulting in a correct dissipation of energy, preservation of the total energy and, ultimately, in the stability of the solution. Accuracy and stability of the method are illustrated through the numerical simulation of the dissipative wave equation and a port-Hamiltonian model of an electric circuit.

math.NA↗

arXiv2017

Structure Preserving Model Reduction of Parametric Hamiltonian Systems

While reduced-order models (ROMs) have been popular for efficiently solving large systems of differential equations, the stability of reduced models over long-time integration is of present challenges. We present a greedy approach for ROM generation of parametric Hamiltonian systems that captures the symplectic structure of Hamiltonian systems to ensure stability of the reduced model. Through the greedy selection of basis vectors, two new vectors are added at each iteration to the linear vector space to increase the accuracy of the reduced basis. We use the error in the Hamiltonian due to model reduction as an error indicator to search the parameter space and identify the next best basis vectors. Under natural assumptions on the set of all solutions of the Hamiltonian system under variation of the parameters, we show that the greedy algorithm converges with exponential rate. Moreover, we demonstrate that combining the greedy basis with the discrete empirical interpolation method also preserves the symplectic structure. This enables the reduction of the computational cost for nonlinear Hamiltonian systems. The efficiency, accuracy, and stability of this model reduction technique is illustrated through simulations of the parametric wave equation and the parametric Schrodinger equation.

math.NA↗

arXiv2016

Projective multiscale time-integration for electrostatic particle-in-cell methods

The simulation of problems in kinetic plasma physics are often challenging due to strongly coupled phenomena across multiple scales. In this work, we propose a wavelet-based coarse-grained numerical scheme, based on the framework of Equation-Free Projective Integration, for a kinetic plasma system modeled by the Vlasov-Poisson equations. A kinetic particle-in-cell (PIC) code is used to simulate the meso scale dynamics for short time intervals. This allows the extrapolation over long time-steps of the behavior of a coarse wavelet-based discretization of the system. To validate the approach and the underlying concepts, we perform two 1D1V numerical experiments: nonlinear propagation and steepening of an ion wave, and the expansion of a plasma slab in vacuum. The direct comparisons to resolved PIC simulations show good agreement. We show that the speedup of the projective integration scheme over the full particle scheme scales linearly with the system size, demonstrating efficiency while taking into account fully kinetic, non-Maxwellian effects. This suggests that the approach is potentially interesting for kinetic plasma problems with a large separation of scales, especially in higher dimensions.

physics.comp-ph↗

arXiv2015

A Preconditioner Based on Low-Rank Approximation of Schur Complements

We introduce a preconditioner based on a hierarchical low-rank compression scheme of Schur complements. The construction is inspired by standard nested dissection, and relies on the assumption that the Schur complements can be approximated, to high precision, by Hierarchically-Semi-Separable matrices. We build the preconditioner as an approximate $LDM^t$ factorization of a given matrix $A$, and no knowledge of $A$ in assembled form is required by the construction. The $LDM^t$ factorization is amenable to fast inversion, and the action of the inverse can be determined fast as well. We investigate the behavior of the preconditioner in the context of DG finite element approximations of elliptic and hyperbolic problems.

math.NA↗

arXiv2014

Local discontinuous Galerkin methods for fractional ordinary differential equations

This paper discusses the upwinded local discontinuous Galerkin methods for the one-term/multi-term fractional ordinary differential equations (FODEs). The natural upwind choice of the numerical fluxes for the initial value problem for FODEs ensures stability of the methods. The solution can be computed element by element with optimal order of convergence $k+1$ in the $L^2$ norm and superconvergence of order $k+1+\min\{k,α\}$ at the downwind point of each element. Here $k$ is the degree of the approximation polynomial used in an element and $α$ ($α\in (0,1]$) represents the order of the one-term FODEs. A generalization of this includes problems with classic $m$'th-term FODEs, yielding superconvergence order at downwind point as $k+1+\min\{k,\max\{α,m\}\}$. The underlying mechanism of the superconvergence is discussed and the analysis confirmed through examples, including a discussion of how to use the scheme as an efficient way to evaluate the generalized Mittag-Leffler function and solutions to more generalized FODE's.

math.NA↗

arXiv2014

Fast prediction and evaluation of gravitational waveforms using surrogate models

[Abridged] We propose a solution to the problem of quickly and accurately predicting gravitational waveforms within any given physical model. The method is relevant for both real-time applications and in more traditional scenarios where the generation of waveforms using standard methods can be prohibitively expensive. Our approach is based on three offline steps resulting in an accurate reduced-order model that can be used as a surrogate for the true/fiducial waveform family. First, a set of m parameter values is determined using a greedy algorithm from which a reduced basis representation is constructed. Second, these m parameters induce the selection of m time values for interpolating a waveform time series using an empirical interpolant. Third, a fit in the parameter dimension is performed for the waveform's value at each of these m times. The cost of predicting L waveform time samples for a generic parameter choice is of order m L + m c_f online operations where c_f denotes the fitting function operation count and, typically, m << L. We generate accurate surrogate models for Effective One Body (EOB) waveforms of non-spinning binary black hole coalescences with durations as long as 10^5 M, mass ratios from 1 to 10, and for multiple harmonic modes. We find that these surrogates are three orders of magnitude faster to evaluate as compared to the cost of generating EOB waveforms in standard ways. Surrogate model building for other waveform models follow the same steps and have the same low online scaling cost. For expensive numerical simulations of binary black hole coalescences we thus anticipate large speedups in generating new waveforms with a surrogate. As waveform generation is one of the dominant costs in parameter estimation algorithms and parameter space exploration, surrogate models offer a new and practical way to dramatically accelerate such studies without impacting accuracy.

gr-qc↗

arXiv2013

Solving Wave Equations on Unstructured Geometries

Waves are all around us--be it in the form of sound, electromagnetic radiation, water waves, or earthquakes. Their study is an important basic tool across engineering and science disciplines. Every wave solver serving the computational study of waves meets a trade-off of two figures of merit--its computational speed and its accuracy. Discontinuous Galerkin (DG) methods fall on the high-accuracy end of this spectrum. Fortuitously, their computational structure is so ideally suited to GPUs that they also achieve very high computational speeds. In other words, the use of DG methods on GPUs significantly lowers the cost of obtaining accurate solutions. This article aims to give the reader an easy on-ramp to the use of this technology, based on a sample implementation which demonstrates a highly accurate, GPU-capable, real-time visualizing finite element solver in about 1500 lines of code.

cs.MS↗

arXiv2012

High-Order Discontinuous Galerkin Methods by GPU Metaprogramming

Discontinuous Galerkin (DG) methods for the numerical solution of partial differential equations have enjoyed considerable success because they are both flexible and robust: They allow arbitrary unstructured geometries and easy control of accuracy without compromising simulation stability. In a recent publication, we have shown that DG methods also adapt readily to execution on modern, massively parallel graphics processors (GPUs). A number of qualities of the method contribute to this suitability, reaching from locality of reference, through regularity of access patterns, to high arithmetic intensity. In this article, we illuminate a few of the more practical aspects of bringing DG onto a GPU, including the use of a Python-based metaprogramming infrastructure that was created specifically to support DG, but has found many uses across all disciplines of computational science.

cs.MS↗