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Author: Eigel, Martin × Subject: Partial differential equations with random coefficients ×

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Now showing 1 - 10 of 10
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    Low-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, Manuel
    This 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.
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    Bayesian inversion with a hierarchical tensor representation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Eigel, Martin; Marschall, Manuel; Schneider, Reinhold
    The statistical Bayesian approach is a natural setting to resolve the ill-posedness of inverse problems by assigning probability densities to the considered calibration parameters. Based on a parametric deterministic representation of the forward model, a sampling-free approach to Bayesian inversion with an explicit representation of the parameter densities is developed. The approximation of the involved randomness inevitably leads to several high dimensional expressions, which are often tackled with classical sampling methods such as MCMC. To speed up these methods, the use of a surrogate model is beneficial since it allows for faster evaluation with respect to calibration parameters. However, the inherently slow convergence can not be remedied by this. As an alternative, a complete functional treatment of the inverse problem is feasible as demonstrated in this work, with functional representations of the parametric forward solution as well as the probability densities of the calibration parameters, determined by Bayesian inversion. The proposed sampling-free approach is discussed in the context of hierarchical tensor representations, which are employed for the adaptive evaluation of a random PDE (the forward problem) in generalized chaos polynomials and the subsequent high-dimensional quadrature of the log-likelihood. This modern compression technique alleviates the curse of dimensionality by hierarchical subspace approximations of the involved low rank (solution) manifolds. All required computations can be carried out efficiently in the low-rank format. A priori convergence is examined, considering all approximations that occur in the method. Numerical experiments demonstrate the performance and verify the theoretical results.
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    An adaptive multi level Monte-Carlo method with stochastic bounds for quantities of interest in groundwater flow with uncertain data
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Eigel, Martin; Merdon, Christian; Neumann, Johannes
    The focus of this work is the introduction of some computable a posteriori error control to the popular multilevel Monte Carlo sampling for PDE with stochastic data. We are especially interested in applications in the geosciences such as groundwater flow with rather rough stochastic fields for the conductive permeability. With a spatial discretisation based on finite elements, a goal functional is defined which encodes the quantity of interest. The devised goal-oriented error estimator enables to determine guaranteed a posteriori error bounds for this quantity. In particular, it allows for the adaptive refinement of the mesh hierarchy used in the multilevel Monte Carlo simulation. In addition to controlling the deterministic error, we also suggest how to treat the stochastic error in probability. Numerical experiments illustrate the performance of the presented adaptive algorithm for a posteriori error control in multilevel Monte Carlo methods. These include a localised goal with problem-adapted meshes and a slit domain example. The latter demonstrates the refinement of regions with low solution regularity based on an inexpensive explicit error estimator in the multilevel algorithm.
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    An adaptive stochastic Galerkin tensor train discretization for randomly perturbed domains
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Eigel, Martin; Marschall, Manuel; Multerer, Michael
    A linear PDE problem for randomly perturbed domains is considered in an adaptive Galerkin framework. The perturbation of the domains boundary is described by a vector valued random field depending on a countable number of random variables in an affine way. The corresponding Karhunen-Loève expansion is approximated by the pivoted Cholesky decomposition based on a prescribed covariance function. The examined high-dimensional Galerkin system follows from the domain mapping approach, transferring the randomness from the domain to the diffusion coefficient and the forcing. In order to make this computationally feasible, the representation makes use of the modern tensor train format for the implicit compression of the problem. Moreover, an a posteriori error estimator is presented, which allows for the problem-dependent iterative refinement of all discretization parameters and the assessment of the achieved error reduction. The proposed approach is demonstrated in numerical benchmark problems.
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    Variational Monte Carlo - Bridging concepts of machine learning and high dimensional partial differential equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Eigel, Martin; Trunschke, Philipp; Schneider, Reinhold; Wolf, Sebastian
    A statistical learning approach for parametric PDEs related to Uncertainty Quantification is derived. The method is based on the minimization of an empirical risk on a selected model class and it is shown to be applicable to a broad range of problems. A general unified convergence analysis is derived, which takes into account the approximation and the statistical errors. By this, a combination of theoretical results from numerical analysis and statistics is obtained. Numerical experiments illustrate the performance of the method with the model class of hierarchical tensors.
