Goal-Oriented Optimal Approximations of Bayesian Linear Inverse Problems

Goal-oriented Optimal Approximations of Bayesian

We propose optimal dimensionality reduction techniques for the solution of goal–oriented linear–Gaussian inverse problems, where the quantity of interest (QoI) is a function of the inversion parameters. These approximations are suitable for large-scale applications. In particular, we study the approximation of the posterior covariance of the QoI as a low-rank negative update of its prior covari...

Goal-Oriented Optimal Design of Experiments for Large-Scale Bayesian Linear Inverse Problems

We develop a framework for goal oriented optimal design of experiments (GOODE) for largescale Bayesian linear inverse problems governed by PDEs. This framework differs from classical Bayesian optimal design of experiments (ODE) in the following sense: we seek experimental designs that minimize the posterior uncertainty in a predicted quantity of interest (QoI) rather than the estimated paramete...

Goal-oriented error estimation for fast approximations of nonlinear problems

Ill-Posed and Linear Inverse Problems

In this paper ill-posed linear inverse problems that arises in many applications is considered. The instability of special kind of these problems and it's relation to the kernel, is described. For finding a stable solution to these problems we need some kind of regularization that is presented. The results have been applied for a singular equation.

A-Optimal Design of Experiments for Infinite-Dimensional Bayesian Linear Inverse Problems with Regularized ℓ0-Sparsification

We present an efficient method for computing A-optimal experimental designs for infinite-dimensional Bayesian linear inverse problems governed by partial differential equations (PDEs). Specifically, we address the problem of optimizing the location of sensors (at which observational data are collected) to minimize the uncertainty in the parameters estimated by solving the inverse problem, where...