Advanced Numerical Methods for CSE

Abstract

This course will focus on teaching different advanced topics in numerical methods for science and engineering. The main aim would be introduce novel algorithms and discuss their implementation.

Objective

* Understanding the mathematical foundations and design principles of a selection of modern numerical methods for challenging problems. * Ability to adapt the presented paradigms and algorithms to modified or new problems arising from applications in computational science and engineering. * Ability to judge the scope, strengths and weaknesses of the numerical methods covered in this course and of methods derived from them. * Skills in translating a high-level description of an algorithm into efficient code.

Content

The course will comprise three main chapters: 1. The Boundary Element Method (BEM): It is a numerical method used to solve boundary value problems for linear PDEs. It focuses only on the boundary, rather than the entire volume of the domain to be modeled. [50%] 2. Hierarchical Matrices (H-matrices): They are an efficient data structure used to approximate dense matrices with a hierarchical block structure, significantly reducing the computational and memory costs for operations like matrix multiplication and inversion. [25%] 3. Hybrid Modeling: The technique combines multiple modeling techniques, such as physics-based models and data-driven approaches, to capitalize on the strengths of each method and improve the accuracy and efficiency of simulations or predictions in complex systems. [25%]

Resources