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    Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) 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.
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    Stochastic topology optimisation with hierarchical tensor reconstruction
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Eigel, Martin; Neumann, Johannes; Schneider, Reinhold; Wolf, Sebastian
    A novel approach for risk-averse structural topology optimization under uncertainties is presented which takes into account random material properties and random forces. For the distribution of material, a phase field approach is employed which allows for arbitrary topological changes during optimization. The state equation is assumed to be a high-dimensional PDE parametrized in a (finite) set of random variables. For the examined case, linearized elasticity with a parametric elasticity tensor is used. Instead of an optimization with respect to the expectation of the involved random fields, for practical purposes it is important to design structures which are also robust in case of events that are not the most frequent. As a common risk-aware measure, the Conditional Value at Risk (CVaR) is used in the cost functional during the minimization procedure. Since the treatment of such high-dimensional problems is a numerically challenging task, a representation in the modern hierarchical tensor train format is proposed. In order to obtain this highly efficient representation of the solution of the random state equation, a tensor completion algorithm is employed which only required the pointwise evaluation of solution realizations. The new method is illustrated with numerical examples and compared with a classical Monte Carlo sampling approach.
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    Local equilibration error estimators for guaranteed error control in adaptive stochastic higher-order Galerkin FEM
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Eigel, Martin; Merdon, Christian
    Equilibration error estimators have been shown to commonly lead to very accurate guaranteed error bounds in the a posteriori error control of finite element methods for second order elliptic equations. Here, we extend previous results by the design of equilibrated fluxes for higher-order finite element methods with nonconstant coefficients and illustrate the favourable performance of different variants of the error estimator within two deterministic benchmark settings. After the introduction of the respective parametric problem with stochastic coefficients and the stochastic Galerkin FEM discretisation, a novel a posteriori error estimator for the stochastic error in the energy norm is devised. The error estimation is based on the stochastic residual and its decomposition into approximation residuals and a truncation error of the stochastic discretisation. Importantly, by using the derived deterministic equilibration techniques for the approximation residuals, the computable error bound is guaranteed for the considered class of problems. An adaptive algorithm allows the simultaneous refinement of the deterministic mesh and the stochastic discretisation in anisotropic Legendre polynomial chaos. Several stochastic benchmark problems illustrate the efficiency of the adaptive process.
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    Adaptive stochastic Galerkin FEM with hierarchical tensor representations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Eigel, Martin; Pfeffer, Max; Schneider, Reinhold
    The solution of PDE with stochastic data commonly leads to very high-dimensional algebraic problems, e.g. when multiplicative noise is present. The Stochastic Galerkin FEM considered in this paper then suffers from the curse of dimensionality. This is directly related to the number of random variables required for an adequate representation of the random fields included in the PDE. With the presented new approach, we circumvent this major complexity obstacle by combining two highly efficient model reduction strategies, namely a modern low-rank tensor representation in the tensor train format of the problem and a refinement algorithm on the basis of a posteriori error estimates to adaptively adjust the different employed discretizations. The adaptive adjustment includes the refinement of the FE mesh based on a residual estimator, the problem-adapted stochastic discretization in anisotropic Legendre Wiener chaos and the successive increase of the tensor rank. Computable a posteriori error estimators are derived for all error terms emanating from the discretizations and the iterative solution with a preconditioned ALS scheme of the problem. Strikingly, it is possible to exploit the tensor structure of the problem to evaluate all error terms very efficiently. A set of benchmark problems illustrates the performance of the adaptive algorithm with higher-order FE. Moreover, the influence of the tensor rank on the approximation quality is investigated.
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    SDE based regression for random PDEs
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Anker, Felix; Bayer, Christian; Eigel, Martin; Ladkau, Marcel; Neumann, Johannes; Schoenmakers, John G.M.
    A simulation based method for the numerical solution of PDE with random coefficients is presented. By the Feynman-Kac formula, the solution can be represented as conditional expectation of a functional of a corresponding stochastic differential equation driven by independent noise. A time discretization of the SDE for a set of points in the domain and a subsequent Monte Carlo regression lead to an approximation of the global solution of the random PDE. We provide an initial error and complexity analysis of the proposed method along with numerical examples illustrating its behaviour.