Sparse Deterministic Approximation of Bayesian Inverse Problems

We present a parametric deterministic formulation of Bayesian inverse problems with input parameter from infinite dimensional, separable Banach spaces. In this formulation, the forward problems are parametric, deterministic elliptic partial differential equations, and the inverse problem is to determine the unknown, parametric deterministic coefficients from noisy observations comprising linear functionals of the solution. We prove a generalized polynomial chaos representation of the posterior density with respect to the prior measure, given noisy observational data. We analyze the sparsity of the posterior density in terms of the summability of the input data's coefficient sequence. To this end, we estimate the fluctuations in the prior. We exhibit sufficient conditions on the prior model in order for approximations of the posterior density to converge at a given algebraic rate, in terms of the number $N$ of unknowns appearing in the parameteric representation of the prior measure. Similar sparsity and approximation results are also exhibited for the solution and covariance of the elliptic partial differential equation under the posterior. These results then form the basis for efficient uncertainty quantification, in the presence of data with noise

Exact shock solution of a coupled system of delay differential equations: a car-following model

In this paper, we present exact shock solutions of a coupled system of delay differential equations, which was introduced as a traffic-flow model called {\it the car-following model}. We use the Hirota method, originally developed in order to solve soliton equations. %While, with a periodic boundary condition, this system has % a traveling-wave solution given by elliptic functions. The relevant delay differential equations have been known to allow exact solutions expressed by elliptic functions with a periodic boundary conditions. In the present work, however, shock solutions are obtained with open boundary, representing the stationary propagation of a traffic jam.Comment: 6 pages, 2 figure

Low rank surrogates for polymorphic fields with application to fuzzy-stochastic partial differential equations

We consider a general form of fuzzy-stochastic PDEs depending on the interaction of probabilistic and non-probabilistic ("possibilistic") influences. Such a combined modelling of aleatoric and epistemic uncertainties for instance can be applied beneficially in an engineering context for real-world applications, where probabilistic modelling and expert knowledge has to be accounted for. We examine existence and well-definedness of polymorphic PDEs in appropriate function spaces. The fuzzy-stochastic dependence is described in a high-dimensional parameter space, thus easily leading to an exponential complexity in practical computations. To aleviate this severe obstacle in practise, a compressed low-rank approximation of the problem formulation and the solution is derived. This is based on the Hierarchical Tucker format which is constructed with solution samples by a non-intrusive tensor reconstruction algorithm. The performance of the proposed model order reduction approach is demonstrated with two examples. One of these is the ubiquitous groundwater flow model with Karhunen-Loeve coefficient field which is generalized by a fuzzy correlation length

Low rank surrogates for polymorphic fields with application to fuzzy-stochastic partial differential equations

We consider a general form of fuzzy-stochastic PDEs depending on the interaction of probabilistic and non-probabilistic ("possibilistic") influences. Such a combined modelling of aleatoric and epistemic uncertainties for instance can be applied beneficially in an engineering context for real-world applications, where probabilistic modelling and expert knowledge has to be accounted for. We examine existence and well-definedness of polymorphic PDEs in appropriate function spaces. The fuzzy-stochastic dependence is described in a high-dimensional parameter space, thus easily leading to an exponential complexity in practical computations. To aleviate this severe obstacle in practise, a compressed low-rank approximation of the problem formulation and the solution is derived. This is based on the Hierarchical Tucker format which is constructed with solution samples by a non-intrusive tensor reconstruction algorithm. The performance of the proposed model order reduction approach is demonstrated with two examples. One of these is the ubiquitous groundwater flow model with Karhunen-Loeve coefficient field which is generalized by a fuzzy correlation length

Computational methods for estimation of parameters in hyperbolic systems

Approximation techniques for estimating spatially varying coefficients and unknown boundary parameters in second order hyperbolic systems are discussed. Methods for state approximation (cubic splines, tau-Legendre) and approximation of function space parameters (interpolatory splines) are outlined and numerical findings for use of the resulting schemes in model "one dimensional seismic inversion' problems are summarized

Energy inequalities and dispersive estimates for wave equations with time-dependent coefficients

We consider wave models with lower order terms and recollect some recent results on energy and dispersive estimates for their solution based on symbolic type estimates for coefficients and partly stabilisation conditions. The exposition is complemented by a collection of open problems.Comment: 11 page