Chapter 2 - Linear and nonlinear dimensionality reduction of biomechanical models

Abstract

This chapter reviews and discusses the theoretical background behind linear and nonlinear dimensionality reduction techniques and their application into a posteriori model reduction. Methodologies for multidimensionality reduction aim at discovering low-dimensional manifolds to which data belong. In particular, the Principal Component Analysis (PCA) and the kernel Principal Component Analysis (kPCA) are considered for linear and nonlinear dimensionality reduction, respectively. The Proper Orthogonal Decomposition (POD) is a reduced order model (ROM) combining PCA with a Reduced Basis approach, in order to reduce the number of degrees of freedom in parametric boundary value problems. Similarly, a kernel Proper Orthogonal Decomposition (kPOD) finds the solution in a kPCA-reduced nonlinear manifold, instead of the PCA linear space used in POD. Both the POD and the kPOD are described, including a detailed algorithm for kPOD. The performance of the presented techniques is assessed in an advection–diffusion problem.