Design and Optimization of Numerical Methods for Solving Inverse Problems

Abstract

. The purpose of inverse problems is to identify the unknown causes or parameters based on observable effects or measurements in a variety of scientific and technical domains. To arrive at precise and trustworthy solutions, numerical algorithms for inverse problems must be designed and optimised. The creation and optimisation of numerical algorithms specifically created for solving inverse issues are presented in detail in this abstract.The formulation of inverse issues and the mathematical models that explain the underlying processes are the primary topics of the first part of this work. We examine many inverse issues, such as ill-posed, parameter estimation, and linear and nonlinear ones. Prior information and regularisation methods must be used to increase the stability and uniqueness of the answers.The creation and application of numerical algorithms for the solution of inverse problems is the focus of the second component. We study in depth a number of iterative techniques, including the conjugate gradient method, Levenberg-Marquardt algorithm, and Gauss-Newton method. There is also discussion of sophisticated methods such variational methods, Bayesian inference, and approaches based on optimisation.The third consideration focuses on optimising numerical techniques to raise their effectiveness and precision. To hasten convergence and lower computational costs, methods like adaptive mesh refinement, parallel processing, and model reduction are examined. Additionally, methods for handling noisy or missing data are looked at, as well as methods for choosing the right regularisation settings.