NUMERICAL SOLUTION OF MULTIDIMENSIONAL SPECTRAL FRACTIONAL DIFFUSION PROBLEMS: FROM CAFFARELLI TO BURA Authors: Bálint Farkas, Béla Nagy and Szilárd Révész, Alfréd Rényi Institute of Mathematics Date: June 6, 2023 Time: 5:30 pm Place: Room 503 of IMI - BAS Abstract: We introduce a general framework to investigate minimax problems for sum of translates functions F(x,t)=\sum_k K(t-x_j) or F(x,t)=J(t)+\sum_k K(t-x_j), where K is a general concave "kernel function" and J is an "outer field" function, t runs [0,1], and x=(x_1,...,x_n) is a set of nodes which are used to translate the kernel. Our setup is very close to logarithmic potential theory, but fixing n and focusing on a more detailed analysis, we obtain new results even for very classical problems. Extending a method of P. Fenton, we reach considerable [...]

NUMERICAL SOLUTION OF MULTIDIMENSIONAL SPECTRAL FRACTIONAL DIFFUSION PROBLEMS: FROM CAFFARELLI TO BURA Lecturer: Prof. Sv. Margenov (IICT - BAS) Time: 3:00 pm Date: May 23, 2023 Place: Room 503 of IMI - BAS Abstract: Fractional diffusion operators appear naturally in many areas in mathematics, physics, ect. The most important property of the related b.v. problems is that they are nonlocal. Let us consider the fractional power of a self-adjoint elliptic operator introduced through its spectral decomposition. It is also self-adjoint but nonlocal. Advanced numerical methods in this area have been heavily influenced by the pioneering work in differential operator theory by Caffarelli and Silvestre, "An Extension Problem Associated with the Fractional Laplacian", 2007. After discretization, nonlocal problems lead to linear systems with dense matrices. In the [...]

A New Walk on Equations Monte Carlo Method for Linear Algebraic Problems Lecturer: Prof. Ivan Dimov Time: 4:00 pm Date: February 14, 2023 Place: Room 503 of IMI - BAS Abstract: A new Walk on Equations (WE) Monte Carlo algorithm for Linear Algebra (LA) problem, namely, functionals of the solution, eigenvalue problems, etc., is proposed and studied. This algorithm relies on a non-discounted sum of an absorbed random walk. It can be applied for either real or complex matrices. Several techniques like simultaneous scoring or the sequential Monte Carlo method are applied to improve the basic algorithm. Numerical tests are performed on examples with matrices of different size and on systems coming from various important applications. Comparisons with standard deterministic, including unimprovable Conjugate Gradient [...]