Reduced-Order Modeling of Cross-Diffusion Systems

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2022-7

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Mülayim, Gülden

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In this thesis, intrusive and non-intrusive reduced-order models (ROMs) are developed for cross-diffusion systems. In the first part, we consider parameter-dependent systems with linear diffusion and cross-diffusion terms. The full-order models (FOMs) are constructed by discretizing them with finite differences in space. The resulting ordinary differential equations (ODEs) in matrix and tensor form are integrated in time with the implicit-explicit Euler (IMEX) method. The reduced bases are constructed non-intrusively with the two-level proper orthogonal decomposition (POD) approach and by applying higher-order singular value decomposition (HOSVD) to the space-time snapshots in tensor form. The reduced coefficients for new parameter values are computed using radial basis function (RBF) interpolation. The efficiency of the proposed method is illustrated through numerical experiments for twodimensional Schnakenberg, three-dimensional Brusselator cross-diffusion equations, and predator-prey problems. In the second part of the thesis, we construct intrusive ROMS with POD-Galerkin projection consider for nonlinear diffusion and crossdiffusion terms such as the Shigesada-Kawasaki-Teramoto (SKT) equation, and a tumor growth model. Finite-difference discretization of these systems in space leads to linear-quadratic ODEs. The FOMs are constructed by integrating in time with the linearly implicit Kahan’s method. ROMs are constructed intrusively applying POD with Galerkin projection. The computation of the reduced-order solutions is accelerated in the tensorial framework so that the offline and online computations are separated. The accuracy and long-term stability of the ROMs are shown for the SKT equation, tumor growth problem, and predator-prey problems. Overall, the spatiotemporal patterns of cross-diffusion systems are accurately approximated by the ROMs with speedup factors of orders two and three over the full-order models.

Subject Keywords

Reaction-Diffusion Systems, Cross-Diffusion, Pattern Formation, Finite Differences, Proper Orthogonal Decomposition, Radial Bases Functions, Singular Value Decomposition, Tensors

URI

https://hdl.handle.net/11511/98171

Collections

Graduate School of Applied Mathematics, Thesis

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Citation Formats

G. Mülayim, “Reduced-Order Modeling of Cross-Diffusion Systems,” Ph.D. - Doctoral Program, Middle East Technical University, 2022.