arXiv subjects

Daniel Peterseim

Publications and source records attributed to Daniel Peterseim.

At least 37 records · Page 2Linked to original sources

arXiv2023

Reconstructing microstructures from statistical descriptors using neural cellular automata

The problem of generating microstructures of complex materials in silico has been approached from various directions including simulation, Markov, deep learning and descriptor-based approaches. This work presents a hybrid method that is inspired by all four categories and has interesting scalability properties. A neural cellular automaton is trained to evolve microstructures based on local information. Unlike most machine learning-based approaches, it does not directly require a data set of reference micrographs, but is trained from statistical microstructure descriptors that can stem from a single reference. This means that the training cost scales only with the complexity of the structure and associated descriptors. Since the size of the reconstructed structures can be set during inference, even extremely large structures can be efficiently generated. Similarly, the method is very efficient if many structures are to be reconstructed from the same descriptor for statistical evaluations. The method is formulated and discussed in detail by means of various numerical experiments, demonstrating its utility and scalability.

cond-mat.mtrl-sci↗

arXiv2023

Super-localised wave function approximation of Bose-Einstein condensates

This paper presents a novel spatial discretisation method for the reliable and efficient simulation of Bose-Einstein condensates modelled by the Gross-Pitaevskii equation and the corresponding nonlinear eigenvector problem. The method combines the high-accuracy properties of numerical homogenisation methods with a novel super-localisation approach for the calculation of the basis functions. A rigorous numerical analysis demonstrates superconvergence of the approach compared to classical polynomial and multiscale finite element methods, even in low regularity regimes. Numerical tests reveal the method's competitiveness with spectral methods, particularly in capturing critical physical effects in extreme conditions, such as vortex lattice formation in fast-rotating potential traps. The method's potential is further highlighted through a dynamic simulation of a phase transition from Mott insulator to Bose-Einstein condensate, emphasising its capability for reliable exploration of physical phenomena.

math.NA↗

arXiv2023

Computational polyconvexification of isotropic functions

Based on the characterization of the polyconvex envelope of isotropic functions by their signed singular value representations, we propose a simple algorithm for the numerical approximation of the polyconvex envelope. Instead of operating on the $d^2$-dimensional space of matrices, the algorithm requires only the computation of the convex envelope of a function on a $d$-dimensional manifold, which is easily realized by standard algorithms. The significant speedup associated with the dimensional reduction from $d^2$ to $d$ is demonstrated in a series of numerical experiments.

math.NA↗

arXiv2023

Computational multiscale methods for nondivergence-form elliptic partial differential equations

This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence-form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit of numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. The rigorous error analysis of one exemplary approach shows that the favorable properties of the LOD methodology known from divergence-form PDEs, i.e., its applicability and accuracy beyond scale separation and periodicity, remain valid for problems in nondivergence-form.

math.NA↗

arXiv2023

Multidimensional rank-one convexification of incremental damage models at finite strains

This paper presents computationally feasible rank-one relaxation algorithms for the efficient simulation of a time-incremental damage model with nonconvex incremental stress potentials in multiple spatial dimensions. While the standard model suffers from numerical issues due to the lack of convexity, the relaxation by rank-one convexification prevents non-existence of minimizers and mesh dependence of the solutions of finite element discretizations. By the combination, modification and parallelization of the underlying convexification algorithms, the novel approach becomes computationally feasible. A descent method and a Newton scheme enhanced by step-size control prevent stability issues related to local minima in the energy landscape and the computation of derivatives. Numerical techniques for the construction of continuous derivatives of the approximated rank-one convex envelope are discussed. A series of numerical experiments demonstrates the ability of the computationally relaxed model to capture softening effects and the mesh independence of the computed approximations. An interpretation in terms of microstructural damage evolution is given, based on the rank-one lamination process.

cs.CE↗

arXiv2022

A reduced basis super-localized orthogonal decomposition for reaction-convection-diffusion problems

This paper presents a method for the numerical treatment of reaction-convection-diffusion problems with parameter-dependent coefficients that are arbitrary rough and possibly varying at a very fine scale. The presented technique combines the reduced basis (RB) framework with the recently proposed super-localized orthogonal decomposition (SLOD). More specifically, the RB is used for accelerating the typically costly SLOD basis computation, while the SLOD is employed for an efficient compression of the problem's solution operator requiring coarse solves only. The combined advantages of both methods allow one to tackle the challenges arising from parametric heterogeneous coefficients. Given a value of the parameter vector, the method outputs a corresponding compressed solution operator which can be used to efficiently treat multiple, possibly non-affine, right-hand sides at the same time, requiring only one coarse solve per right-hand side.

