Lecturer: Christos Pervolianakis
Office: Ernst-Abbe-Platz 2, Room 3533
Email: christos.pervolianakis (AT) uni-jena.de
Dates: Tuesday and Thursday 12-14Uhr
Office hours: Tuesday and Thursday 15-17Uhr.
Description
Many physical and engineering problems are modeled using partial differential equations (PDEs). However, their solutions often cannot be expressed in closed form, requiring numerical approximation. Discretization methods such as finite differences and finite elements transform PDEs into large-scale linear systems that must be solved efficiently. This course delves into iterative solvers, essential for efficiently handling these systems, particularly for sparse and structured matrices. We will explore fundamental iterative methods such as Jacobi, Gauss-Seidel and multigrid methods, as well as minimization techniques like Krylov subspace methods. Additionally, we will examine their convergence properties and preconditioning techniques to accelerate computations.
Recommended knowledge
Course Evaluation
01.04.2025 I would like to inform you that the lectures for this course will begin on April 15, 2025, as I will be attending a conference next week. The two lectures scheduled for next week will be rescheduled in the near future.
15.04.2025 During the lectures on 24.04, 06.05, 20.05, 03.06, 01.06, and 08.07, we will solve exercises.
30.04.2025 Here you will find a list of potential topics for the optional project, along with instructions.
27.05.2025 I would like to inform you that on 02.06.2025, there will be a replacement lecture in Seminar Room 3517, at Ernst-Abbe-Platz 2, from 14:00 to 16:00.
13.06.2025 The exercise session on 17.06 is rescheduled to 08.07. Additionally, during the lecture on 17.06, we will discuss the possibility of an extra exercise session. The date and time will be announced afterwards. The exercise set 4 will be announced on 19.06 after the lecture.
15-04-2025 Two boundary value problem. Maximum principle. Finite Difference Method for solving the two boudary value problem.
17-04-2025 Finite Difference Method for solving the two boudary value problem. The discrete maximum principle.
22-04-2025 Finite Difference Method for solving the Dirichlet problem for Poisson equation. The discrete maximum principle.
24-04-2025 Solution of the exercise 4 from the Exercise set 1.
28-04-2025 Introduction to iterative methods. General iterative method with the matrix splitting \(\mathbf{A} = \mathbf{M} - \mathbf{N}\). Jordan normal form.
29-04-2025 Necessary and sufficient condition for the convergence of the general iterative method. Classical iterative methods (e.g., Jacobi and Gauẞ-Seidel methods)
06-05-2025 Solution of the exercises 1,2,5,6,7 from the Exercise set 1.
08-05-2025 No lecture.
12-05-2025 Convergence of Jacobi and Gauss-Seidel iteration.
13-05-2025 Convergence of Jacobi iteration for the linear system derived from FDM method in \(\mathbb{R}^2\) with 5 point stencil.
15-05-2025 Continue from the previous lecture. Introduction to relaxation techniques.
20-05-2025 Solution of the exercises 1,2,3 from the Exercise set 2 and the exercise 3 from Exercise set 1.
22-05-2025 Relaxation techniques. Optimal parameter for a symmetrizable iteration. Successive Over-Relaxation. Necessary criterion for the convergence. Also, a necessary and sufficient criterion for Hermite positive definite matrices.
27-05-2025 Relaxation techniques. Successive Over-Relaxation. Optimal value of SOR.
02-06-2025 Symmetric Successive Over-Relaxation (SSOR). Nececcary condition for its convergence. Sufficient and necessary condition for its convergence for \(\mathbf{A}\in\mathbb{C}^{n,n}\) Hermite and positive definite. Introduction to minimization methods. Minimization property: For an s.d.p matrix \(\mathbf{A}\in\mathbb{R}^{n,n}\), the vector \(\mathbf{x}\in\mathbb{R}^n\) solves the linear system \(\mathbf{A}\mathbf{x} = \mathbf{b}\) iff \(f(\mathbf{z}) = \min_{\mathbf{z}\in\mathbb{R}^n}f(\mathbf{z})\) where the functional is defined as \(f\,:\,\mathbb{R}^n \mapsto \mathbb{R},\,f(\mathbf{z}) = \frac{1}{2}\mathbf{z}^T\mathbf{A}\mathbf{z} - \mathbf{z}^T\mathbf{b}\).
03-06-2025 Steepest Descent Method. Error estimation. Generalized Directions Method. An interesting 3D visualization of the steepest descent method can be found in the following video https://www.youtube.com/watch?v=iudXf5n_3ro.
05-06-2025 Conjugate Directions Method. Proof that the conjugate direction method ends in at most \(n\) iterations. Further, we proved that minimizes the functional \(f(\mathbf{z}) = \frac{1}{2}\mathbf{z}^T\mathbf{A}\mathbf{z} - \mathbf{z}^T\mathbf{b}\) along the direction \(\mathbf{p}^{(\ell)}\) and at the same time in the subspace \(\mathrm{span}\{\mathbf{p}^{(1)}, \mathbf{p}^{(2)},\ldots,\mathbf{p}^{(\ell)}\}\).
10-06-2025 Solution of the exercises 4-7 from the Exercise set 2 and the exercise 1 from Exercise set 3.
12-06-2025 Proof that the conjugate direction method ends in at most \(n\) iterations. Further, we proved that minimizes the functional \(f(\mathbf{z}) = \frac{1}{2}\mathbf{z}^T\mathbf{A}\mathbf{z} - \mathbf{z}^T\mathbf{b}\) along the direction \(\mathbf{p}^{(\ell)}\) and at the same time in the subspace \(\mathrm{span}\{\mathbf{p}^{(1)}, \mathbf{p}^{(2)},\ldots,\mathbf{p}^{(\ell)}\}\). Conjugate Gradient Method. We proved also some preparatory results.
17-06-2025 Conjugate Gradient Method: Error estimation. Comparison of the convergence rate of CG with Steepest Descent method.
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