Higher order Quasi-Monte Carlo integration for Bayesian Estimation

We analyze combined Quasi-Monte Carlo quadrature and Finite Element approximations in Bayesian estimation of solutions to countably-parametric operator equations with holomorphic dependence on the parameters as considered in [Cl.~Schillings and Ch.~Schwab: Sparsity in Bayesian Inversion of Parametric Operator Equations. Inverse Problems, {\bf 30}, (2014)]. Such problems arise in numerical uncertainty quantification and in Bayesian inversion of operator equations with distributed uncertain inputs, such as uncertain coefficients, uncertain domains or uncertain source terms and boundary data. We show that the parametric Bayesian posterior densities belong to a class of weighted Bochner spaces of functions of countably many variables, with a particular structure of the QMC quadrature weights: up to a (problem-dependent, and possibly large) finite dimension $S$ product weights can be used, and beyond this dimension, weighted spaces with so-called SPOD weights are used to describe the solution regularity. We establish error bounds for higher order Quasi-Monte Carlo quadrature for the Bayesian estimation based on [J.~Dick, Q.T.~LeGia and Ch.~Schwab, Higher order Quasi-Monte Carlo integration for holomorphic, parametric operator equations, Report 2014-23, SAM, ETH Z\"urich]. It implies, in particular, regularity of the parametric solution and of the countably-parametric Bayesian posterior density in SPOD weighted spaces. This, in turn, implies that the Quasi-Monte Carlo quadrature methods in [J. Dick, F.Y.~Kuo, Q.T.~Le Gia, D.~Nuyens, Ch.~Schwab, Higher order QMC Galerkin discretization for parametric operator equations, SINUM (2014)] are applicable to these problem classes, with dimension-independent convergence rates \calO(N^{-1/p}) of $N$-point HoQMC approximated Bayesian estimates, where $0<p<1$ depends only on the sparsity class of the uncertain input in the Bayesian estimation.Comment: arXiv admin note: text overlap with arXiv:1409.218

Primal-Dual Reduced Basis Methods for Convex Minimization Variational Problems: Robust True Solution A Posteriori Error Certification and Adaptive Greedy Algorithms

In this paper, with the parametric symmetric coercive elliptic boundary value problem as an example of the primal-dual variational problems satisfying the strong duality, we develop primal-dual reduced basis methods (PD-RBM) with robust true error certifications and discuss three versions of greedy algorithms to balance the finite element error, the exact reduced basis error, and the adaptive mesh refinements. For a class of convex minimization variational problems which has corresponding dual problems satisfying the strong duality, the primal-dual gap between the primal and dual functionals can be used as a posteriori error estimator. This primal-dual gap error estimator is robust with respect to the parameters of the problem, and it can be used for both mesh refinements of finite element methods and the true RB error certification. With the help of integrations by parts formula, the primal-dual variational theory is developed for the symmetric coercive elliptic boundary value problems with non-homogeneous boundary conditions by both the conjugate function and Lagrangian theories. A generalized Prager-Synge identity, which is the primal-dual gap error representation for this specific problem, is developed. RBMs for both the primal and dual problems with robust error estimates are developed. The dual variational problem often can be viewed as a constraint optimization problem. In the paper, different from the standard saddle-point finite element approximation, the dual RBM is treated as a Galerkin projection by constructing RB spaces satisfying the homogeneous constraint. Inspired by the greedy algorithm with spatio-parameter adaptivity of \cite{Yano:18}, adaptive balanced greedy algorithms with primal-dual finite element and reduced basis error estimators are discussed. Numerical tests are presented to test the PD-RBM with adaptive balanced greedy algorithms.Comment: 40 pages. The paper is accepted by SIAM Journal on Scientific Computin

Robust a posteriori error estimation for stochastic Galerkin formulations of parameter-dependent linear elasticity equations

The focus of this work is a posteriori error estimation for stochastic Galerkin approximations of parameter-dependent linear elasticity equations. The starting point is a three-field PDE model in which the Young's modulus is an affine function of a countable set of parameters. We analyse the weak formulation, its stability with respect to a weighted norm and discuss approximation using stochastic Galerkin mixed finite element methods (SG-MFEMs). We introduce a novel a posteriori error estimation scheme and establish upper and lower bounds for the SG-MFEM error. The constants in the bounds are independent of the Poisson ratio as well as the SG-MFEM discretisation parameters. In addition, we discuss proxies for the error reduction associated with certain enrichments of the SG-MFEM spaces and we use these to develop an adaptive algorithm that terminates when the estimated error falls below a user-prescribed tolerance. We prove that both the a posteriori error estimate and the error reduction proxies are reliable and efficient in the incompressible limit case. Numerical results are presented to validate the theory. All experiments were performed using open source (IFISS) software that is available online.Comment: 23 pages, 5 figure

Optimal Experimental Design for Infinite-dimensional Bayesian Inverse Problems Governed by PDEs: A Review