math.NA↗

arXiv2022

Multi-resolution Localized Orthogonal Decomposition for Helmholtz problems

We introduce a novel multi-resolution Localized Orthogonal Decomposition (LOD) for time-harmonic acoustic scattering problems that can be modeled by the Helmholtz equation. The method merges the concepts of LOD and operator-adapted wavelets (gamblets) and proves its applicability for a class of complex-valued, non-hermitian and indefinite problems. It computes hierarchical bases that block-diagonalize the Helmholtz operator and thereby decouples the discretization scales. Sparsity is preserved by a novel localization strategy that improves stability properties even in the elliptic case. We present a rigorous stability and a-priori error analysis of the proposed method for homogeneous media. In addition, we investigate the fast solvability of the blocks by a standard iterative method. A sequence of numerical experiments illustrates the sharpness of the theoretical findings and demonstrates the applicability to scattering problems in heterogeneous media.

math.NA↗

arXiv2022

Super-localization of elliptic multiscale problems

Numerical homogenization aims to efficiently and accurately approximate the solution space of an elliptic partial differential operator with arbitrarily rough coefficients in a $d$-dimensional domain. The application of the inverse operator to some standard finite element space defines an approximation space with uniform algebraic approximation rates with respect to the mesh size parameter $H$. This holds even for under-resolved rough coefficients. However, the true challenge of numerical homogenization is the localized computation of a localized basis for such an operator-dependent approximate solution space. This paper presents a novel localization technique that enforces the super-exponential decay of the basis relative to $H$. This shows that basis functions with supports of width $\mathcal O(H|\log H|^{(d-1)/d})$ are sufficient to preserve the optimal algebraic rates of convergence in $H$ without pre-asymptotic effects. A sequence of numerical experiments illustrates the significance of the new localization technique when compared to the so far best localization to supports of width $\mathcal O(H|\log H|)$.

math.NA↗

arXiv2022

A Super-Localized Generalized Finite Element Method

This paper presents a novel multi-scale method for elliptic partial differential equations with arbitrarily rough coefficients. In the spirit of numerical homogenization, the method constructs problem-adapted ansatz spaces with uniform algebraic approximation rates. Localized basis functions with the same super-exponential localization properties as the recently proposed Super-Localized Orthogonal Decomposition enable an efficient implementation. The method's basis stability is enforced using a partition of unity approach. A natural extension to higher order is presented, resulting in higher approximation rates and enhanced localization properties. We perform a rigorous a priori and a posteriori error analysis and confirm our theoretical findings in a series of numerical experiments. In particular, we demonstrate the method's applicability for challenging high-contrast channeled coefficients.

math.NA↗

arXiv2022

Neural network approximation of coarse-scale surrogates in numerical homogenization

Coarse-scale surrogate models in the context of numerical homogenization of linear elliptic problems with arbitrary rough diffusion coefficients rely on the efficient solution of fine-scale sub-problems on local subdomains whose solutions are then employed to deduce appropriate coarse contributions to the surrogate model. However, in the absence of periodicity and scale separation, the reliability of such models requires the local subdomains to cover the whole domain which may result in high offline costs, in particular for parameter-dependent and stochastic problems. This paper justifies the use of neural networks for the approximation of coarse-scale surrogate models by analyzing their approximation properties. For a prototypical and representative numerical homogenization technique, the Localized Orthogonal Decomposition method, we show that one single neural network is sufficient to approximate the coarse contributions of all occurring coefficient-dependent local sub-problems for a non-trivial class of diffusion coefficients up to arbitrary accuracy. We present rigorous upper bounds on the depth and number of non-zero parameters for such a network to achieve a given accuracy. Further, we analyze the overall error of the resulting neural network enhanced numerical homogenization surrogate model.

math.NA↗

arXiv2022

Super-localized orthogonal decomposition for convection-dominated diffusion problems

This paper presents a multi-scale method for convection-dominated diffusion problems in the regime of large Péclet numbers. The application of the solution operator to piecewise constant right-hand sides on some arbitrary coarse mesh defines a finite-dimensional coarse ansatz space with favorable approximation properties. For some relevant error measures, including the $L^2$-norm, the Galerkin projection onto this generalized finite element space even yields $\varepsilon$-independent error bounds, $\varepsilon$ being the singular perturbation parameter. By constructing an approximate local basis, the approach becomes a novel multi-scale method in the spirit of the Super-Localized Orthogonal Decomposition (SLOD). The error caused by basis localization can be estimated in an a-posteriori way. In contrast to existing multi-scale methods, numerical experiments indicate $\varepsilon$-independent convergence without preasymptotic effects even in the under-resolved regime of large mesh Péclet numbers.

math.NA↗

arXiv2022

Energy-adaptive Riemannian optimization on the Stiefel manifold

This paper addresses the numerical solution of nonlinear eigenvector problems such as the Gross-Pitaevskii and Kohn-Sham equation arising in computational physics and chemistry. These problems characterize critical points of energy minimization problems on the infinite-dimensional Stiefel manifold. To efficiently compute minimizers, we propose a novel Riemannian gradient descent method induced by an energy-adaptive metric. Quantified convergence of the methods is established under suitable assumptions on the underlying problem. A non-monotone line search and the inexact evaluation of Riemannian gradients substantially improve the overall efficiency of the method. Numerical experiments illustrate the performance of the method and demonstrates its competitiveness with well-established schemes.

