Decomposition Interpolation Research Articles

Empirical Interpolation Decomposition

Many physical problems need a multidimensional description and involve high dimensional spaces. Standard discretization techniques often lead to an excessive computation time. To solve this problem, we develop in this paper an empirical interpolation decomposition (EID) for multivariate functions. This method provides an approximate representation of a given function in separate form. Error estimates of the developed EID are derived and some properties are given.

Performance Evaluation of the Fast Spherical Harmonic Transform Algorithm in the Yin–He Global Spectral Model

Abstract In this paper, we describe an implementation of the fast spherical harmonic transform (SHT) algorithm in the Yin–He global spectral model (YHGSM). A new analysis method is proposed to study the potential instability of interpolative decomposition and to evaluate the performance of fast SHT on the MilkyWay-2 supercomputer. The novel aspect of the proposed method is the use of the computational complexity analysis method by studying the properties of interpolative decompositions. Furthermore, three test cases are conducted to verify fast SHT in YHGSM. The results demonstrate that fast SHT is feasible and efficient for reducing the computational complexity and memory cost of SHT while still keeping enough accuracy.

Wide Angular Sweeping of Dynamic Electromagnetic Responses From Large Targets by MPI Parallel Skeletonization

It has been revealed that the interpolative decomposition (ID) skeletonization can greatly accelerate the wide angular sweeping, which involves repeatedly solving many/massive incidents. However, the peak memory usage required by skeletonization may be a bottleneck of that algorithm. In this communication, a message passing interface parallel skeletonization scheme is developed to alleviate the memory usage and to incorporate with the parallel multilevel fast multipole algorithm (MLFMA) for large targets, which are hundreds of wavelengths in size. The parallel scheme includes two ingredients. One is to feed into the ID, a size-reduced matrix to find skeleton right-hand sides. This matrix is produced upon skeleton RWGs instead of their original counterpart. The other is to integrate the parallel ID into the parallel MLFMA to alleviate the peak memory usage of skeletonization on each participating process. Numerical experiments show that the peak memory usage for the skeletonization can be less than 2 GB in each process for the target about 300 wavelengths in size.

A Finite Element Model Order Reduction Technique for Multiscale Electromagnetic Problems

A fast finite element method (FEM) is proposed for electromagnetic computation of recently developed hyperuniform disordered materials having multiscale geometry features. The main idea of our fast FEM is to construct a small dimensional local solution space to approximate the solution space of a multiscale subdomain that requires a large number of unknowns in the conventional FEM. The approximation efficiency is guaranteed by the fact that the multiscale subdomain is usually electrically small at its working frequency and is densely meshed due to the fine geometry and material features. To further reduce the size of the solution space, the interpolative decomposition approach is employed to adaptively choose the number of skeleton vectors for a given accuracy. The solution space is then used as basis functions for the multiscale subdomain instead of using conventional ones in the finite element analysis to reduce the total number of unknowns. We present some numerical examples to demonstrate the accuracy and efficiency of our method.

Literature survey on low rank approximation of matrices

Low rank approximation of matrices has been well studied in literature. Singular value decomposition, QR decomposition with column pivoting, rank revealing QR factorization, Interpolative decomposition, etc. are classical deterministic algorithms for low rank approximation. But these techniques are very expensive ( operations are required for matrices). There are several randomized algorithms available in the literature which are not so expensive as the classical techniques (but the complexity is not linear in n). So, it is very expensive to construct the low rank approximation of a matrix if the dimension of the matrix is very large. There are alternative techniques like Cross/Skeleton approximation which gives the low-rank approximation with linear complexity in n. In this article we review low rank approximation techniques briefly and give extensive references of many techniques.

An Efficient Multicore Implementation of a Novel HSS-Structured Multifrontal Solver Using Randomized Sampling

We present a sparse linear system solver that is based on a multifrontal variant of Gaussian elimination and exploits low-rank approximation of the resulting dense frontal matrices. We use hierarchically semiseparable (HSS) matrices, which have low-rank off-diagonal blocks, to approximate the frontal matrices. For HSS matrix construction, a randomized sampling algorithm is used together with interpolative decompositions. The combination of the randomized compression with a fast ULV HSS factorization leads to a solver with lower computational complexity than the standard multifrontal method for many applications, resulting in speedups up to sevenfold for problems in our test suite. The implementation targets many-core systems by using task parallelism with dynamic runtime scheduling. Numerical experiments show performance improvements over state-of-the-art sparse direct solvers. The implementation achieves high performance and good scalability on a range of modern shared memory parallel systems, including ...

