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- ItemLocal surrogate responses in the Schwarz alternating method for elastic problems on random voided domains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Drieschner, Martin; Gruhlke, Robert; Petryna, Yuri; Eigel, Martin; Hömberg, DietmarImperfections and inaccuracies in real technical products often influence the mechanical behavior and the overall structural reliability. The prediction of real stress states and possibly resulting failure mechanisms is essential and a real challenge, e.g. in the design process. In this contribution, imperfections in elastic materials such as air voids in adhesive bonds between fiber-reinforced composites are investigated. They are modeled as arbitrarily shaped and positioned. The focus is on local displacement values as well as on associated stress concentrations caused by the imperfections. For this purpose, the resulting complex random one-scale finite element model is numerically solved by a new developed surrogate model using an overlapping domain decomposition scheme based on Schwarz alternating method. Here, the actual response of local subproblems associated with isolated material imperfections is determined by a single appropriate surrogate model, that allows for an accelerated propagation of randomness. The efficiency of the method is demonstrated for imperfections with elliptical and ellipsoidal shape in 2D and 3D and extended to arbitrarily shaped voids. For the latter one, a local surrogate model based on artificial neural networks (ANN) is constructed. Finally, a comparison to experimental results validates the numerical predictions for a real engineering problem.
- ItemOn the convergence of adaptive stochastic collocation for elliptic partial differential equations with affine diffusion(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Eigel, Martin; Ernst, Oliver; Sprungk, Björn; Tamellini, LorenzoConvergence of an adaptive collocation method for the stationary parametric diffusion equation with finite-dimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residual-based reliable a posteriori error estimator. For the convergence proof, a strategy recently used for a stochastic Galerkin method with an hierarchical error estimator is transferred to the collocation setting.
- ItemEfficient approximation of high-dimensional exponentials by tensor networks(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Eigel, Martin; Farchmin, Nando; Heidenreich, Sebastian; Trunschke, PhilippIn this work a general approach to compute a compressed representation of the exponential exp(h) of a high-dimensional function h is presented. Such exponential functions play an important role in several problems in Uncertainty Quantification, e.g. the approximation of log-normal random fields or the evaluation of Bayesian posterior measures. Usually, these high-dimensional objects are intractable numerically and can only be accessed pointwise in sampling methods. In contrast, the proposed method constructs a functional representation of the exponential by exploiting its nature as a solution of an ordinary differential equation. The application of a Petrov--Galerkin scheme to this equation provides a tensor train representation of the solution for which we derive an efficient and reliable a posteriori error estimator. Numerical experiments with a log-normal random field and a Bayesian likelihood illustrate the performance of the approach in comparison to other recent low-rank representations for the respective applications. Although the present work considers only a specific differential equation, the presented method can be applied in a more general setting. We show that the composition of a generic holonomic function and a high-dimensional function corresponds to a differential equation that can be used in our method. Moreover, the differential equation can be modified to adapt the norm in the a posteriori error estimates to the problem at hand.
- ItemConvergence bounds for empirical nonlinear least-squares(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Eigel, Martin; Trunschke, Philipp; Schneider, ReinholdWe consider best approximation problems in a nonlinear subset of a Banach space of functions. The norm is assumed to be a generalization of the L2 norm for which only a weighted Monte Carlo estimate can be computed. The objective is to obtain an approximation of an unknown target function by minimizing the empirical norm. In the case of linear subspaces it is well-known that such least squares approximations can become inaccurate and unstable when the number of samples is too close to the number of parameters. We review this statement for general nonlinear subsets and establish error bounds for the empirical best approximation error. Our results are based on a restricted isometry property (RIP) which holds in probability and we show sufficient conditions for the RIP to be satisfied with high probability. Several model classes are examined where analytical statements can be made about the RIP. Numerical experiments illustrate some of the obtained stability bounds.
- ItemAdaptive non-intrusive reconstruction of solutions to high-dimensional parametric PDEs(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Eigel, Martin; Farchmin, Nando; Heidenreich, Sebastian; Trunschke, PhilippNumerical methods for random parametric PDEs can greatly benefit from adaptive refinement schemes, in particular when functional approximations are computed as in stochastic Galerkin and stochastic collocations methods. This work is concerned with a non-intrusive generalization of the adaptive Galerkin FEM with residual based error estimation. It combines the non-intrusive character of a randomized least-squares method with the a posteriori error analysis of stochastic Galerkin methods. The proposed approach uses the Variational Monte Carlo method to obtain a quasi-optimal low-rank approximation of the Galerkin projection in a highly efficient hierarchical tensor format. We derive an adaptive refinement algorithm which is steered by a reliable error estimator. Opposite to stochastic Galerkin methods, the approach is easily applicable to a wide range of problems, enabling a fully automated adjustment of all discretization parameters. Benchmark examples with affine and (unbounded) lognormal coefficient fields illustrate the performance of the non-intrusive adaptive algorithm, showing best-in-class performance
- ItemNumerical upscaling of parametric microstructures in a possibilistic uncertainty framework with tensor trains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Eigel, Martin; Gruhlke, Robert; Moser, DieterWe develop a new fuzzy arithmetic framework for efficient possibilistic uncertainty quantification. The considered application is an edge detection task with the goal to identify interfaces of blurred images. In our case, these represent realisations of composite materials with possibly very many inclusions. The proposed algorithm can be seen as computational homogenisation and results in a parameter dependent representation of composite structures. For this, many samples for a linear elasticity problem have to be computed, which is significantly sped up by a highly accurate low-rank tensor surrogate. To ensure the continuity of the underlying effective material tensor map, an appropriate diffeomorphism is constructed to generate a family of meshes reflecting the possible material realisations. In the application, the uncertainty model is propagated through distance maps with respect to consecutive symmetry class tensors. Additionally, the efficacy of the best/worst estimate analysis of the homogenisation map as a bound to the average displacement for chessboard like matrix composites with arbitrary star-shaped inclusions is demonstrated.