We present a review of methods for optimal experimental design (OED) for Bayesian inverse problems governed by partial differential equations with infinite-dimensional parameters. The focus is on problems where one seeks to optimize the placement of measurement points, at which data are collected, such that the uncertainty in the estimated parameters is minimized. We present the mathematical foundations of OED in this context and survey the computational methods for the class of OED problems under study. We also outline some directions for future research in this area.Comment: 37 pages; minor revisions; added more references; article accepted for publication in Inverse Problem

A posteriori error estimation in a finite element method for reconstruction of dielectric permittivity

We present a posteriori error estimates for finite element approximations in a minimization approach to a coefficient inverse problem. The problem is that of reconstructing the dielectric permittivity $\varepsilon = \varepsilon(\mathbf{x})$, $\mathbf{x}\in\Omega\subset\mathbb{R}^3$, from boundary measurements of the electric field. The electric field is related to the permittivity via Maxwell's equations. The reconstruction procedure is based on minimization of a Tikhonov functional where the permittivity, the electric field and a Lagrangian multiplier function are approximated by peicewise polynomials. Our main result is an estimate for the difference between the computed coefficient $\varepsilon_h$ and the true minimizer $\varepsilon$, in terms of the computed functions.Comment: 17 pages, 1 figur

Nitsche-XFEM for optimal control problems governed by elliptic PDEs with interfaces

For the optimal control problem governed by elliptic equations with interfaces, we present a numerical method based on the Hansbo's Nitsche-XFEM. We followed the Hinze's variational discretization concept to discretize the continuous problem on a uniform mesh. We derive optimal error estimates of the state, co-state and control both in mesh dependent norm and L2 norm. In addition, our method is suitable for the model with non-homogeneous interface condition. Numerical results confirmed our theoretical results, with the implementation details discussed

Adaptive Estimation for Nonlinear Systems using Reproducing Kernel Hilbert Spaces

This paper extends a conventional, general framework for online adaptive estimation problems for systems governed by unknown nonlinear ordinary differential equations. The central feature of the theory introduced in this paper represents the unknown function as a member of a reproducing kernel Hilbert space (RKHS) and defines a distributed parameter system (DPS) that governs state estimates and estimates of the unknown function. This paper 1) derives sufficient conditions for the existence and stability of the infinite dimensional online estimation problem, 2) derives existence and stability of finite dimensional approximations of the infinite dimensional approximations, and 3) determines sufficient conditions for the convergence of finite dimensional approximations to the infinite dimensional online estimates. A new condition for persistency of excitation in a RKHS in terms of its evaluation functionals is introduced in the paper that enables proof of convergence of the finite dimensional approximations of the unknown function in the RKHS. This paper studies two particular choices of the RKHS, those that are generated by exponential functions and those that are generated by multiscale kernels defined from a multiresolution analysis.Comment: 24 pages, Submitted to CMAM

Sparse approximate inverses of Gramians and impulse response matrices of large-scale interconnected systems

In this paper we show that inverses of well-conditioned, finite-time Gramians and impulse response matrices of large-scale interconnected systems described by sparse state-space models, can be approximated by sparse matrices. The approximation methodology established in this paper opens the door to the development of novel methods for distributed estimation, identification and control of large-scale interconnected systems. The novel estimators (controllers) compute local estimates (control actions) simply as linear combinations of inputs and outputs (states) of local subsystems. The size of these local data sets essentially depends on the condition number of the finite-time observability (controllability) Gramian. Furthermore, the developed theory shows that the sparsity patterns of the system matrices of the distributed estimators (controllers) are primarily determined by the sparsity patterns of state-space matrices of large-scale systems. The computational and memory complexity of the approximation algorithms are $O(N)$, where $N$ is the number of local subsystems of the interconnected system. Consequently, the proposed approximation methodology is computationally feasible for interconnected systems with an extremely large number of local subsystems.Comment: 11 pages, 4 figures. Submitted to IEEE Transactions on Automatic Contro

Spatial Mat\'ern fields driven by non-Gaussian noise

The article studies non-Gaussian extensions of a recently discovered link between certain Gaussian random fields, expressed as solutions to stochastic partial differential equations (SPDEs), and Gaussian Markov random fields. The focus is on non-Gaussian random fields with Mat\'ern covariance functions, and in particular we show how the SPDE formulation of a Laplace moving average model can be used to obtain an efficient simulation method as well as an accurate parameter estimation technique for the model. This should be seen as a demonstration of how these techniques can be used, and generalizations to more general SPDEs are readily available

pyMOR - Generic Algorithms and Interfaces for Model Order Reduction

Reduced basis methods are projection-based model order reduction techniques for reducing the computational complexity of solving parametrized partial differential equation problems. In this work we discuss the design of pyMOR, a freely available software library of model order reduction algorithms, in particular reduced basis methods, implemented with the Python programming language. As its main design feature, all reduction algorithms in pyMOR are implemented generically via operations on well-defined vector array, operator and discretization interface classes. This allows for an easy integration with existing open-source high-performance partial differential equation solvers without adding any model reduction specific code to these solvers. Besides an in-depth discussion of pyMOR's design philosophy and architecture, we present several benchmark results and numerical examples showing the feasibility of our approach