$\mathcal {O}(N)$ Nested Skeletonization Scheme for the Analysis of Multiscale Structures Using the Method of Moments

We present a scheme to compress the method of moments (MoM) matrix, which is linear in complexity for low to intermediate frequency problems in electromagnetics. The method is fully kernel independent and easy to implement using existing codes. The O(N) complexity for both memory and time (setup and matrix-vector product) is achieved thanks to the application of a recursive skeletonization to the H 2 matrix structure of the MoM matrix that uses the nested nature of the far interactions. The interpolative decomposition is applied in a novel manner in order to compress the “far-field signature” of the groups of basis functions. Moreover the scheme is fully characterized and it proves itself well suited for the analysis of multiscale structures.

Fast Solution of Linear Systems With Many Right-Hand Sides Based on Skeletonization

For applications with many right-hand sides (RHSs), to avoid the solution of m different RHSs repeatedly, previous work has employed the interpolative decomposition (ID) to figure out skeleton RHSs by exploiting the rank deficient property of the n ×m RHS matrix B for a system with n unknowns. However, the peak memory requirement for the skeletonization may degrade the performance of the previously developed algorithms when n becomes very large. To alleviate the associated difficulty, a strategy is proposed to construct a submatrix of B to figure out skeleton RHSs. Numerical experiments on different applications show the accuracy and efficiency of the proposed algorithms.

Far-field compression for fast kernel summation methods in high dimensions

Abstract We consider fast kernel summations in high dimensions: given a large set of points in d dimensions (with d ≫ 3 ) and a pair-potential function (the kernel function), we compute a weighted sum of all pairwise kernel interactions for each point in the set. Direct summation is equivalent to a (dense) matrix–vector multiplication and scales quadratically with the number of points. Fast kernel summation algorithms reduce this cost to log-linear or linear complexity. Treecodes and Fast Multipole Methods (FMMs) deliver tremendous speedups by constructing approximate representations of interactions of points that are far from each other. In algebraic terms, these representations correspond to low-rank approximations of blocks of the overall interaction matrix. Existing approaches require an excessive number of kernel evaluations with increasing d and number of points in the dataset. To address this issue, we use a randomized algebraic approach in which we first sample the rows of a block and then construct its approximate, low-rank interpolative decomposition. We examine the feasibility of this approach theoretically and experimentally. We provide a new theoretical result showing a tighter bound on the reconstruction error from uniformly sampling rows than the existing state-of-the-art. We demonstrate that our sampling approach is competitive with existing (but prohibitively expensive) methods from the literature. We also construct kernel matrices for the Laplacian, Gaussian, and polynomial kernels—all commonly used in physics and data analysis. We explore the numerical properties of blocks of these matrices, and show that they are amenable to our approach. Depending on the data set, our randomized algorithm can successfully compute low rank approximations in high dimensions. We report results for data sets with ambient dimensions from four to 1,000.

Randomized interpolative decomposition of separated representations

We introduce an algorithm to compute tensor interpolative decomposition (dubbed CTD-ID) for the reduction of the separation rank of Canonical Tensor Decompositions (CTDs). Tensor ID selects, for a user-defined accuracy ϵ, a near optimal subset of terms of a CTD to represent the remaining terms via a linear combination of the selected terms. CTD-ID can be used as an alternative to or in combination with the Alternating Least Squares (ALS) algorithm. We present examples of its use within a convergent iteration to compute inverse operators in high dimensions. We also briefly discuss the spectral norm as a computational alternative to the Frobenius norm in estimating approximation errors of tensor ID.We reduce the problem of finding tensor IDs to that of constructing interpolative decompositions of certain matrices. These matrices are generated via randomized projection of the terms of the given tensor. We provide cost estimates and several examples of the new approach to the reduction of separation rank.

