Lambert Theisen

Accelerating iterative eigenvalue algorithms is often achieved by employing a spectral shifting strategy. Unfortunately, improved shifting typically leads to a smaller eigenvalue for the resulting shifted operator, which in turn results in a high condition number of the underlying solution matrix, posing a major challenge for iterative linear solvers. This paper introduces a two-level domain decomposition preconditioner that addresses this issue for the linear Schrödinger eigenvalue problem, even in the presence of a vanishing eigenvalue gap in non-uniform, expanding domains. Since the quasi-optimal shift, which is already available as the solution to a spectral cell problem, is required for the eigenvalue solver, it is logical to also use its associated eigenfunction as a generator to construct a coarse space. We analyze the resulting two-level additive Schwarz preconditioner and obtain a condition number bound that is independent of the domain's anisotropy, despite the need for only one basis function per subdomain for the coarse solver. Several numerical examples are presented to illustrate its flexibility and efficiency.

Keywords: Schrödinger equation, iterative methods, preconditioning, domain decomposition, coarse spaces, finite element method

This paper provides a provably quasi-optimal preconditioning strategy of the linear Schrödinger eigenvalue problem with periodic potentials for a possibly nonuniform spatial expansion of the domain. The quasi-optimality is achieved by having the iterative eigenvalue algorithms converge in a constant number of iterations for different domain sizes. In the analysis, we derive an analytic factorization of the spectrum and asymptotically describe it using concepts from the homogenization theory. This decomposition allows us to express the eigenpair as an easy-to-calculate cell problem solution combined with an asymptotically vanishing remainder. We then prove that the easy-to-calculate limit eigenvalue can be used in a shift-and-invert preconditioning strategy to bound the number of eigensolver iterations uniformly. Several numerical examples illustrate the effectiveness of this quasi-optimal preconditioning strategy.

Keywords: periodic Schrödinger equation, iterative eigenvalue solvers, preconditioner, asymptotic eigenvalue analysis, factorization principle, directional homogenization

We present a mixed finite element solver for the linearized regularized 13-moment equations of non-equilibrium gas dynamics. The Python implementation builds upon the software tools provided by the FEniCS computing platform. We describe a new tensorial approach utilizing the extension capabilities of FEniCS’ Unified Form Language to define required differential operators for tensors above second degree. The presented solver serves as an example for implementing tensorial variational formulations in FEniCS, for which the documentation and literature seem to be very sparse. Using the software abstraction levels provided by the Unified Form Language allows an almost one-to-one correspondence between the underlying mathematics and the resulting source code. Test cases support the correctness of the proposed method using validation with exact solutions. To justify the usage of extended gas flow models, we discuss typical application cases involving rarefaction effects. We provide the documented and validated solver publicly.

Keywords: tensorial mixed finite element method, R13 equations, FEniCS project, continuous interior penalty

We covered topics from the real life including: How does GPS work?; Why is the Netflix recommendation algorithm so good?; How to optimize solar plants?; What's the algorithm to detect traffic lights?; How is data compression done?; What was the idea of the initial PageRank algorithm by Google? We used the learning concept from CAMMP (https://www.cammp.online) in a workshop style and reflected the mathematical concepts from the higher perspective and how to embedded them in the mathematics class in the school .

Linear algebra is the study of the basic concepts and techniques involving vectors and matrices. The course topics include logic, numbers and sets; vectors and vector spaces; systems of linear equations; linear transformations and their properties; eigenvalues and eigenvectors. The course objectives are to develop students’ skills in reasoning, modeling and problem-solving with vectors and matrices.

This course explores the use of mathematical concepts in computational chemistry, specifically in creating and breaking down models of molecules. We will take a mathematical approach to theoretical chemistry, covering topics such as electric charge interactions between molecular systems, the transition from classical to quantum mechanics, the Hartree-Fock model, and its breakdown. If time allows, we will also examine some aspects of Density Functional Theory (DFT). By the end of the course, students will have a deep understanding of the mathematical principles underlying computational chemistry and will be able to apply these principles to their own research in the field.

This course introduces variational calculus, which is a branch of mathematics that deals with finding the best solution to a problem involving functions. It also teaches how to integrate functions in several variables and on different types of spaces, such as curves and surfaces. The course also covers numerical methods for solving ordinary differential equations, which are equations that relate a function and its derivatives. Moreover, the course explores optimization techniques for finding the minimum or maximum value of a function and eigenvalue computation methods for finding the characteristic values of a matrix. The course aims to help students acquire and apply these mathematical tools in various fields of science and engineering.

This course introduces variational calculus, which is a branch of mathematics that deals with finding the best solution to a problem involving functions. It also teaches how to integrate functions in several variables and on different types of spaces, such as curves and surfaces. The course also covers numerical methods for solving ordinary differential equations, which are equations that relate a function and its derivatives. Moreover, the course explores optimization techniques for finding the minimum or maximum value of a function and eigenvalue computation methods for finding the characteristic values of a matrix. The course aims to help students acquire and apply these mathematical tools in various fields of science and engineering.

In this course, students will explore the theory and numerics of partial differential equations (PDEs), which are mathematical models of phenomena involving rates of change in multiple variables. The course will cover various aspects of PDEs, such as their classification by type and basic characteristics, their elementary solution methods for some classical examples, their generalization by using distributions and Sobolev spaces to define weak derivatives, their analysis by applying Fourier and other integral transformations to different domains, their discretization by finite difference methods on grids, and their numerical solution by efficient techniques such as FFT or filtering. The course will combine theoretical lectures with practical exercises using MATLAB or Julia.

In this course, you will learn about various aspects of linear algebra and analysis of functions of several variables. You will explore how to solve eigenvalue problems and transform matrices into diagonal or normal forms. You will also learn how to use singular value decomposition, rank determination and regularization concepts. You will apply differentiation, Taylor expansion, inverse and implicit functions to analyze and optimize multivariable functions. You will use iterative methods such as Newton’s method or Gauss-Newton method to solve nonlinear systems of equations and least squares problems. You will understand how to interpolate data using polynomials and how to perform numerical differentiation and integration using Newton-Cotes formulas, Gauss quadrature and extrapolation. Finally, you will get an introduction to the theory of ordinary differential equations.

I executed the self exercise and supervised students in the course on partial differential equations (PDEs). I learned and applied the theory and numerics of PDEs, such as their types, characteristics, solutions, generalizations, analysis, discretization and numerical solution. I also helped the students understand and practice these concepts and methods using MATLAB or Julia.

This course covers various aspects of the variational formulation for elliptic problems, such as the Galerkin technique and the Lax-Milgram theorem. It also introduces the finite element method for elliptic problems and some modern iterative methods, such as PCG and multigrid method. The course then extends to parabolic problems and shows how to use the method of lines for their discretization. It also presents the finite volume method as another discretization technique. The course then deals with saddle point problems and their application to Stokes equations. Finally, it discusses the Navier-Stokes equation for incompressible fluids. The main goals of this course are to help students understand the basic principles of discretizing partial differential equations and to teach them how to use different numerical methods for solving them. The students will also learn how to evaluate the results of these methods and how to adapt them to new tasks. The students will acquire confidence in using discretization techniques such as finite elements and finite volume methods, as well as iterative solution methods such as PCG and multigrid method.