DMI, Spiegelgasse 5, Basel, Seminarraum 05.001

## Doktoratskolloquium 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.

Der Vortrag ist universitätsöffentlich.

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