Chapter 5

Numerical Methods

Quantitative analyses are essential elements in solving forward and inverse problems. To utilize computers, we should devise numerically implementable algorithms to solve given problems, which include step-by-step procedures to obtain final answers. Formulation of a forward as well as an inverse problem should be done bearing in mind that we will adopt a certain numerical algorithm to solve them. After reviewing the basics of numerical computations, we will introduce various methods to solve a linear system of equations, which are most commonly used to obtain numerical solutions of both forward and inverse problems. Considering that most forward problems are formulated by using partial differential equations, we will study numerical techniques such as the finite difference method and finite element method. The accuracy or consistency of a numerical solution as well as the convergence and stability of an algorithm to obtain the solution need to be investigated.

5.1 Iterative Method for Nonlinear Problem

Recall the abstract form of the inverse problem in section 4.1, where we tried to reconstruct a material property P from knowledge of the input data X and output data Y using the forward problem (4.1). It is equivalent to the following root-finding problem:

images

where X and Y are given data.

We assume that both the material property P and F(P, X) are expressed ...

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