Monte Carlo Algorithms for the Extracting of Electrical Capacitance

We present new Monte Carlo algorithms for extracting mutual capacitances for a system of conductors embedded in inhomogeneous isotropic dielectrics. We represent capacitances as functionals of the solution of the external Dirichlet problem for the Laplace equation. Unbiased and low-biased estimators for the capacitances are constructed on the trajectories of the Random Walk on Spheres or the Random Walk on Hemispheres. The calculation results show that the accuracy of these new algorithms does not exceed the statistical error of estimators, which is easily determined in the course of calculations. The algorithms are based on mean value formulas for harmonic functions in different domains and do not involve a transition to a difference problem. Hence, they do not need a lot of storage space.

Kuratowski MNC method on a generalized fractional Caputo Sturm–Liouville–Langevin q-difference problem with generalized Ulam–Hyers stability

In this work, we consider a generalized quantum fractional Sturm–Liouville–Langevin difference problem with terminal boundary conditions. The relevant results rely on Mönch’s fixed point theorem along with a theoretical method by terms of Kuratowski measure of noncompactness (MNC) and the Banach contraction principle (BCP). Furthermore, two dynamical notions of Ulam–Hyers (UH) and generalized Ulam–Hyers (GUH) stability are addressed for solutions of the supposed Sturm–Liouville–Langevin quantum boundary value problem (q-FBVP). Two examples are presented to show the validity and also the effectiveness of theoretical results. In the last part of the paper, we conclude our exposition with some final remarks and observations.

On the Lie symmetries of the boundary value problems for differential and difference sine-Gordon equations

In general, due to the nature of the Lie group theory, symmetry analysis is applied to single equations rather than boundary value problems. In this paper boundary value problems for the sine-Gordon equations under the group of Lie point symmetries are obtained in both differential and difference forms. The invariance conditions for the boundary value problems and their solutions are obtained. The invariant discretization of the difference problem corresponding to the boundary value problem for sine-Gordon equation is studied. In the differential case an unbounded domain is considered and in the difference case a lattice with points lying in the plane and stretching in all directions with no boundaries is considered.

Fast VMZ : code enhancements for video mosaicing and summarization

Video Mosaicing and Summarization (VMZ) is a novel image processing pipeline that summarizes the content of a long sequence of geospatial or biomedical videos using a few coverage maps or mini mosaics. The existing VMZ algorithm uses Normalized Cross-Correlation (NCC), Structure Tensor (ST), Affine-Invariant SIFT (ASIFT), Speeded up robust features for its feature matching and homography estimation pipeline, which are the most computationally expensive modules in the VMZ pipeline. Due to these long-running compute-intensive modules, the VMZ pipeline is not suitable for real-time mosaic formation in drones or UAVs. For instance, VMZ takes around 4 hours to generate mini-mosaics from an image sequence containing 9291 image frames. The blending algorithms used for mini-mosaic generation suffer from illumination variation due to the illumination difference in image frames. Such illumination inconsistency causes severe problems for biomedical scene understanding where curvilinear or tiny biological structures are present. VMZ pipeline is also dependent on 3rd party libraries not aligned with the flow of VMZ, which introduces redundant computation. One of the main reasons for the slow processing of the VMZ pipeline is not leveraging any parallel processing techniques and available graphics processing hardware. Therefore, the objective of this thesis is mainly three-fold: (i) speeding up the computeintensive and long-running modules in the VMZ pipeline, (ii) modifying the existing libraries and interfaces for better alignment with VMZ workflow, and (iii) resolving the illumination difference problem of the blending algorithms. Selected longrunning modules with the most impact on the overall run-time have been improved using CPU-based Multi-Threading, GPU-based Parallelization, and better integration with the existing VMZ pipeline. An illumination-matched blending algorithm has been proposed to improve the illumination problem. Besides, to evaluate the performance of different blending algorithms, a novel metric named Maximum Overall Illumination Difference (MOID) has been proposed. The improvement of VMZ modules has resulted in more than 100x speed-up in certain modules, with a 4.4x speed-up for the total VMZ run-time. The novel illumination matched blending resulted in a better MOID value for image sequences not having illumination variance in a single frame.

