Search Results
A local hybrid surrogate-based finite element tearing interconnecting dual-primal method for nonsmooth random partial differential equations
2021, Eigel, Martin, Gruhlke, Robert
A domain decomposition approach for high-dimensional random partial differential equations exploiting the localization of random parameters is presented. To obtain high efficiency, surrogate models in multielement representations in the parameter space are constructed locally when possible. The method makes use of a stochastic Galerkin finite element tearing interconnecting dual-primal formulation of the underlying problem with localized representations of involved input random fields. Each local parameter space associated to a subdomain is explored by a subdivision into regions where either the parametric surrogate accuracy can be trusted or where instead one has to resort to Monte Carlo. A heuristic adaptive algorithm carries out a problem-dependent hp-refinement in a stochastic multielement sense, anisotropically enlarging the trusted surrogate region as far as possible. This results in an efficient global parameter to solution sampling scheme making use of local parametric smoothness exploration for the surrogate construction. Adequately structured problems for this scheme occur naturally when uncertainties are defined on subdomains, for example, in a multiphysics setting, or when the Karhunen–Loève expansion of a random field can be localized. The efficiency of the proposed hybrid technique is assessed with numerical benchmark problems illustrating the identification of trusted (possibly higher order) surrogate regions and nontrusted sampling regions. © 2020 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.
Dynamical low-rank approximations of solutions to the Hamilton--Jacobi--Bellman equation
2021, Eigel, Martin, Schneider, Reinhold, Sommer, David
We 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.
Adaptive non-intrusive reconstruction of solutions to high-dimensional parametric PDEs
2021, Eigel, Martin, Farchmin, Nando, Heidenreich, Sebastian, Trunschke, Philipp
Numerical 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
Convergence bounds for empirical nonlinear least-squares
2020, Eigel, Martin, Trunschke, Philipp, Schneider, Reinhold
We 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.
Convergence bounds for empirical nonlinear least-squares
2022, Eigel, Martin, Schneider, Reinhold, Trunschke, Philipp
We consider best approximation problems in a nonlinear subset ℳ of a Banach space of functions (𝒱,∥•∥). The norm is assumed to be a generalization of the L 2-norm for which only a weighted Monte Carlo estimate ∥•∥n can be computed. The objective is to obtain an approximation v ∈ ℳ of an unknown function u ∈ 𝒱 by minimizing the empirical norm ∥u − v∥n. We consider this problem 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 is independent of the specified nonlinear least squares setting. Several model classes are examined and the analytical statements about the RIP are compared to existing sample complexity bounds from the literature. We find that for well-studied model classes our general bound is weaker but exhibits many of the same properties as these specialized bounds. Notably, we demonstrate the advantage of an optimal sampling density (as known for linear spaces) for sets of functions with sparse representations.
Local surrogate responses in the Schwarz alternating method for elastic problems on random voided domains
2022, Drieschner, Martin, Gruhlke, Robert, Petryna, Yuri, Eigel, Martin, Hömberg, Dietmar
Imperfections 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.
Low-rank Wasserstein polynomial chaos expansions in the framework of optimal transport
2022, Gruhlke, Robert, Eigel, Martin
A 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.
Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations
2020, Eigel, Martin, Marschall, Manuel, Pfeffer, Max, Schneider, Reinhold
Stochastic Galerkin methods for non-affine coefficient representations are known to cause major difficulties from theoretical and numerical points of view. In this work, an adaptive Galerkin FE method for linear parametric PDEs with lognormal coefficients discretized in Hermite chaos polynomials is derived. It employs problem-adapted function spaces to ensure solvability of the variational formulation. The inherently high computational complexity of the parametric operator is made tractable by using hierarchical tensor representations. For this, a new tensor train format of the lognormal coefficient is derived and verified numerically. The central novelty is the derivation of a reliable residual-based a posteriori error estimator. This can be regarded as a unique feature of stochastic Galerkin methods. It allows for an adaptive algorithm to steer the refinements of the physical mesh and the anisotropic Wiener chaos polynomial degrees. For the evaluation of the error estimator to become feasible, a numerically efficient tensor format discretization is developed. Benchmark examples with unbounded lognormal coefficient fields illustrate the performance of the proposed Galerkin discretization and the fully adaptive algorithm.
Efficient approximation of high-dimensional exponentials by tensor networks
2021, Eigel, Martin, Farchmin, Nando, Heidenreich, Sebastian, Trunschke, Philipp
In 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.
Pricing high-dimensional Bermudan options with hierarchical tensor formats
2021, Bayer, Christian, Eigel, Martin, Sallandt, Leon, Trunschke, Philipp
An 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.