math.NA↗

arXiv2022

Operator Compression with Deep Neural Networks

This paper studies the compression of partial differential operators using neural networks. We consider a family of operators, parameterized by a potentially high-dimensional space of coefficients that may vary on a large range of scales. Based on existing methods that compress such a multiscale operator to a finite-dimensional sparse surrogate model on a given target scale, we propose to directly approximate the coefficient-to-surrogate map with a neural network. We emulate local assembly structures of the surrogates and thus only require a moderately sized network that can be trained efficiently in an offline phase. This enables large compression ratios and the online computation of a surrogate based on simple forward passes through the network is substantially accelerated compared to classical numerical upscaling approaches. We apply the abstract framework to a family of prototypical second-order elliptic heterogeneous diffusion operators as a demonstrating example.

math.NA↗

arXiv2021

Super-localized Orthogonal Decomposition for high-frequency Helmholtz problems

We propose a novel variant of the Localized Orthogonal Decomposition (LOD) method for time-harmonic scattering problems of Helmholtz type with high wavenumber $κ$. On a coarse mesh of width $H$, the proposed method identifies local finite element source terms that yield rapidly decaying responses under the solution operator. They can be constructed to high accuracy from independent local snapshot solutions on patches of width $\ell H$ and are used as problem-adapted basis functions in the method. In contrast to the classical LOD and other state-of-the-art multi-scale methods, the localization error decays super-exponentially as the oversampling parameter $\ell$ is increased. This implies that optimal convergence is observed under the substantially relaxed oversampling condition $\ell \gtrsim (\log \tfracκ{H})^{(d-1)/d}$ with $d$ denoting the spatial dimension. Numerical experiments demonstrate the significantly improved offline and online performance of the method also in the case of heterogeneous media and perfectly matched layers.

math.NA↗

arXiv2021

Localization and delocalization of ground states of Bose-Einstein condensates under disorder

This paper studies the localization behaviour of Bose-Einstein condensates in disorder potentials, modeled by a Gross-Pitaevskii eigenvalue problem on a bounded interval. In the regime of weak particle interaction, we are able to quantify exponential localization of the ground state, depending on statistical parameters and the strength of the potential. Numerical studies further show delocalization if we leave the identified parameter range, which is in agreement with experimental data. These mathematical and numerical findings allow the prediction of physically relevant regimes where localization of ground states may be observed experimentally.

cond-mat.quant-gas↗

arXiv2020

The $J$-method for the Gross-Pitaevskii eigenvalue problem

This paper studies the $J$-method of [E. Jarlebring, S. Kvaal, W. Michiels. SIAM J. Sci. Comput. 36-4:A1978-A2001, 2014] for nonlinear eigenvector problems in a general Hilbert space framework. This is the basis for variational discretization techniques and a mesh-independent numerical analysis. A simple modification of the method mimics an energy-decreasing discrete gradient flow. In the case of the Gross-Pitaevskii eigenvalue problem, we prove global convergence towards an eigenfunction for a damped version of the $J$-method. More importantly, when the iterations are sufficiently close to an eigenfunction, the damping can be switched off and we recover a local linear convergence rate previously known from the discrete setting. This quantitative convergence analysis is closely connected to the~$J$-method's unique feature of sensitivity with respect to spectral shifts. Contrary to classical gradient flows, this allows both the selective approximation of excited states as well as the amplification of convergence beyond linear rates in the spirit of the Rayleigh quotient iteration for linear eigenvalue problems. These advantageous convergence properties are demonstrated in a series of numerical experiments involving exponentially localized states under disorder potentials and vortex lattices in rotating traps.

math.NA↗

arXiv2020

A priori error analysis of a numerical stochastic homogenization method

This paper provides an a~priori error analysis of a localized orthogonal decomposition method (LOD) for the numerical stochastic homogenization of a model random diffusion problem. If the uniformly elliptic and bounded random coefficient field of the model problem is stationary and satisfies a quantitative decorrelation assumption in form of the spectral gap inequality, then the expected $L^2$ error of the method can be estimated, up to logarithmic factors, by $H+(\varepsilon/H)^{d/2}$; $\varepsilon$ being the small correlation length of the random coefficient and $H$ the width of the coarse finite element mesh that determines the spatial resolution. The proof bridges recent results of numerical homogenization and quantitative stochastic homogenization.

math.NA↗

arXiv2020

Reconstruction of quasi-local numerical effective models from low-resolution measurements

We consider the inverse problem of reconstructing an effective model for a prototypical diffusion process in strongly heterogeneous media based on coarse measurements. The approach is motivated by quasi-local numerical effective forward models that are provably reliable beyond periodicity assumptions and scale separation. The goal of this work is to show that an identification of the matrix representation related to these effective models is possible. On the one hand, this provides a reasonable surrogate in cases where a direct reconstruction is unfeasible due to a mismatch between the coarse data scale and the microscopic quantities to be reconstructed. On the other hand, the approach allows us to investigate the requirement for a certain non-locality in the context of numerical homogenization. Algorithmic aspects of the inversion procedure and its performance are illustrated in a series of numerical experiments.

math.NA↗