Regular Domain Problems Research Articles

Modification of the dimensional doma method for linear elliptic equations

Currently, one of the most common methods for the numerical solution of boundary value problems are difference methods (grid method, finite difference method). These methods are well suited for solving problems in regular domains. In areas of complex structure, their use leads to a number of difficulties both in the construction of difference schemes, and in the implementation on a computer. One of the methods to overcome these difficulties is the method of fictitious regions. There are four interrelated directions in the development of the method of fictitious regions: study of various ways to continue the original problems in a fictitious area; obtaining unimprovable estimates in stronger metrics; extension of the class of problems on the application of the method of fictitious areas; construction of effective difference schemes for solving auxiliary problems constructed by the method of fictitious areas. For the first time, the fictitious domain method for the Navier-Stokes equation was studied in the work of A.N. Bugrova, Sh. Smagulov, Smagulov Sh. This article discusses the method of additional areas, which is an analogue of the method of fictitious areas for solving linear boundary value problems, but in the auxiliary problem of this method there is no small perimeter. This method is based on the variational principle of solving linear boundary value problems of elliptic type in an arbitrary region. The existence of a solution is proved by the Galerkin uniqueness method.

Subspace decomposition based DNN algorithm for elliptic type multi-scale PDEs

While deep learning algorithms demonstrate a great potential in scientific computing, its application to multi-scale problems remains to be a big challenge. This is manifested by the “frequency principle” that neural networks tend to learn low frequency components first. Novel architectures such as multi-scale deep neural network (MscaleDNN) were proposed to alleviate this problem to some extent. In this paper, we construct a subspace decomposition based DNN (dubbed SD2NN) architecture for a class of multi-scale problems by combining MscaleDNN algorithms with traditional numerical analysis ideas. The proposed architecture includes one low frequency normal DNN submodule, and one (or a few) high frequency MscaleDNN submodule(s), which are designed to capture the smooth part and the oscillatory part of the multi-scale solutions simultaneously. We demonstrate that the SD2NN outperforms existing models such as MscaleDNN, through several benchmark multi-scale problems in regular or perforated domains.

A Hybrid Local Radial Basis Function Method for the Numerical Modeling of Mixed Diffusion and Wave-Diffusion Equations of Fractional Order Using Caputo’s Derivatives

This article presents an efficient method for the numerical modeling of time fractional mixed diffusion and wave-diffusion equations with two Caputo derivatives of order 0<α<1, and 1<β<2. The numerical method is based on the Laplace transform technique combined with local radial basis functions. The method consists of three main steps: (i) first, the Laplace transform is used to transform the given time fractional model into an equivalent time-independent inhomogeneous problem in the frequency domain; (ii) in the second step, the local radial basis functions method is utilized to obtain an approximate solution for the reduced problem; (iii) finally, the Stehfest method is employed to convert the obtained solution from the frequency domain back to the time domain. The use of the Laplace transform eliminates the need for classical time-stepping techniques, which often require very small time steps to achieve accuracy. Additionally, the application of local radial basis functions helps overcome issues related to ill-conditioning and sensitivity to shape parameters typically encountered in global radial basis function methods. To validate the efficiency and accuracy of the proposed method, several test problems in regular and irregular domains with uniform and non-uniform nodes are considered.

Subspace Decomposition Based Dnn Algorithm for Elliptic-Type Multi-Scale Pdes

While deep learning algorithms demonstrate a great potential in scientific computing, its application to multi-scale problems remains to be a big challenge. This is manifested by the "frequency principle" that neural networks tend to learn low frequency components first. Novel architectures such as multi-scale deep neural network (MscaleDNN) were proposed to alleviate this problem to some extent. In this paper, we construct a subspace decomposition based DNN (dubbed SD$^2$NN) architecture for a class of multi-scale problems by combining traditional numerical analysis ideas and MscaleDNN algorithms. The proposed architecture includes one low frequency normal DNN submodule, and one (or a few) high frequency MscaleDNN submodule(s), which are designed to capture the smooth part and the oscillatory part of the multi-scale solutions, respectively. In addition, a novel trigonometric activation function is incorporated in the SD$^2$NN model. We demonstrate the performance of the SD$^2$NN architecture through several benchmark multi-scale problems in regular or irregular geometric domains. Numerical results show that the SD$^2$NN model is superior to existing models such as MscaleDNN.

