Abstract. We consider the problem of estimating the uncertainty in large-scale linear statistical inverse problems with high-dimensional parameter spaces within the framework of Bayesian inference. When the noise and prior probability densities are Gaussian, the solution to the inverse problem is also Gaussian, and is thus characterized by the mean and covariance matrix of the posterior probability density. Unfortunately, explicitly computing the posterior covariance matrix requires as many forward solutions as there are parameters, and is thus prohibitive when the forward problem is expensive and the parameter dimension is large. However, for many ill-posed inverse problems, the Hessian matrix of the data misfit term has a spectrum that collapses rapidly to zero. We present a fast method for computation of an approximation to the posterior covariance that exploits the lowrank structure of the preconditioned (by the prior covariance) Hessian of the data misfit. Analysis of an infinite-dimensional model convection-diffusion problem, and numerical experiments on largescale 3D convection-diffusion inverse problems with up to 1.5 million parameters, demonstrate that the number of forward PDE solves required for an accurate low-rank approximation is independent of the problem dimension. This permits scalable estimation of the uncertainty in large-scale ill-posed linear inverse problems at a small multiple (independent of the problem dimension) of the cost of solving the forward problem.

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Abstract. Optimal design, optimal control, and parameter estimation of systems governed by partial differential equations (PDE) give rise to a class of problems known as PDE-constrained optimization. The size and complexity of the discretized PDEs often pose significant challenges for contemporary optimization methods. Recent advances in algorithms, software, and high performance computing systems have resulted in PDE simulations that can often scale to millions of variables, thousands of processors, and multiple physics interactions. As PDE solvers mature, there is increasing interest in industry and the national labs in solving optimization problems governed by such large-scale simulations. This article provides a brief introduction and overview to the Lecture Notes in Computational Science and Engineering volume entitled Large-Scale PDE-Constrained Optimization.This volume contains nineteen articles that were initially presented at the First Sandia Workshop on Large-Scale PDE-Constrained Optimization. The articles in this volume assess the state-of-the-art in PDE-constrained optimization, identify challenges to optimization presented by modern highly parallel PDE simulation codes and discuss promising algorithmic and software approaches to address them. These contributions represent current research of two strong scientific computing communities, in optimization and PDE simulation. This volume merges perspectives in these two different areas and identifies interesting open questions for further research. We hope that this volume leads to greater synergy and collaboration between these communities. Algorithmic challenges for PDE-constrained optimizationPDE simulation is widespread in science and engineering applications. Moreover, with increasing development and application of supercomputing hardware and advances in numerical methods, very large-scale and detailed simulations can now be considered. An essential sequel to simulation is its application in design, control, data assimilation, and inversion. Most of these tasks are naturally stated as continuous variable optimization problems (i.e.,t Sandia is a multi program laboratory operated by Sandia Corporation, a

Surrogate and reduced-order modeling: a comparison of approaches for large-scale statistical inverse problems [Chapter 7]The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. CitationFrangos, M., Y. Marzouk, K. Willcox, and B. van Bloemen Waanders (2010). Surrogate and reduced-order modeling: a comparison of approaches for large-scale statistical inverse problems.