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1. S. Zampini. NON-OVERLAPPING DOMAIN DECOMPOSITION METHODS FOR CARDIAC REACTION-DIFFUSION MODELS AND APPLICATIONS.

Degree: 2010, Università degli Studi di Milano

In this thesis we consider different aspects related to the mathematical modeling of cardiac electrophysiology, either from the cellular or from the tissue perspective, and we develope novel numerical methods for the parallel iterative solution of the resulting reaction-diffusion models. In Chapter one we develope and validate the HHRd model, which accounts for transmural cellular heterogeneities of the canine left ventricle. Next, we introduce the reaction-diffusion models describing the spread of excitation in cardiac tissue, namely the anisotropic Bidomain and Monodomain models. For their discretization, we consider trilinear isoparametric finite elements in space and a semi-implicit (IMEX) method in time. In order to reduce the computational costs of parallel three-dimensional cardiac simulations, in Chapter three we consider different strategies to accelerate convergence of the Preconditioned Conjugate Gradient method. We consider novel choices for the Krylov initial guess in order to reduce the number of iterations per time step, either lagrangian interpolants in time or the Proper Orthogonal Decomposion technique combined with a usual Galerkin projection. In the last three chapters we construct non-overlapping domain decomposition methods for both cardiac reaction-diffusion models. In Chapter four we deal with preconditioners of the Neumann-Neumann type, in particular we consider the additive Neumann-Neumann method for the Monodomain model and the Balancing Neumann-Neumann method for the Bidomain model. In Chapter five we construct a Balancing Domain Decomposition by Constraint (BDDC) method for the Bidomain model, whereas in Chapter six we investigate the use of an approximate BDDC method for the Bidomain model. For all preconditioners considered, we develope novel theoretical estimates for the condition number of the preconditioned systems with respect to the spatial discretization, to the subdomains' diameter and to the time step, also in case of discontinuity in the conductivity coefficients of the cardiac tissue, with jumps aligned with the interface among subdomains. We prove scalability and quasi-optimality for the balancing methods, providing parallel numerical results confirming the theoretical estimates. Advisors/Committee Members: advisor: Luca F. Pavarino, program coordinator: Vincenzo Capasso, PAVARINO, LUCA FRANCO, CAPASSO, VINCENZO.

Subjects/Keywords: CARDIAC ELECTROPHYSIOLOGY; TRANSMURAL CELLULAR HETEROGENEITY; BIDOMAIN MODEL; CHOICHE OF KRYLOV INITIAL GUESS; PROPER ORTHOGONAL DECOMPOSITION; NON-OVERLAPPING DOMAIN DECOMPOSITION; BALANCING NEUMANN NEUMANN; BDDC; APPROXIMATE BDDC; PARALLEL COMPUTING; Settore MAT/08 - Analisi Numerica

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Zampini, S. (2010). NON-OVERLAPPING DOMAIN DECOMPOSITION METHODS FOR CARDIAC REACTION-DIFFUSION MODELS AND APPLICATIONS. (Thesis). Università degli Studi di Milano. Retrieved from http://hdl.handle.net/2434/150076

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Zampini, S.. “NON-OVERLAPPING DOMAIN DECOMPOSITION METHODS FOR CARDIAC REACTION-DIFFUSION MODELS AND APPLICATIONS.” 2010. Thesis, Università degli Studi di Milano. Accessed February 28, 2021. http://hdl.handle.net/2434/150076.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Zampini, S.. “NON-OVERLAPPING DOMAIN DECOMPOSITION METHODS FOR CARDIAC REACTION-DIFFUSION MODELS AND APPLICATIONS.” 2010. Web. 28 Feb 2021.