A meshless multiple-scale polynomial method for numerical solution of 3D convection–diffusion problems with variable coefficients

In this paper numerical solution of 3D convection–diffusion problems both with high Reynolds (Re) numbers and variable coefficients are investigated via a meshless method based on polynomial basis. It is well known that using polynomial basis directly for solving partial differential equations may be unsafe due to ill-conditioned resultant coefficients matrix that formed after discretization process. Therefore to get rid of highly ill-conditioned coefficient matrix we took advantage of multiple-scale method which is essentially based on the idea of equating norm of each column of resultant coefficients matrix. Through this approach we can reduce the condition number greatly. The proposed method is a truly meshless method since there is no need for meshing or any node connectivity and implementation of the method is simple and straightforward. The performance of the proposed method is measured by four test problems in both regular and irregular computational domains. Numerical results corroborate efficiency of meshless multiple-scale polynomial method for 3D convection–diffusion problems as well show its stability in case of large noise effect.

Kansa-RBF algorithms for elliptic problems in regular polygonal domains

We propose matrix decomposition algorithms for the efficient solution of the linear systems arising from Kansa radial basis function discretizations of elliptic boundary value problems in regular polygonal domains. These algorithms exploit the symmetry of the domains of the problems under consideration which lead to coefficient matrices possessing block circulant structures. In particular, we consider the Poisson equation, the inhomogeneous biharmonic equation, and the inhomogeneous Cauchy-Navier equations of elasticity. Numerical examples demonstrating the applicability of the proposed algorithms are presented.

An inhomogeneous polyharmonic Dirichlet problem with boundary data in the upper half-plane

In this paper, an inhomogeneous Dirichlet problem with data for the polyharmonic equation is investigated in the upper half-plane. By using higher order Poisson kernels and Pompeiu operators, which are respectively due to Du et al. [Du Z, Qian T, Wang J. polyharmonic Dirichlet problems in regular domains II: the upper half plane. J Differ Equ. 2012;252:1789–1812] as well as Begehr and Hile [Begehr H, Hile G. A hierarchy of integral operators. Rocky Mountain J Math. 1997;27:669–706], an integral representation for the unique solution is given under certain assumptions.

Solving Singularly Perturbed Multipantograph Delay Equations Based on the Reproducing Kernel Method

A numerical method is presented for solving the singularly perturbed multipantograph delay equations with a boundary layer at one end point. The original problem is reduced to boundary layer and regular domain problems. The regular domain problem is solved by combining the asymptotic expansion and the reproducing kernel method (RKM). The boundary layer problem is treated by the method of scaling and the RKM. Two numerical examples are provided to illustrate the effectiveness of the present method. The results from the numerical example show that the present method can provide very accurate analytical approximate solutions.

Application of Radial Basis Function Method for Solving Nonlinear Integral Equations

The radial basis function (RBF) method, especially the multiquadric (MQ) function, was proposed for one- and two-dimensional nonlinear integral equations. The unknown function was firstly interpolated by MQ functions and then by forming the nonlinear algebraic equations by the collocation method. Finally, the coefficients of RBFs were determined by Newton’s iteration method and an approximate solution was obtained. In implementation, the Gauss quadrature formula was employed in one-dimensional and two-dimensional regular domain problems, while the quadrature background mesh technique originated in mesh-free methods was introduced for irregular situation. Due to the superior interpolation performance of MQ function, the method can acquire higher accuracy with fewer nodes, so it takes obvious advantage over the Gaussian RBF method which can be revealed from the numerical results.

