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TANA15 Lectures and Seminars

(*) For Lectures the slides are available as PDFs and the the material are chapters in the course book. The seminars are for problem solving and the exercises are found in the problem collection.
Activity(*) Description Material
L1 Introduction

Introduction. Matrix-Matrix Multiply. Operation counts. Basic concepts such as Rank and Null Space.

Wendland 1.1-1.4, 2.2
L2 Linear Systems

The LU and Cholesky decompositions. The Condition number. Sensitivity analysis. Linear Least Squares problems. The QR decomposition. Application: Projections in Computer graphics.

Wendland 3.1-3.6, 2.3-2.4

1.1-1.2, 1.4-1.6, 1.10,
2.2-2.5. 3.1-3.3.
L3 QR Decomposition

Computing the QR decompositon. Refletions and Rotations. Row updating. Applications: Circle fitting, Tikhonov regularization, Image deblurring.

Wendland 3.6
L4 Eigenvalues

Basic theory of eigenvalues. Rayleigh quotient. The Power method and Inverse iteration. Localization and Sensitivity.

Wendland 5.1-5.4

3.4-3.6, 3.8, 3.9, 3.12-3.13,
4.1, 4.3, 4.5, 4.10
L5 QR Algorithm

Similarity transformations. Decoupling. The Schur and Hessenberg decompositions. The QR algorithm. Application: Google pageRank.

Wendland 3.7, 5.5-5.6

4.12,4.14-4.18 4.20-4.22
L6 Non-Linear

Non-linear equations. The Contraction mapping theorem. Taylor series and linearization. Newton's method. Updating methods. Application: Image inpainting. Trajectory of a Soccer ball.


L7 Optimization

Non-Linear Least Squares. Existance and Uniquenesas results. The Newton and Gauss-Newton methods. Application: Data Assimilation. The Singular Value Decomposition. Application: Linear Systems of Equations. The Condition Number.

Heath 6.5-6.6
L8 Singular values

The singular value decomposition. Fundamental Subspaces. Projections. Computing the SVD. Applications: Low rank approximation, Total least squares, Classification of Hand written digits.

Wendland 1.5, 5.7.

5.1, 5.7-5.9, 6.1-6.5
L9 Sparse

Integral Equations. Application: Remote sensing. Sparse matrices. Stationary iterative methods.

Wendland 4.1-4.3

6.6-6.13, 7.1-7.2
L10 Krylov

The projection method. Optimality Results. Krylov subspaces. GMRES and CG methods. Least Squares problems.

Wendland 6.1-6.3

7.3-5, 7.7-7.8

Page responsible: Fredrik Berntsson
Last updated: 2023-01-13