Accurate and Efficient Evaluation of Spatial Electromagnetic Responses of Large Scale Targets

An error controllable algorithm is proposed to efficiently evaluate two-dimensional (2-D) monostatic radar cross section (RCS) from electrically large and complex targets. The algorithm employs interpolative decomposition (ID) to conduct low-rank decomposition on the excitation matrix consisting of multiple right-hand-sides (RHS's) to figure out the so-called skeleton incidents. After the solutions corresponding to skeletons are obtained, the complete angular responses can be recovered efficiently. The proposed algorithm is error controllable because the accuracy/resolution of the excitation matrix and ID can both be manipulated. The algorithm is efficient in terms of CPU time due to the high efficiency of ID. For large scale problems, the number of unknowns and that of incidents become large, which, in turn, would lead to a huge excitation matrix. A multiinterval variation is proposed to overcome the possible bottle-neck on memory usage. The proposed method is user-friendly because all the tunable parameters are problem independent. Numerical experiments on electrically large and complex targets have been performed to show the performance of the proposed algorithm.

Efficient Wide-Band Evaluation of Electromagnetic Wave Scattering From Complex Targets

A skeleton-based wide-band algorithm (SBWBA) is proposed to efficiently evaluate radar cross section (RCS) from complex targets over a wide frequency band. The algorithm begins with generating a matrix, ~I, from which skeleton frequencies (SFs) are extracted by the proposed skeleton frequency extraction (SFE) strategy. The SFE strategy employs an adaptive manner to select SFs through the interpolative decomposition (ID) skeletonization. After the RCS results associated with the SFs are obtained, the complete wide-band response can be recovered through the ID efficiently. An inherent requirement for the employment of ~I and the SFE strategy is that the equivalent currents vary smoothly and slowly with frequency, which is satisfied only when the frequency band is sufficiently narrow. A multiinterval SFE strategy is developed as a remedy by automatically dividing the wide frequency band into several relatively narrow ones. Numerical experiments on complex targets have been performed to show efficiency and accuracy of the proposed SBWBA.

Multidomain, sparse, spectral-tau method for helically symmetric flow

We consider the application of a multidomain, sparse, and modal spectral-tau method to the helically reduced Navier Stokes equations describing pipe flow. This work (i) formulates the corresponding modal approximations, (ii) describes improved boundary conditions for the helically reduced equations, and (iii) constructs iterative solutions of the corresponding elliptic problem that arises in the reduction. Regarding (iii), we also present and test a method for preconditioning the matching conditions between subdomains, a method based on statistical sampling and the interpolative decomposition. Although the following application is only discussed in our concluding section, a partial motivation for this work has been our ongoing development of similar spectral methods for the construction binary neutron star spacetimes.

Prediction of radiation pressure force exerted on moving particles by the two-level skeletonization.

A fast full-wave method for computing radiation pressure force (RPF) exerted by shaped light beams on moving particles is presented. The problem of evaluating RPF exerted on a moving particle by a single excitation beam is converted into that of computing RPF's exerted on a static particle by multiple beams. The discretization of different beams leads to distinct right hand sides (RHS's) for the matrix system. To avoid solving each RHS by the brute-force manner, the algorithm conducts low-rank decomposition on the excitation matrix consisting of all RHS's to figure out the so-called skeleton light beams by interpolative decomposition (ID). The peak memory requirement of the skeletonization is a bottle-neck if the particle is large. A two-level skeletonization scheme is proposed to solve this problem. Some numerical experiments on arbitrarily shaped homogeneous particles are performed to illustrate the performance and capability of the developed method.

Parallel Implementation of Fast Randomized Algorithms for Low Rank Matrix Decomposition

We analyze the parallel performance of randomized interpolative decomposition by decomposing low rank complex-valued Gaussian random matrices of about 100 GB. We chose a Cray XMT supercomputer as it provides an almost ideal PRAM model permitting quick investigation of parallel algorithms without obfuscation from hardware idiosyncrasies. We obtain that on non-square matrices performance scales almost linearly with runtime about 100 times faster on 128 processors. We also verify that numerically discovered error bounds still hold on matrices two orders of magnitude larger than those previously tested.