L p Polyharmonic Dirichlet problems in regular domains I: the unit disc

In this article, we consider a class of Dirichlet problems with L p boundary data for polyharmonic functions in the unit disc. Using higher order Poisson kernels due to Begehr et al. [H. Begehr, J. Du, and Y. Wang, A Dirichlet problem for polyharmonic functions, Ann. Mat. Pura Appl. 187(4) (2008), pp. 435–457; Z. Du, G. Guo, and N. Wang, Decompositions of functions and Dirichlet problems in the unit disc, J. Math. Anal. Appl. 362(1) (2010), pp. 1–16], we give the integral representation solutions of the problems.

A numerical method for singularly perturbed turning point problems with an interior layer

The objective of this paper is to present a numerical method for solving singularly perturbed turning point problems exhibiting an interior layer. The method is based on the asymptotic expansion technique and the reproducing kernel method (RKM). The original problem is reduced to interior layer and regular domain problems. The regular domain problems are solved by using the asymptotic expansion method. The interior layer problem is treated by the method of stretching variable and the RKM. Four numerical examples are provided to illustrate the effectiveness of the present method. The results of numerical examples show that the present method can provide very accurate approximate solutions.

Analytical Method for Heat Conduction Problem with Internal Heat Source in Irregular Domains

In engineering field, analytical methods can only be applied to classic heat transfer problems with regular geometric boundaries. It is difficult to apply analytical methods in solving the mathematical and physical equations in nonorthogonal boundary of irregular domains. In this paper, we presented a method by conformal mapping from the solution for heat conduction problems in regular domains to solve problems in irregular domains.

Analytical diagonal elements of regularized meshless method for regular domains of [formula omitted] Dirichlet Laplace problems

This note is to present a simple approach to derive the analytical formula of the diagonal elements of the interpolation matrix of the regularized meshless method (RMM) for regular domain problems, which is a very new boundary-type numerical discretization technique. In literature, these diagonal elements are mostly calculated numerically by the desingular technique, except for the circular domain problems. Our numerical experiments show that the analytical diagonal elements can improve the solution accuracy of the RMM for some regular domain problems, and the diagonal elements are critical to the solution accuracy of the RMM. Thus, a searching process is employed to find the optimal diagonal elements for RMM.

Efficient MFS algorithms in regular polygonal domains

In this work, we apply the Method of Fundamental Solutions (MFS) to harmonic and biharmonic problems in regular polygonal domains. The matrices resulting from the MFS discretization possess a block circulant structure. This structure is exploited to produce efficient Fast Fourier Transform–based Matrix Decomposition Algorithms for the solution of these problems. The proposed algorithms are tested numerically on several examples.

Finite state machine-based optimization of data parallel regular domain problems applied in low-level image processing

A popular approach to providing nonexperts in parallel computing with an easy-to-use programming model is to design a software library consisting of a set of preparallelized routines, and hide the intricacies of parallelization behind the library's API. However, for regular domain problems (such as simple matrix manipulations or low-level image processing applications-in which all elements in a regular subset of a dense data field are accessed in turn) speedup obtained with many such library-based parallelization tools is often suboptimal. This is because interoperation optimization (or: time-optimization of communication steps across library calls) is generally not incorporated in the library implementations. We present a simple, efficient, finite state machine-based approach for communication minimization of library-based data parallel regular domain problems. In the approach, referred to as lazy parallelization, a sequential program is parallelized automatically at runtime by inserting communication primitives and memory management operations whenever necessary. Apart from being simple and cheap, lazy parallelization guarantees to generate legal, correct, and efficient parallel programs at all times. The effectiveness of the approach is demonstrated by analyzing the performance characteristics of two typical regular domain problems obtained from the field of low-level image processing. Experimental results show significant performance improvements over nonoptimized parallel applications. Moreover, obtained communication behavior is found to be optimal with respect to the abstraction level of message passing programs.

