Nonlinear reaction-diffusion equations are an important and interesting class of problems. However, we almost never know the solution of a nonlinear reaction-diffusion model and consequently numerical computation is a principal tool for analysis. But the same features that make analysis difficult also make numerics difficult. To be on a firm scientific footing, a numerical analysis of a differential equation must be accompanied by a discussion of the error in the results. In this paper, we are interested in obtaining accurate estimates of the error of a numerical solution, as opposed to inaccurate analytic upper bounds. In the first part of the paper, we describe a computational method for estimating error. The remainder of the paper is devoted to analyzing the method and applying it to some well-known models. After spending a few years at Stanford, Mats now has a Postdoc with Claes Johnson at Chalmers. Roy is at the Center for Advanced Computing Research at Caltech.

No abstract

No abstract

We consider the inverse sensitivity analysis problem of quantifying the uncertainty of inputs to a deterministic map given specified uncertainty in a linear functional of the output of the map. This is a version of the model calibration or parameter estimation problem for a deterministic map. We assume that the uncertainty in the quantity of interest is represented by a random variable with a given distribution, and we use the law of total probability to express the inverse problem for the corresponding probability measure on the input space. Assuming that the map from the input space to the quantity of interest is smooth, we solve the generally ill-posed inverse problem by using the implicit function theorem to derive a method for approximating the set-valued inverse that provides an approximate quotient space representation of the input space. We then derive an efficient computational approach to compute a measure theoretic approximation of the probability measure on the input space imparted by the approximate set-valued inverse that solves the inverse problem.

We consider inverse problems for a deterministic model in which the dimension of the output quantities of interest computed from the model is smaller than the dimension of the input quantities into the model. In this case, the inverse problem admits set-valued solutions (equivalence classes of solutions). We devise a method for approximating a representation of the set-valued solutions in the parameter domain. We then consider a stochastic version of the inverse problem in which a probability distribution on the output quantities is specified. We construct a measure theoretic formulation of the stochastic inverse problem, then develop the existence and structure of the solution using measure theory and the Disintegration Theorem. We also develop and analyze an approximate solution method for the stochastic inverse problem based on measure-theoretic techniques. We demonstrate the numerical implementation of the theory on a high-dimensional storm surge application where simulated noisy surge data from Hurricane Katrina is used to determine the spatially variable bathymetry fields of highest probability.

One well-known approach to a posteriori analysis of finite element solutions of elliptic problems estimates the error in a quantity of interest in terms of residuals and a generalized Green's function. The generalized Green's function solves the adjoint problem with data related to a quantity of interest and measures the effects of stability, including any decay of influence characteristic of elliptic problems. We show that consideration of the generalized Green's function can be used to improve the efficiency of the solution process when the goal is to compute multiple quantities of interest and/or to compute quantities of interest that involve globally supported information such as average values and norms. In the latter case, we introduce a solution decomposition in which we solve a set of problems involving localized information and then recover the desired information by combining the local solutions. By treating each computation of a quantity of interest independently, the maximum number of elements required to achieve the desired accuracy can be decreased significantly.

No abstract

In this paper, we perform an a posteriori error analysis of a multiscale operator decomposition finite element method for the solution of a system of coupled elliptic problems. The goal is to compute accurate error estimates that account for the effects arising from multiscale discretization via operator decomposition. Our approach to error estimation is based on a well-known a posteriori analysis involving variational analysis, residuals, and the generalized Green's function. Our method utilizes adjoint problems to deal with several new features arising from the multiscale operator decomposition. In part I of this paper, we focus on the propagation of errors arising from the solution of one component to another and the transfer of information between different representations of solution components. We also devise an adaptive discretization strategy based on the error estimates that specifically controls the effects arising from operator decomposition. In part II of this paper, we address issues related to the iterative solution of a fully coupled nonlinear system.