PhD Defense Computer Science: Dmytro Shulga

Tensor B-Spline Numerical Method for PDEs: a High Performance Approach

ABSTRACT:

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.

The talk is open to members of the university.

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