arXiv subjects

Petr N. Vabishchevich

Publications and source records attributed to Petr N. Vabishchevich.

At least 19 recordsLinked to original sources

arXiv2026

Splitting schemes for problems with memory

The paper considers the Cauchy problem for a first-order integro-differential equation with memory in a finite-dimensional Hilbert space. The main computational difficulty of such problems is the need to store and process the solution at all previous time levels. To overcome this difficulty, an approach is used that approximates the memory kernel by a sum of exponentials, which reduces the original nonlocal problem to a local one -- a system of weakly coupled evolution equations with additional ordinary differential equations for auxiliary functions. The problem is formulated in vector form on the direct sum of Hilbert spaces. Unconditional stability of two-level operator-difference schemes with weights is proved under standard restrictions. Splitting schemes are proposed and investigated by separating the local and integral operators of the problem. Possibilities for constructing similar schemes for other nonlocal problems, in particular for the equation with memory of the time derivative of the solution, are noted.

math.NA↗

arXiv2024

Operator-difference approximations on two-dimensional merged Voronoi-Delaunay grids

Formulating boundary value problems for multidimensional partial derivative equations in terms of invariant operators of vector (tensor) analysis is convenient. Computational algorithms for approximate solutions are based on constructing grid analogs of vector analysis operators. This is most easily done by dividing the computational domain into rectangular cells when the grid nodes coincide with the cell vertices or are the cell centers. Grid operators of vector analysis for irregular regions are constructed using Delaunay triangulations or Voronoi partitions. This paper uses two-dimensional merged Voronoi-Delaunay grids to represent the grid cells as orthodiagonal quadrilaterals. Consistent approximations of the gradient, divergence, and rotor operators are proposed. On their basis, operator-difference approximations for typical stationary scalar and vector problems are constructed.

math.NA↗

arXiv2024

Decoupling technology for systems of evolutionary equations

Numerical methods of approximate solution of the Cauchy problem for coupled systems of evolution equations are considered. Separating simpler subproblems for individual components of the solution achieves simplification of the problem at a new level in time. The decoupling method, a significant approach to simplifying the problem, is based on the decomposition of the problem's operator matrix. The approximate solution is constructed based on the linear composition of solutions to auxiliary problems. The paper investigates decoupling variants based on extracting the diagonal part of the operator matrix and the lower and upper triangular submatrices. The study introduces a new decomposition approach, which involves splitting the operator matrix into rows and columns. The composition stage utilizes various variants of splitting schemes, showcasing the versatility of the approach. In additive operator-difference schemes, we can distinguish explicit-implicit schemes, factorized schemes for two-component splitting, and regularized schemes for general multicomponent splitting. The study of stability of two- and three-level decoupling composition schemes is carried out using the theory of stability (correctness) of operator-difference schemes for finite-dimensional Hilbert spaces. The theoretical results of the decoupling technique for systems of evolution equations are illustrated on a test two-dimensional problem for a coupled system of two diffusion equations with inhomogeneous self- and cross-diffusion coefficients.

math.NA↗

arXiv2023

Nonlinear approximation of functions based on non-negative least squares solver

In computational practice, most attention is paid to rational approximations of functions and approximations by the sum of exponents. We consider a wide enough class of nonlinear approximations characterized by a set of two required parameters. The approximating function is linear in the first parameter; these parameters are assumed to be positive. The individual terms of the approximating function represent a fixed function that depends nonlinearly on the second parameter. A numerical approximation minimizes the residual functional by approximating function values at individual points. The second parameter's value is set on a more extensive set of points of the interval of permissible values. The proposed approach's key feature consists in determining the first parameter on each separate iteration of the classical non-negative least squares method. The computational algorithm is used to rational approximate the function $x^{-α}, \ 0 < α< 1, \ x \geq 1$. The second example concerns the approximation of the stretching exponential function $\exp(- x^α ), \ \ \quad 0 < α< 1$ at $ x \geq 0$ by the sum of exponents.

