European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
15:45   Error Estimation and Analysis (Part 1)
Chair: Fred Vermolen
25 mins
Image Denoising by a Two-Step Finite Element Method with Mesh Adaptivity for Perona-Malik Equation
Tharana Yosprakob, Khamron Mekchay
Abstract: Perona-Malik equation is one of the partial differential equations (PDEs) that has been widely used in computer vision and image processing technology. It performs particularly for edge detection and image denoising. In order to reduce a computation cost during a denoising process, we propose an algorithm of finite element method with mesh adaptivity for solving the Perona-Malik equation. We divide a full-time implicit method into two steps. First, the brightness function is estimated by a half step computation as suggested by He and Yang (2016). Then the full-time step is computed by using the estimation. We simulate denoised images from our scheme. Then the quality of our results and use of the computational resource are analyzed. We found that our algorithm does not only shows the decreasing of computation time but also indicates less memory requirement of nodes on a mesh for the computation.
25 mins
High-order Two and Three Level Schemes for Solving Fractional Powers of Elliptic Operators
Raimondas Ciegis, Petr Vabishchevich
Abstract: In this paper we develop and investigate numerical algorithms for solving nonstationary equa\-tions of the first order for approximations of the fractional powers of discrete elliptic operators: $$ A^\alpha y = \varphi, \;\; 0< \alpha < 1, $$ $\varphi \in V_h$ with $V_h$ a finite element or discrete appro\-xi\-ma\-tion space. Here $\mathcal A$ is an isomorphism $H^1_0(\Omega) \to H^{-1}(\Omega)$ given by $u \to a (u, \cdot )$ and \begin{equation*} %\label{1} a (u, v) := \int_\Omega \big( k(x) \nabla u \cdot \nabla v + c(x) u v \big) d x, \quad \forall v \in V. \end{equation*} The main goal is to consider two different approaches to construct efficient high order time stepping schemes for the implementation of the Cauchy problem method (Petr N. Vabishchevich, Journal of Computational Physics. 2015, Vol. 282, No.1, pp. 289--302). The first approach is based on a solution of the related time-dependent pseudo-parabolic equation. The second approach is based on diagonal Pad{\'e} appro\-xi\-ma\-tions to $(1 + x)^{-\alpha}$ when the relation of the solution on two times levels is used. The second and fourth order approximations are constructed by both approaches. In order to increase the accuracy of approximations the graded time grid is constructed which compensates the singular behavior of the solution for $t$ close to $0$. Results of numerical experiments are presented, they agree well with the theoretical results.
25 mins
Moment Approximation for the Joint Probability Density Function of the Jump-Diffusion Approximation
Derya Altıntan, Heinz Koeppl
Abstract: Biochemical reaction networks are very heteregeneous in terms of reaction rates and abundance of species. To model these systems with multi-scale nature, we proposed jump-diffusion approximation. The strategy of this hybrid model is to partition reactions into fast and slow groups and approximate the fast reaction group using chemical Langevin equation while Markov chain representation is used for slow reactions. We show that joint probability density function of the jump diffusion approximation satisfies the hybrid master equation which is a summation of the Fokker-Planck Equation and Chemical Master Equation. In this talk, we propose an algorithm to approximate the solution of the hybrid master equation. The first step of the algorithm is to construct an ordinary differential equation (ode) system for the moments of counters of fast reactions given the counters of slow reactions. In the second step of the algorithm, based on the solution of the corresponding ode system, we construct a constrained optimization problem. Solving optimization problem via maximum entropy approach gives the conditional probability density functions of the counters of fast reactions given the counters of slow reactions which in turn helps us to construct the joint probability density function solving hybrid master equation. In our algorithm, to retain the dimensionality of the corresponding optimization problem appropriate for the CVX toolbox of the MATLAB, we extend the state space iteratively based on the idea of sliding window method. We implement the algorithm to a reaction network. This work is supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) Program no:3501 Grant, no. 115E252.