STE 6239 Simulering

1. STE 6239 Simulering Friday, Week 1: 5. Scientific computing: basic solvers

2. 5.1. Scientific computing • The science of inventing, adapting, improving and optimizing algorithms for computational implementation of mathematical models, depending on the available computational resources. • Some of its branches: • Numerical analysis • Numerical linear algebra • Computatrional geometry • Computational version of virtually every exact natural and engineering science • Originates in ancient times, but becomes a key factor for the development of science and technology after the invention of the first digital computer • The inventor of the digital computer: Prof. John V. Atanasoff

3. John Vincent Atanasoff- the visionary who gave us the digital computer The memorial monument of John Vincent Atanasoff in the Bulgarian capital Sofia

4. 5.2. One challenge of scientific computing • Challenge: One important measure of intellectual progress is the range of important problems for which we have feasible closed-form theoretical solutions. It increases steadily, but at a much slower rate than the rapidly widening horizon of theoretical and practical challenges accompanying technological progress • Current solution: use computer-aided simulation based on scientific computing • Challenge: The process of maturing of the human individual and society is a process of deepening understanding how non-linear the true nature of our Universe is. However, we have (more or less) closed-form feasible solutions only for the linear models and a small range of special non-linear models (e.g., solvable models based on Galois theory or Lie-Backlund transformation group theory). • Solution: Reduce the general non-linear problems to a sequence of small steps (iterations) on which only solvable linear or simple non-linear problems are studied. For this purpose, use approximation to achieve local linearization of the problem. • Conclusion: we need: • A good knowledge of linear solvers (andsimple non-linear solvers). • A good knowledge of iterative algorithms.

5. 5.3. Linear solvers • The approximate dicretization of a continual mathematical model follows the scheme: • (Boundary problem for) (a system of) differential equation(s) [or other operator equation(s), e.g., integral equation(s)]  • Discretization – e.g., using FDM or FEM, (or other methods like BEM, FVM, SM, etc.)  • A system of linear equations (= a matrix equation) • Question: in some cases (e.g., typically in solid mechanics) we end up with systems of linear equations involving multidimensional arrays (tensors). What to do then? • Answer: this case can is also reduced to the case of a matrix equation by using the block matrix approach. • Conclusion: one way to improve computational performance (using the same computational resources) is to adapt the solver to the structure of the matrix. The importance of this becomes crucial for 2D, 3D and higher dimensional problems. • From this point of view, general rectagular (m x n) matrices can be classified, as follows: • Full • Dense (almost full) (sometimes ’dense’ is used simply as a synonim and replacement for ’full’) • Sparse (with a great diversity of sub-types within this type) • The solvers for full and sparse differ a lot; those for dense coincide with the ones for full or, more rarely, for very regular dense patterns, with the ones for sparse, depending on the context. Remarks: more to come

6. 5.4. Linear solvers for full (+ dense) matrices • A mysterious example of a dense matrix • Gaussian elimination and its various upgrades (more to come)

7. 5.5. Linear solvers for sparse matrices • A brief outline • A more advanced exposition (more to come)

8. 5.6. Multilevel techniques • Idea about Multigrid (more to come)

9. 5.7. Nonlinear techniques • Fixed points • Contraction mappings • Banach contraction mapping principle • Some applications • Iterative solutions of nonlinear systems of equations • Monotone nonlinear operators • Contraction mapping principle for operaror equations with monotone operators (more to come)

10. 5.8. Approximation via discretization • More to come

11. 5.9. Exercises More to come