math.NA↗

arXiv2022

Subdomain solution decomposition method for nonstationary problems

The reduction of computational costs in the numerical solution of nonstationary problems is achieved through splitting schemes. In this case, solving a set of less computationally complex problems provides the transition to a new level in time. The traditional construction approach of splitting schemes is based on an additive representation of the problem operator(s) and uses explicit-implicit approximations for individual terms. Recently (Y. Efendiev, P.N. Vabishchevich. Splitting methods for solution decomposition in nonstationary problems. \textit{Applied Mathematics and Computation}. \textbf{397}, 125785, 2021), a new class of methods of approximate solution of nonstationary problems has been introduced based on decomposition not of operators but of the solution itself. This new approach with subdomain solution selection is used in this paper to construct domain decomposition schemes. The boundary value problem for a second-order parabolic equation in a rectangle with a difference approximation in space is typical. Two and three-level schemes for decomposition of the domain with and without overlapping subdomains are investigated. Our numerical experiments complement the theoretical results.

math.NA↗

arXiv2022

Nonlocal transport equations in multiscale media. Modeling, dememorization, and discretizations

In this paper, we consider a class of convection-diffusion equations with memory effects. These equations arise as a result of homogenization or upscaling of linear transport equations in heterogeneous media and play an important role in many applications. First, we present a dememorization technique for these equations. We show that the convection-diffusion equations with memory effects can be written as a system of standard convection-diffusion-reaction equations. This allows removing the memory term and simplifying the computations. We consider a relation between dememorized equations and micro-scale equations, which do not contain memory terms. We note that dememorized equations differ from micro-scale equations and constitute a macroscopic model. Next, we consider both implicit and partially explicit methods. The latter is introduced for problems in multiscale media with high-contrast properties. Because of high contrast, explicit methods are restrictive and require time steps that are very small (scales as the inverse of the contrast). We show that, by appropriately decomposing the space, we can treat only a few degrees of freedom implicitly and the remaining degrees of freedom explicitly. We present a stability analysis. Numerical results are presented that confirm our theoretical findings of partially explicit schemes applied to dememorized systems of equations.

math.NA↗

arXiv2021

Numerical solution of the heat conduction problem with memory

It is necessary to use more general models than the classical Fourier heat conduction law to describe small-scale thermal conductivity processes. The effects of heat flow memory and heat capacity memory (internal energy) in solids are considered in first-order integrodifferential evolutionary equations with difference-type kernels. The main difficulties in applying such nonlocal in-time mathematical models are associated with the need to work with a solution throughout the entire history of the process. The paper develops an approach to transforming a nonlocal problem into a computationally simpler local problem for a system of first-order evolution equations. Such a transition is applicable for heat conduction problems with memory if the relaxation functions of the heat flux and heat capacity are represented as a sum of exponentials. The correctness of the auxiliary linear problem is ensured by the obtained estimates of the stability of the solution concerning the initial data and the right-hand side in the corresponding Hilbert spaces. The study's main result is to prove the unconditional stability of the proposed two-level scheme with weights for the evolutionary system of equations for modeling heat conduction in solid media with memory. In this case, finding an approximate solution on a new level in time is not more complicated than the classical heat equation. The numerical solution of a model one-dimensional in space heat conduction problem with memory effects is presented.

math.NA↗

arXiv2021

Approximate solution of the Cauchy problem for a first-order integrodifferential equation with solution derivative memory

We consider the Cauchy problem for a first-order evolution equation with memory in a finite-dimensional Hilbert space when the integral term is related to the time derivative of the solution. The main problems of the approximate solution of such nonlocal problems are due to the necessity to work with the approximate solution for all previous time moments. We propose a transformation of the first-order integrodifferential equation to a system of local evolutionary equations. We use the approach known in the theory of Voltaire integral equations with an approximation of the difference kernel by the sum of exponents. We formulate a local problem for a weakly coupled system of equations with additional ordinary differential equations. We have given estimates of the stability of the solution by initial data and the right-hand side for the solution of the corresponding Cauchy problem. The primary attention is paid to constructing and investigating the stability of two-level difference schemes, which are convenient for computational implementation. The numerical solution of a two-dimensional model problem for the evolution equation of the first order, when the Laplace operator conditions the dependence on spatial variables, is presented.

math.NA↗

arXiv2021

Numerical solution of the Cauchy problem for Volterra integrodifferential equations with difference kernels

