A Short Course on Duality, Adjoint Operators, Green’s Functions, and A Posteriori Error Analysis

Continuous optimization, data assimilation, determining model sensitivity, uncertainty quantification, and a posteriori estimation of computational error are fundamentally important problems in mathematical modeling of the physical world. There has been some substantial progress on solving these problems in recent years, and some of these solution techniques are entering mainstream computational science. A powerful framework for tackling all of these problems rests on the notion of duality and an adjoint operator. In the first part of this short course, we will discuss duality, adjoint operators, and Green′s functions; covering both the theoretical underpinnings and practical examples. We will motivate these ideas by explaining the fundamental role of the adjoint operator in the solution of linear problems, working both on the level of linear algebra and differential equations. This will lead in a natural way to the definition of the Green′s function. In the second part of the course, we will describe how a generalization of the idea of a Green′s function is connected to a powerful technique for a posteriori error analysis of finite element methods. This technique is widely employed to obtain accurate and reliable error estimates in “quantities of interest”. We will also discuss the use of these estimates for adaptive error control. Finally, in the third part of the course, we will describe some applications of these analytic techniques. In the first, we will use the properties of Green′s functions to improve the efficiency of the solution process for an elliptic problem 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. Time permitting, we will also discuss applications to a posteriori estimation of the effects of operator splitting in a multi-physics problem, estimation of the effect of random variation in parameters in a deterministic model (without using Monte-Carlo), and extensions to nonlinear problems. The research activities of D. Estep are partially supported by the Department of Energy through grant 90143, the National Aeronautics and Space Administration through grant NNG04GH63G, the National Science Foundation through grants DMS-0107832, DGE-0221595003, and MSPA-CSE-0434354, the Sandia Corporation through contract number PO299784, and the United States Department of Agriculture through contract 58-5402-3-306.