SIMULATION OF AN UNSTEADY INCOMPRESSIBLE FLUID FLOW THROUGH A PERFORATED PIPELINE

A mathematical model of the unsteady flow of an incompressible viscous fluid through a perforated pipeline is proposed, which is described by a system of nonlinear partial differential equations. In the framework of the model, the purpose is to determine the pressure and the flow rate of the fluid at the pipeline inlet, providing the flow rate and the pressure required at the pipeline outlet. By combining the system of the equations, the original problem is reduced to a boundary-value inverse problem for a nonlinear parabolic equation with respect to fluid flow rate. To solve the boundary inverse problem, the method of nonlocal perturbation of boundary conditions is proposed. A discrete analog of the inverse problem is obtained using the finitedifference approximation, and a special approach is suggested for solving the resulting system of difference equations. As a result, the difference problem for each discrete value of the time variable splits into two second-order difference problems and a linear equation with respect to an approximate value of the desired flow rate at the pipeline inlet. The absolutely stable Thomas method is used to numerically solve the obtained difference problems. After determining the flow rate distribution along the entire pipeline, the pressure at the pipeline inlet is also calculated using an explicit formula. Based on the proposed computational algorithm, the numerical experiments are performed for benchmark problems.

Numerical solution to elliptic inverse problem with Neumann-type integral condition and overdetermination

In modeling various real processes, an important role is played by methods of solution source identification problem for partial differential equation. The current paper is devoted to approximate of elliptic over determined problem with integral condition for derivatives. In the beginning, inverse problem is reduced to some auxiliary nonlocal boundary value problem with integral boundary condition for derivatives. The parameter of equation is defined after solving that auxiliary nonlocal problem. The second order of accuracy difference scheme for approximately solving abstract elliptic overdetermined problem is proposed. By using operator approach existence of solution difference problem is proved. For solution of constructed difference scheme stability and coercive stability estimates are established. Later, obtained abstract results are applied to get stability estimates for solution Neumann-type overdetermined elliptic multidimensional difference problems with integral conditions. Finally, by using MATLAB program, we present numerical results for two dimensional and three dimensional test examples with short explanation on realization on computer.

Study of a Numerical Method for Solving a Boundary Value Problem for a Differential Equation with a Fractional Time Derivative

Recently, there has been an increased interest in the problem of numerical implementation of multiphase filtration models due to its enormous economic importance in the oil industry, hydrology, and nuclear waste management. In contrast to the classical models of filtration, filtration models in highly porous fractured formations with the fractal geometry of wells are not fully understood. The solution to this problem reduces to solving a system of differential equations with fractional derivatives. In the paper, a finite-difference scheme is constructed for solving the initial-boundary value problem for the convection-diffusion equation with a fractional time derivative in the sense of Caputo-Fabrizio. A priori estimates are obtained for solving a difference problem under the assumption that there is a solution to the problem in the class of sufficiently smooth functions that prove the uniqueness of the solution and the stability of the difference scheme. The convergence of the solution of the difference problem to the solution of the original differential problem with the second order in time and space variables is shown. The results of computational experiments confirming the reliability of theoretical analysis are presented.

Approximation of the Multidimensional Optimal Control Problem for the Heat Equation (Applicable to Computational Fluid Dynamics (CFD))

This work is devoted to finding an estimate of the convergence rate of an algorithm for numerically solving the optimal control problem for the three-dimensional heat equation. An important aspect of the work is not only the establishment of convergence of solutions of a sequence of discrete problems to the solution of the original differential problem, but the determination of the order of convergence, which plays a very important role in applications. The paper uses the discretization method of the differential problem and the method of integral estimates. The reduction of a differential multidimensional mixed problem to a difference one is based on the approximation of the desired solution and its derivatives by difference expressions, for which the error of such an approximation is known. The idea of using integral estimates is typical for such problems, but in the multidimensional case significant technical difficulties arise. To estimate errors, we used multidimensional analogues of the integration formula by parts, Friedrichs and Poincare inequalities. The technique used in this work can be applied under some additional assumptions, and for nonlinear multidimensional mixed problems of parabolic type. To find a numerical solution, the variable direction method is used for the difference problem of a parabolic type equation. The resulting algorithm is implemented using program code written in the Python 3.7 programming language.

M-MATRICES AND CONVERGENCE OF FINITE DIFFERENCE SCHEME FOR PARABOLIC EQUATION WITH INTEGRAL BOUNDARY CONDITION

In the paper, the stability and convergence of difference schemes approximating semilinear parabolic equation with a nonlocal condition are considered. The proof is based on the properties of M-matrices, not requiring the symmetry or diagonal predominance of difference problem. The main presumption is that all the eigenvalues of the corresponding difference problem with nonlocal conditions are positive.