We consider the problems of the numerical solution of the Cauchy problem for an evolutionary equation with memory when the kernel of the integral term is a difference one. The computational implementation is associated with the need to work with an approximate solution for all previous points in time. In this paper, the considered nonlocal problem is transformed into a local one; a loosely coupled equation system with additional ordinary differential equations is solved. This approach is based on the approximation of the difference kernel by the sum of exponentials. Estimates for the stability of the solution concerning the initial data and the right-hand side for the corresponding Cauchy problem are obtained. Two-level schemes with weights with convenient computational implementation are constructed and investigated. The theoretical consideration is supplemented by the results of the numerical solution of the integrodifferential equation when the kernel is the stretching exponential function.

math.NA↗

arXiv2021

Some methods for solving equations with an operator function and applications for problems with a fractional power of an operator

Several applied problems are characterized by the need to numerically solve equations with an operator function (matrix function). In particular, in the last decade, mathematical models with a fractional power of an elliptic operator and numerical methods for their study have been actively discussed. Computational algorithms for such non-standard problems are based on approximations by the operator function. The most widespread are the approaches using various options for rational approximation. Also, we note the methods that relate to approximation by exponential sums. In this paper, the possibility of using approximation by exponential products is noted. The solution of an equation with an operator function is based on the transition to standard stationary or evolutionary problems. General approaches are illustrated by a problem with a fractional power of the operator. The first class of methods is based on the integral representation of the operator function under rational approximation, approximation by exponential sums, and approximation by exponential products. The second class of methods is associated with solving an auxiliary Cauchy problem for some evolutionary equation.

math.NA↗

arXiv2021

Temporal Splitting algorithms for non-stationary multiscale problems

In this paper, we study temporal splitting algorithms for multiscale problems. The exact fine-grid spatial problems typically require some reduction in degrees of freedom. Multiscale algorithms are designed to represent the fine-scale details on a coarse grid and, thus, reduce the problems' size. When solving time-dependent problems, one can take advantage of the multiscale decomposition of the solution and perform temporal splitting by solving smaller-dimensional problems, which is studied in the paper. In the proposed approach, we consider the temporal splitting based on various low dimensional spatial approximations. Because a multiscale spatial splitting gives a "good" decomposition of the solution space, one can achieve an efficient implicit-explicit temporal discretization. We present a recently developed theoretical result in our earlier work and adopt it in this paper for multiscale problems. Numerical results are presented to demonstrate the efficiency of the proposed splitting algorithm.

math.NA↗

arXiv2021

Contrast-independent partially explicit time discretizations for multiscale wave problems

In this work, we design and investigate contrast-independent partially explicit time discretizations for wave equations in heterogeneous high-contrast media. We consider multiscale problems, where the spatial heterogeneities are at subgrid level and are not resolved. In our previous work, we have introduced contrast-independent partially explicit time discretizations and applied to parabolic equations. The main idea of contrast-independent partially explicit time discretization is to split the spatial space into two components: contrast dependent (fast) and contrast independent (slow) spaces defined via multiscale space decomposition. Using this decomposition, our goal is further appropriately to introduce time splitting such that the resulting scheme is stable and can guarantee contrast-independent discretization under some suitable (reasonable) conditions. In this paper, we propose contrast-independent partially explicitly scheme for wave equations. The splitting requires a careful design. We prove that the proposed splitting is unconditionally stable under some suitable conditions formulated for the second space (slow). This condition requires some type of non-contrast dependent space and is easier to satisfy in the "slow" space. We present numerical results and show that the proposed methods provide results similar to implicit methods with the time step that is independent of the contrast.

math.NA↗

arXiv2021

Contrast-independent partially explicit time discretizations for multiscale flow problems

Many multiscale problems have a high contrast, which is expressed as a very large ratio between the media properties. The contrast is known to introduce many challenges in the design of multiscale methods and domain decomposition approaches. These issues to some extend are analyzed in the design of spatial multiscale and domain decomposition approaches. However, some of these issues remain open for time dependent problems as the contrast affects the time scales, particularly, for explicit methods. For example, in parabolic equations, the time step is $dt=H^2/κ_{max}$, where $κ_{max}$ is the largest diffusivity. In this paper, we address this issue in the context of parabolic equation by designing a splitting algorithm. The proposed splitting algorithm treats dominant multiscale modes in the implicit fashion, while the rest in the explicit fashion. The unconditional stability of these algorithms require a special multiscale space design, which is the main purpose of the paper. We show that with an appropriate choice of multiscale spaces we can achieve an unconditional stability with respect to the contrast. This could provide computational savings as the time step in explicit methods is adversely affected by the contrast. We discuss some theoretical aspects of the proposed algorithms. Numerical results are presented.