Incorporating memory layout in the modeling of message passing programs

One of the most fundamental tasks of an automatic parallelization tool is to find an optimal domain decomposition for a given application. For regular domain problems (such as simple matrix manipulations) this task may seem trivial. However, communication costs in message passing programs often significantly depend on the memory layout of data blocks to be transmitted. As a consequence, straightforward domain decompositions may be non-optimal. In this paper we introduce a new point-to-point communication model (called P-3PC) that is specifically designed to overcome this problem. In comparison with related models (e.g., LogGP) P-3PC is similar in complexity, but more accurate in many situations. Although the model is aimed at MPI's standard point-to-point operations, it is applicable to similar message passing definitions as well. The effectiveness of the model is tested in a framework for automatic parallelization of imaging applications. Experiments are performed on two Beowulf-type systems, each having a different interconnection network, and a different MPI implementation. Results show that, where other models frequently fail, P-3PC correctly predicts the communication costs related to any type of domain decomposition.

P-3PC: a point-to-point communication model for automatic and optimal decomposition of regular domain problems

One of the most fundamental problems automatic parallelization tools are confronted with is to find an optimal domain decomposition for a given application. For regular domain problems (such as simple matrix manipulations), this task may seem trivial. However, communication costs in message-passing programs often depend significantly on the memory layout of data blocks to be transmitted. As a consequence, straightforward domain decompositions may be non-optimal. In this paper, we introduce a new point-to-point communication model, called P-3PC (Parameterized model based on the Three Paths of Communication), that is specifically designed to overcome this problem. In comparison with related models (e.g. LogGP), P-3PC is similar in complexity, but more accurate in many situations. Although the model is aimed at MPI's standard point-to-point operations, it is applicable to similar message-passing definitions as well. The effectiveness of the model is tested in a framework for automatic parallelization of low-level image processing applications. Experiments are performed on two Beowulf-type systems, each having a different interconnection network and a different MPI implementation. The results show that, where other models frequently fail, P-3PC correctly predicts the communication costs related to any type of domain decomposition.

The Riemann complex boundary element method for the solutions of two-dimensional Elliptic equations

In this paper, a new boundary integral equation model Riemann complex boundary element method (RCBEM), is proposed based on the boundary element method (BEM) and the theory of Vekua and its modification as well as complex Riemann function as the fundamental solution. The RCBEM method is used to solve the linear, second order, elliptical partial differential equations in the fluid flow problems. In comparison to the generally used BEM, for RCBEM, there are two distinct differences. First one is that, RCBEM applies complex Riemann function as the fundamental solution of the adjoint operator while in direct BEM, on the other hand Green function is used. The second one is that the governing equations should be transformed into complex domain because there exist two characteristics in complex plane for elliptic systems, while in the direct BEM is not, since the Green function is adopted instead. The singular problem occurring in direct BEM can be avoided in RCBEM, especially for regular domain problems. The efficiency and accuracy of the RCBEM depends on the complex variable integration. To verify the feasibility and accuracy of RCBEM, the model is applied to different case studies of potential flows, Helmholtz equation problem and advection–diffusion problem and results are compared with analytical solutions and other numerical models. The results are satisfactory and prove the applicability of RCBEM for various two-dimensional elliptic equation problems.

THE EXPONENTIAL BEHAVIOUR OF THE GREEN FUNCTION IN A DIHEDRAL ANGLE

We study the behaviour of the Green function of the heat problem in a dihedral angle with oblique (or Neumann) conditions on the faces. The uniform estimates obtained for the Green function and its derivatives show, in particular, the exponential decay at infinity, as in the parabolic problems in regular domains.

Data locality exploitation in the decomposition of regular domain problems

The aim of this paper is to study the effect of local memory hierarchy and communication network exploitation on message sending and the influence of this effect on the decomposition of regular applications. In particular, we have considered two different parallel computers, a Cray T3E-900 and an SGI Origin 2000. In both systems, the bandwidth reduction due to non-unit-stride memory access is quite significant and could be more important than the reduction due to contention in the network. These conclusions affect the choice of optimal decompositions for regular domains problems. Thus, although traditional 3D decompositions lead to lower inherent communication-to-computation ratios and could exploit more efficiently the interconnection network, lower dimensional decompositions are found to be more efficient due to the data decomposition effects on the spatial locality of the messages to be communicated. This increasing importance of local optimisations has also been shown using a well-known communication-computation overlapping technique which increases execution time, instead of reducing it as we could expect, due to poor cache memory exploitation.