Tensor B-spline numerical method for PDEs : a high performance approach

Shulga, Dmytro. Tensor B-spline numerical method for PDEs : a high performance approach. 2019, Doctoral Thesis, University of Basel, Faculty of Science.

Available under License CC BY-NC-ND (Attribution-NonCommercial-NoDerivatives).


Official URL: http://edoc.unibas.ch/diss/DissB_13724

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Solutions of Partial Differential Equations (PDEs) form the basis of many mathematical models in physics and medicine. In this work, a novel Tensor B-spline methodology for numerical solutions of linear second-order PDEs is proposed. The methodology applies the B-spline signal processing framework and computational tensor algebra in order to construct high-performance numerical solvers for PDEs. The method allows high-order approximations, is mesh-free, matrix-free and computationally and memory efficient.
The first chapter introduces the main ideas of the Tensor B-spline method, depicts the main contributions of the thesis and outlines the thesis structure.
The second chapter provides an introduction to PDEs, reviews the numerical methods for solving PDEs, introduces splines and signal processing techniques with B-splines, and describes tensors and the computational tensor algebra.
The third chapter describes the principles of the Tensor B-spline methodology. The main aspects are 1) discretization of the PDE variational formulation via B-spline representation of the solution, the coefficients, and the source term, 2) introduction to the tensor B-spline kernels, 3) application of tensors and computational tensor algebra to the discretized variational formulation of the PDE, 4) tensor-based analysis of the problem structure, 5) derivation of the efficient computational techniques, and 6) efficient boundary processing and numerical integration procedures.
The fourth chapter describes 1) different computational strategies of the Tensor B-spline solver and an evaluation of their performance, 2) the application of the method to the forward problem of the Optical Diffusion Tomography and an extensive comparison with the state-of-the-art Finite Element Method on synthetic and real medical data, 3) high-performance multicore CPU- and GPU-based implementations, and 4) the solution of large-scale problems on hardware with limited memory resources.
Advisors:Hunziker, Patrick and Unser, Michael
Faculties and Departments:05 Faculty of Science > Departement Biozentrum
03 Faculty of Medicine > Departement Biomedizin > Associated Research Groups > Nanomedicine Research Group (Hunziker)
Item Type:Thesis
Thesis Subtype:Doctoral Thesis
Thesis no:13724
Thesis status:Complete
Number of Pages:1 Online-Ressource (100 Seiten)
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edoc DOI:
Last Modified:22 Oct 2020 07:31
Deposited On:22 Oct 2020 07:28

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