Inverse Problems and Model Reduction Techniques

Abstract

Real problems come from engineering, industry, science and technology. Inverse problems for real applications usually have a large number of parameters to be reconstructed due to the accuracy needed to make accurate data predictions. This feature makes these problems highly underdetermined and ill-posed. Good prior information and regularization techniques are needed when using local optimization methods but only linear model appraisal (uncertainty) around the solution can be performed. The large number of parameters precludes the use of global sampling methods to approach inverse problem solution and appraisal. In this paper we show how to construct different kinds of reduced bases using Principal Component Analysis (PCA), Singular Value Decomposition (SVD), Discrete Cosine Transform (DCT) and Discrete Wavelet Transform (DWT). The use of a reduced base helps us to regularize the inverse problem and to find the set of equivalent models that fit the data within a prescribed tolerance and are compatible with the model prior.