Combined fast multipole-QR compression technique for solving electrically small to large structures for broadband applications

An approach that efficiently solves for a desired parameter of a system or device that can include both electrically large fast multipole method (FMM) elements, and electrically small QR elements. The system or device is setup as an oct-tree structure that can include regions of both the FMM type and the QR type. An iterative solver is then used to determine a first matrix vector product for any electrically large elements, and a second matrix vector product for any electrically small elements that are included in the structure. These matrix vector products for the electrically large elements and the electrically small elements are combined, and a net delta for a combination of the matrix vector products is determined. The iteration continues until a net delta is obtained that is within predefined limits. The matrix vector products that were last obtained are used to solve for the desired parameter

Technique for Solving Electrically Small to Large Structures for Broadband Applications

Fast iterative algorithms are often used for solving Method of Moments (MoM) systems, having a large number of unknowns, to determine current distribution and other parameters. The most commonly used fast methods include the fast multipole method (FMM), the precorrected fast Fourier transform (PFFT), and low-rank QR compression methods. These methods reduce the O(N) memory and time requirements to O(N log N) by compressing the dense MoM system so as to exploit the physics of Green s Function interactions. FFT-based techniques for solving such problems are efficient for spacefilling and uniform structures, but their performance substantially degrades for non-uniformly distributed structures due to the inherent need to employ a uniform global grid. FMM or QR techniques are better suited than FFT techniques; however, neither the FMM nor the QR technique can be used at all frequencies. This method has been developed to efficiently solve for a desired parameter of a system or device that can include both electrically large FMM elements, and electrically small QR elements. The system or device is set up as an oct-tree structure that can include regions of both the FMM type and the QR type. The system is enclosed with a cube at a 0- th level, splitting the cube at the 0-th level into eight child cubes. This forms cubes at a 1st level, recursively repeating the splitting process for cubes at successive levels until a desired number of levels is created. For each cube that is thus formed, neighbor lists and interaction lists are maintained. An iterative solver is then used to determine a first matrix vector product for any electrically large elements as well as a second matrix vector product for any electrically small elements that are included in the structure. These matrix vector products for the electrically large and small elements are combined, and a net delta for a combination of the matrix vector products is determined. The iteration continues until a net delta is obtained that is within the predefined limits. The matrix vector products that were last obtained are used to solve for the desired parameter. The solution for the desired parameter is then presented to a user in a tangible form; for example, on a display

Some Experiments and Issues to Exploit Multicore Parallelism in a Distributed-Memory Parallel Sparse Direct Solver

MUMPS is a parallel sparse direct solver, using message passing (MPI) for parallelism. In this report we experiment how thread parallelism can help taking advantage of recent multicore architectures. The work done consists in testing multithreaded BLAS libraries and inserting OpenMP directives in the routines revealed to be costly by profiling, with the objective to avoid any deep restructuring or rewriting of the code. We report on various aspects of this work, present some of the benefits and difficulties, and show that 4 threads per MPI process is generally a good compromise. We then discuss various issues that appear to be critical in a mixed MPI-OpenMP environment

Precise Error Bounds for Numerical Approximations of Fractional HJB Equations

We prove precise rates of convergence for monotone approximation schemes of fractional and nonlocal Hamilton-Jacobi-Bellman (HJB) equations. We consider diffusion corrected difference-quadrature schemes from the literature and new approximations based on powers of discrete Laplacians, approximations which are (formally) fractional order and 2nd order methods. It is well-known in numerical analysis that convergence rates depend on the regularity of solutions, and here we consider cases with varying solution regularity: (i) Strongly degenerate problems with Lipschitz solutions, and (ii) weakly non-degenerate problems where we show that solutions have bounded fractional derivatives of order between 1 and 2. Our main results are optimal error estimates with convergence rates that capture precisely both the fractional order of the schemes and the fractional regularity of the solutions. For strongly degenerate equations, these rates improve earlier results. For weakly non-degenerate problems of order greater than one, the results are new. Here we show improved rates compared to the strongly degenerate case, rates that are always better than 1/2

Discretization of fractional fully nonlinear equations by powers of discrete Laplacians

We study discretizations of fractional fully nonlinear equations by powers of discrete Laplacians. Our problems are parabolic and of order $\sigma\in(0,2)$ since they involve fractional Laplace operators $(-\Delta)^{\sigma/2}$. They arise e.g.~in control and game theory as dynamic programming equations, and solutions are non-smooth in general and should be interpreted as viscosity solutions. Our approximations are realized as finite-difference quadrature approximations and are 2nd order accurate for all values of $\sigma$. The accuracy of previous approximations depend on $\sigma$ and are worse when $\sigma$ is close to $2$. We show that the schemes are monotone, consistent, $L^\infty$-stable, and convergent using a priori estimates, viscosity solutions theory, and the method of half-relaxed limits. We present several numerical examples.Comment: 19 pages, 5 figures, 1 tabl