math.NA↗

arXiv2020

Splitting Schemes for Non-Stationary Problems with a Rational Approximation for Fractional Powers of the Operator

Problems of the numerical solution of the Cauchy problem for a first-order differential-operator equation are discussed. A fundamental feature of the problem under study is that the equation includes a fractional power of the self-adjoint positive operator. In computational practice, rational approximations of the fractional power operator are widely used in various versions. The purpose of this work is to construct special approximations in time when the transition to a new level in time provided a set of standard problems for the operator and not for the fractional power operator. Stable splitting schemes with weights parameters are proposed for the additive representation of rational approximation for a fractional power operator. Possibilities of using similar time approximations for other problems are noted. The numerical solution of a two-dimensional non-stationary problem with a fractional power of the Laplace operator is also presented.

math.NA↗

arXiv2020

Splitting Schemes for Some Second-Order Evolution Equations

We consider the Cauchy problem for a second-order evolution equation, in which the problem operator is the sum of two self-adjoint operators. The main feature of the problem is that one of the operators is represented in the form of the product of operator A by its conjugate A*. Time approximations are carried out so that the transition to a new level in time was associated with a separate solution of problems for operators A and A*, not their products. The construction of unconditionally stable schemes is based on general results of the theory of stability (correctness) of operator-difference schemes in Hilbert spaces and is associated with the multiplicative perturbation of the problem operators, which lead to stable implicit schemes. As an example, the problem of the dynamics of a thin plate on an elastic foundation is considered.

math.NA↗

arXiv2020

Numerical recovery of the piecewise constant leading coefficient of an elliptic equation

We propose a numerical algorithm for the reconstruction of a piecewise constant leading coefficient of an elliptic problem. The inverse problem is reduced to a shape reconstruction problem. The proposed algorithm is based on the minimization of a cost functional where a control function is the right-hand side of an auxiliary elliptic equation for a level set representation of unknown shape. The numerical implementation is based on the finite element method and the open-source computing platform FEniCS. The performance of the algorithm is demonstrated on computationally simulated data.

math.NA↗

arXiv2020

Splitting methods for solution decomposition in nonstationary problems

In approximating solutions of nonstationary problems, various approaches are used to compute the solution at a new time level from a number of simpler (sub-)problems. Among these approaches are splitting methods. Standard splitting schemes are based on one or another additive splitting of the operator into "simpler" operators that are more convenient/easier for the computer implementation and use inhomogeneous (explicitly-implicit) time approximations. In this paper, a new class of splitting schemes is proposed that is characterized by an additive representation of the solution instead of the operator corresponding to the problem (called problem operator). A specific feature of the proposed splitting is that the resulting coupled equations for individual solution components consist of the time derivatives of the solution components. The proposed approaches are motivated by various applications, including multiscale methods, domain decomposition, and so on, where spatially local problems are solved and used to compute the solution. Unconditionally stable splitting schemes are constructed for a first-order evolution equation, which is considered in a finite-dimensional Hilbert space. In our splitting algorithms, we consider the decomposition of both the main operator of the system and the operator at the time derivative. Our goal is to provide a general framework that combines temporal splitting algorithms and spatial decomposition and its analysis. Applications of the framework will be studied separately.

math.NA↗

arXiv2020

Modified Bee Colony optimization algorithm for computational parameter identification for pore scale transport in periodic porous media

This paper discusses an optimization method called Modified Bee Colony algorithm (MBC) based on a particular intelligent behavior of honeybee swarms. The algorithm was checked in a few benchmarks like Shekel, Rozenbroke, Himmelblau and Rastrigin functions, then was applied to parameter identification for reactive flow problems in periodic porous media. The simulation results show that the performance and efficiency of MBC algorithm are comparable to the other parameter identification methods and strategies, at the same time it is able to better capture local minima for the considered class of problems. The proposed identification approach is applicable for different geometries (random and periodic) and for a range of process parameters. In this paper the potential of the approach is demonstrated in identifying parameters of Langmuir isotherm for low Peclet and high Damkoler numbers reactive flow in a 2D periodic porous media with circular inclusions. Finite element approximation in space and implicit time discretization are exploited.

cs.CE↗