Model Reduction by Proper Orthogonal Decomposition (POD)

Mathematical models for human tissue and blood flow both represent time dependent nonlinear partial differential equations in three space dimensions. Their numerical solution based on appropriate space/time discretizations requires computational times that even when using state-of-the-art algorithmic solvers are far from being acceptable for real time OR scenarios. A way to overcome this difficulty is to use reduced order models (ROMs) where the dimension of the ROM is by orders of magnitude less than the dimension of the full order model while still reflecting the essential dynamics of the underlying physiological processes. Suitable model order reduction techniques include balanced truncation (BT), proper orthogonal decomposition (POD), and reduced basis methods (RBM) (cf., e.g., [4, 6, 7, 17, 19, 26, 33, 36]). In this project, we will focus on POD in combination with the discrete empirical interpolation method (DEIM) [9, 10] which has been particularly designed for nonlinear problems and has been shown to result in substantial savings of computational time compared to classical POD.