A composite preconditioner for the electromagnetic scattering from a large cavity

We solve the variable coefficient Cauchy---Navier equations of elasticity in the unit square, for Dirichlet and Dirichlet-Neumann boundary conditions, using second order finite difference schemes. The resulting linear systems are solved by the preconditioned conjugate gradient (PCG) method with preconditioners corresponding to to the Laplace operator. The multiplication of a vector by the matrices of the resulting systems and the solution of systems with the preconditioners are performed at optimal and nearly optimal costs, respectively. For the case of Dirichlet boundary conditions, we prove the second order accuracy of the scheme in the discrete $$H^1$$ H 1 norm, symmetry of the resulting matrix and its spectral equivalence to the preconditioner. For the case of Dirichlet---Neumann boundary conditions, we prove symmetry of the resulting matrix. Numerical tests demonstrating the convergence properties of the schemes and PCG are presented.

Groundwater flow simulation has become one of the top international issues in new generation of environmental applications. When managing large-scale groundwater flow problems, the intensive computational ability and large amounts of memory space required for modeling are the main bottlenecks for researchers. In order to solve three-dimensional large-scale groundwater flow problems more rapidly, the Open MP was adopted to parallelize the preconditioned conjugate gradient (PCG) algorithm in this paper. And this paper carried out a numerical experiment of the three-dimensional groundwater flow model on a computer with four cores. Based on the numerical experiment, it is found that the execution time of the original serial PCG program is about 1.74 to 2.86 times of the parallel PCG program executed with different number of threads. The experimental results also demonstrate that the PCG solver based on Open MP is an effective way for solving large-scale three-dimensional groundwater flow problem.

Compact optimal quadratic spline collocation methods for the Helmholtz equation

A matching Schur complement preconditioning technique for inverse source problems

A full 3D finite element analysis using adaptive refinement and PCG solver with back interpolation

A parallel linelet preconditioner has been implemented to accelerate finite element (FE) solvers for incompressible flows when highly anisotropic meshes are used. The convergence of the standard preconditioned conjugate gradient (PCG) solver that is commonly used to solve the discrete pressure equations, greatly deteriorates due to the presence of highly distorted elements, which are of mandatory use for high Reynolds‐number flows. The linelet preconditioner notably accelerates the convergence rate of the PCG solver in such situations, saving an important amount of CPU time. Unlike other more sophisticated preconditioners, parallelization of the linelet preconditioner is almost straighforward. Numerical examples and some comparisons with other preconditioners are presented to demonstrate the performance of the proposed preconditioner.

A sixth-order fast algorithm for the electromagnetic scattering from large open cavities

Summary form only given as follows. A mathematical formulation for the guided modes of the weakly guiding optical fibers is derived from the two-dimensional Helmholtz equation. The original problem is reduced to a spectral problem for the Helmholtz equation in a bounded domain with nonlocal boundary condition. This formulation leads to a nonlinear eigenvalue problem for a family of self-adjoint compact operators. The main spectral properties of these operators are established. Then the min-max principle provides an expression of the nonlinear dispersion relation, which connects the propagation constants of guided modes with the frequency. Various existence results are proved and a complete description of dispersion curves (monotonicity, asymptotic behavior, existence of cutoff values) is carried out. A finite element scheme follows from the operator formulation of the problem. This scheme leads to a nonlinear eigenvalue problem for a family of self-adjoint matrix operators. The existence results for this problem are proved. For its solution we propose an efficient method. Practical opportunities of this algorithm are shown by the results of numerical experiments.

We consider the problem of scheduling sparse linear application on heterogeneous (CPUs/GPUs) platform. More specifically, we focus on the preconditioned conjugate gradient solver (PCG) since it exhibits the main features of such problems. Indeed, the relative performance of CPU and GPU highly depends on the sub-routine: GPUs are for instance much more efficient to process regular kernels such as matrix vector multiplications rather than more irregular kernels such as matrix factorization. In this context, one solution consists in relying on dynamic scheduling and resource allocation mechanisms such as the ones provided by StarPU. In this paper we evaluate the performance of dynamic schedulers proposed by StarPU, and we analyse the scalability of PCG algorithm. We show how effectively we can choose the best combination of resources in order to improve their performance.

In this paper we study existence, uniqueness, and solution estimates to the mixed problem $\nabla\cdot\sigma\nabla u=0$ with Dirichlet to Neumann map boundary conditions and Neumann boundary conditions. We then show how this can be used in the reconstruction of $\sigma$, given the relationship between u and its normal derivative on the boundary portion where we do not apply the Dirichlet to Neumann map. A numerical reconstruction scheme is also derived.

In this paper, we firstly introduce two BPX-like preconditioners \(B_J^1\) and \(B_J^3\), and present an equivalent but more robust BPX-like preconditioner \(B_J^2\) for the solution of the linear finite element discretization of Poisson problem. Secondly, we implement these preconditioners and their preconditioned conjugate gradient (PCG) solvers \(B_l^p\)-CG(\(l=1,2,3\)) under Compute Unified Device Architecture (CUDA), where we exploit the hierarchical and the overall storage structure, take advantage of the multicolored Gauss–Seidel smoother. Finally, comparisons are made among these PCG solvers and the state-of-the-art SA-AMG preconditioned CG solver (SA-CG) in CUSP library. Numerical results demonstrate that the iteration numbers of \(B_2^p\)-CG holds the weakest dependence on the grid size, while \(B_3^p\)-CG is the most efficient solver. Furthermore, \(B_3^p\)-CG possesses considerable advantages over SA-CG in computational capability and efficiency. In particular, \(B_3^p\)-CG runs 3.67 times faster than SA-CG when solving a problem with about one-million unknowns.

Transformation optics constructions have allowed the design of cloaking devices that steer electromagnetic, acoustic and quantum waves around a region without penetrating it, so that this region is hidden from external observations. The material pa- rameters used to describe these devices are anisotropic, and singular at the interface between the cloaked and uncloaked regions, making physical realization a challenge. These singular material parameters correspond to singular coefficient functions in the partial differential equations modeling these constructions and the presence of these singularities causes various mathematical problems and physical effects on the interface surface. In this paper, we analyze the two dimensional cloaking for Helmholtz equation when there are sources or sinks present inside the cloaked region. In particular, we consider nonsingular approximate invisibility cloaks based on the truncation of the singular transformations. Using such truncation we analyze the limit when the approximate cloaking approaches the ideal cloaking. We show that, surprisingly, a non-local boundary condition appears on the inner cloak interface. This effect in the two dimensional (or cylindrical) invisibility cloaks, which seems to be caused by the infinite phase velocity near the interface between the cloaked and uncloaked regions, is very different from the earlier studied behavior of the solutions in the three dimensional cloaks.

This paper deals with a singular two dimensional initial boundary value problem for a Caputo time fractional parabolic equation supplemented by Neumann and non-local boundary conditions. The well posedness of the posed problem is demonstrated in a fractional weighted Sobolev space. The used method based on some functional analysis tools has been successfully showed its efficiency in proving the existence, uniqueness and continuous dependence of the solution upon the given data of the considered problem. More precisely, for proving the uniqueness of the solution of the posed problem, we established an energy inequality for the solution from which we deduce the uniqueness. For the existence, we proved that the range of the operator generated by the considered problem is dense.

A proof of convergence for the combination technique for the Laplace equation using tools of symbolic computation

Helmholtz equation in parabolic rotational coordinates: application to wave problems in quantum mechanics and acoustics