- ItemDynamical low-rank approximations of solutions to the Hamilton--Jacobi--Bellman equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Eigel, Martin; Schneider, Reinhold; Sommer, DavidWe present a novel method to approximate optimal feedback laws for nonlinar optimal control basedon low-rank tensor train (TT) decompositions. The approach is based on the Dirac-Frenkel variationalprinciple with the modification that the optimisation uses an empirical risk. Compared to currentstate-of-the-art TT methods, our approach exhibits a greatly reduced computational burden whileachieving comparable results. A rigorous description of the numerical scheme and demonstrations ofits performance are provided.
- ItemLow-rank tensor reconstruction of concentrated densities with application to Bayesian inversion(Dordrecht [u.a.] : Springer Science + Business Media B.V, 2022) Eigel, Martin; Gruhlke, Robert; Marschall, ManuelThis paper presents a novel method for the accurate functional approximation of possibly highly concentrated probability densities. It is based on the combination of several modern techniques such as transport maps and low-rank approximations via a nonintrusive tensor train reconstruction. The central idea is to carry out computations for statistical quantities of interest such as moments based on a convenient representation of a reference density for which accurate numerical methods can be employed. Since the transport from target to reference can usually not be determined exactly, one has to cope with a perturbed reference density due to a numerically approximated transport map. By the introduction of a layered approximation and appropriate coordinate transformations, the problem is split into a set of independent approximations in seperately chosen orthonormal basis functions, combining the notions h- and p-refinement (i.e. “mesh size” and polynomial degree). An efficient low-rank representation of the perturbed reference density is achieved via the Variational Monte Carlo method. This nonintrusive regression technique reconstructs the map in the tensor train format. An a priori convergence analysis with respect to the error terms introduced by the different (deterministic and statistical) approximations in the Hellinger distance and the Kullback–Leibler divergence is derived. Important applications are presented and in particular the context of Bayesian inverse problems is illuminated which is a main motivation for the developed approach. Several numerical examples illustrate the efficacy with densities of different complexity and degrees of perturbation of the transport to the reference density. The (superior) convergence is demonstrated in comparison to Monte Carlo and Markov Chain Monte Carlo methods.
- ItemLow-rank Wasserstein polynomial chaos expansions in the framework of optimal transport(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Gruhlke, Robert; Eigel, MartinA unsupervised learning approach for the computation of an explicit functional representation of a random vector Y is presented, which only relies on a finite set of samples with unknown distribution. Motivated by recent advances with computational optimal transport for estimating Wasserstein distances, we develop a new Wasserstein multi-element polynomial chaos expansion (WPCE). It relies on the minimization of a regularized empirical Wasserstein metric known as debiased Sinkhorn divergence. As a requirement for an efficient polynomial basis expansion, a suitable (minimal) stochastic coordinate system X has to be determined with the aim to identify ideally independent random variables. This approach generalizes representations through diffeomorphic transport maps to the case of non-continuous and non-injective model classes M with different input and output dimension, yielding the relation Y=M(X) in distribution. Moreover, since the used PCE grows exponentially in the number of random coordinates of X, we introduce an appropriate low-rank format given as stacks of tensor trains, which alleviates the curse of dimensionality, leading to only linear dependence on the input dimension. By the choice of the model class M and the smooth loss function, higher order optimization schemes become possible. It is shown that the relaxation to a discontinuous model class is necessary to explain multimodal distributions. Moreover, the proposed framework is applied to a numerical upscaling task, considering a computationally challenging microscopic random non-periodic composite material. This leads to tractable effective macroscopic random field in adopted stochastic coordinates.
- ItemPricing high-dimensional Bermudan options with hierarchical tensor formats(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Bayer, Christian; Eigel, Martin; Sallandt, Leon; Trunschke, PhilippAn efficient compression technique based on hierarchical tensors for popular option pricing methods is presented. It is shown that the ``curse of dimensionality" can be alleviated for the computation of Bermudan option prices with the Monte Carlo least-squares approach as well as the dual martingale method, both using high-dimensional tensorized polynomial expansions. This discretization allows for a simple and computationally cheap evaluation of conditional expectations. Complexity estimates are provided as well as a description of the optimization procedures in the tensor train format. Numerical experiments illustrate the favourable accuracy of the proposed methods. The dynamical programming method yields results comparable to recent Neural Network based methods.