Vancouver:

Zampini S. NON-OVERLAPPING DOMAIN DECOMPOSITION METHODS FOR CARDIAC REACTION-DIFFUSION MODELS AND APPLICATIONS. [Internet] [Thesis]. Università degli Studi di Milano; 2010. [cited 2021 Feb 28]. Available from: http://hdl.handle.net/2434/150076.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Zampini S. NON-OVERLAPPING DOMAIN DECOMPOSITION METHODS FOR CARDIAC REACTION-DIFFUSION MODELS AND APPLICATIONS. [Thesis]. Università degli Studi di Milano; 2010. Available from: http://hdl.handle.net/2434/150076

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

2. Wang, Bin. Balancing domain decomposition by constraints algorithms for incompressible Stokes equations with nonconforming finite element discretizations.

Degree: PhD, Mathematics, 2017, University of Kansas

Hybridizable Discontinuous Galerkin (HDG) is an important family of methods, which combine the advantages of both Discontinuous Galerkin in terms of flexibility and standard finite elements in terms of accuracy and efficiency. The impact of this method is partly evidenced by the prolificacy of research work in this area. Weak Galerkin (WG) is a relatively newly proposed method by introducing weak functions and generalizing the differential operator for them. This method has also drawn remarkable interests from both numerical practitioners and analysts recently. HDG and WG are different but closely related. BDDC algorithms are developed for numerical solution of elliptic problems with both methods. We prove that the optimal condition number estimate for BDDC operators with standard finite element methods can be extended to the counterparts arising from the HDG and WG methods, which are nonconforming finite element methods. Numerical experiments are conducted to verify the theoretical analysis. Further, we propose BDDC algorithms for the saddle point system arising from the Stokes equations using both HDG and WG methods. By design of the preconditioner, the iterations are restricted to a benign subspace, which makes the BDDC operator effectively positive definite thus solvable by the conjugate gradient method. We prove that the algorithm is scalable in the number of subdomains with convergence rate only dependent on subdomain problem size. The condition number bound for the BDDC preconditioned Stokes system is the same as the optimal bound for the elliptic case. Numerical results confirm the theoretical analysis. Advisors/Committee Members: Tu, Xuemin (advisor), Huang, Weizhang (cmtemember), Van Vleck, Erik (cmtemember), Xu, Hongguo (cmtemember), Wang, Z.J. (cmtemember).

Subjects/Keywords: Mathematics; BDDC; domain decomposition; hybridizable discontinuous Galerkin; saddle point problems; Stokes; weak Galerkin

…List of Tables 4.1 Condition number estimates and iteration counts for the BDDC… …4.2 Condition number estimates and iteration counts for the BDDC preconditioned operator… …57 Condition number estimates and iteration counts for the BDDC preconditioned operator… …4.6 56 Condition number estimates and iteration counts for the BDDC preconditioned… …counts for the BDDC preconditioned operator with changing subdomains numbers. 4.4 56 57… 

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Wang, B. (2017). Balancing domain decomposition by constraints algorithms for incompressible Stokes equations with nonconforming finite element discretizations. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/27005

Chicago Manual of Style (16th Edition):

Wang, Bin. “Balancing domain decomposition by constraints algorithms for incompressible Stokes equations with nonconforming finite element discretizations.” 2017. Doctoral Dissertation, University of Kansas. Accessed February 28, 2021. http://hdl.handle.net/1808/27005.

MLA Handbook (7th Edition):

Wang, Bin. “Balancing domain decomposition by constraints algorithms for incompressible Stokes equations with nonconforming finite element discretizations.” 2017. Web. 28 Feb 2021.

Vancouver:

Wang B. Balancing domain decomposition by constraints algorithms for incompressible Stokes equations with nonconforming finite element discretizations. [Internet] [Doctoral dissertation]. University of Kansas; 2017. [cited 2021 Feb 28]. Available from: http://hdl.handle.net/1808/27005.

Council of Science Editors:

Wang B. Balancing domain decomposition by constraints algorithms for incompressible Stokes equations with nonconforming finite element discretizations. [Doctoral Dissertation]. University of Kansas; 2017. Available from: http://hdl.handle.net/1808/27005

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