Application of Interpolative Decomposition to FE-BI-MLFMA for Fast Computation of Monostatic Scattering from 3-D Complex Composite Objects

Interpolative decomposition (ID) algorithm is applied to the hybrid finite element-boundary integral-multilevel fast multipole algorithm (FE-BI-MLFMA) for fast computation of monostatic scattering from 3-D complex composite objects in this letter. The sparse skeleton directions are first selected from dense incident directions by employing ID. Then, the scattering fields from these skeleton directions are computed by FE-BI-MLFMA. Finally, scattering fields from any other interested directions can be constructed by the scattering fields from the skeleton directions. The numerical performance of ID for FE-BI-MLFMA has been investigated in detail through computing monostatic scattering by typical complex composite objects. It is observed that the skeleton directions by using ID are nonuniform and far less than directions uniformly selected. Furthermore, the result of skeleton directions by using ID is very robust and almost independent of procedures of using ID. Based on it, a two-level approach of using ID is first presented to reduce memory requirement in using ID in this letter.

Статистичне моделювання сейсмічного шуму в багатовимірній області змінних для визначення частотних характеристик геологічного середовища

The paper deals with the theory and methods of statistical simulation of random processes and fields based on their spectral decomposition and Kotelnikov-Shennon modified interpolation sums, as well as applying these methods for environmental geophysical monitoring. Statistical simulation of multivariate random fields (those homogeneous in time and homogeneous isotropic in n other variables) are considered to be essential for seismological research into frequency characteristics of geological media. A statistical model and a numerical algorithm of simulating random fields are built on the basis of Kotelnikov-Shennon modified interpolation decomposition to generate adequate realizations of seismic noise. The paper examines real-valued random fields ξ(t,x),tϵÎR,xÎRn, those homogeneous in time and homogeneous isotropic ones relative to spatial variables in the multidimensional space. It also considers approximation of random fields by the random fields with a bounded spectrum. There is made an analogue of the Kotelnikov–Shannon theorem for random fields with a bounded spectrum. Besides, there are obtained estimates of the mean-square approximation of random fields in the space R´Rn by a model constructed with the help of spectral decomposition and Kotelnikov–Shannon interpolation formula. The paper provides a mechanism for statistical simulation of Gaussian random fields with a bounded spectrum; namely, those homogeneous in time and homogeneous isotropic ones relative to spatial variables in the multidimensional space. Proved have been the theorems of the mean-square approximation of random fields (those homogeneous in time and homogeneous isotropic ones relative to n- other variables) by special partial sums. A simulation method was used to formulate an algorithm of numerical simulation by means of these theorems. There are also considered ways to carry out spectral analysis of generated seismic noise realizations. Finally, there have been developed universal methods of statistical simulation (Monte Carlo methods) of multi-parameter seismology data for generating seismic noise on 2D and 3D grids of the required detail and regularity.

Preconditioning Technique in the Interpolative Decomposition Multilevel Fast Multipole Algorithm

The potential of the interpolative decomposition multilevel fast multipole algorithm (ID-MLFMA) on developing an effective preconditioning technique for multiscale, dynamic electromagnetic problems are analyzed. The preconditioner based on multilevel inverse-based ILU is developed for ID-MLFMA. The proposed preconditioning technique is investigated by numerical experiments on complex targets.

HIERARCHICAL INTERPOLATIVE DECOMPOSITION MULTILEVEL FAST MULTIPOLE ALGORITHM FOR DYNAMIC ELECTROMAGNETIC SIMULATIONS

A hierarchical interpolative decomposition multilevel fast multipole algorithm (ID-MLFMA) is proposed to handle multiscale, dynamic electromagnetic problems. The hierarchical scheme to conduct the ID skeletonization and to implement the matrix vector multiplication is discussed. A strategy to improve the e-ciency of ID skeletonization is developed. The hierarchical ID-MLFMA are investigated by numerical experiments on complex targets, demonstrating the capability of the hierarchical ID-MLFMA.

A fast algorithm for multiscale electromagnetic problems using interpolative decomposition and multilevel fast multipole algorithm

The interpolative decomposition (ID) is combined with the multilevel fast multipole algorithm (MLFMA), denoted by ID‐MLFMA, to handle multiscale problems. The ID‐MLFMA first generates ID levels by recursively dividing the boxes at the finest MLFMA level into smaller boxes. It is specifically shown that near‐field interactions with respect to the MLFMA, in the form of the matrix vector multiplication (MVM), are efficiently approximated at the ID levels. Meanwhile, computations on far‐field interactions at the MLFMA levels remain unchanged. Only a small portion of matrix entries are required to approximate coupling among well‐separated boxes at the ID levels, and these submatrices can be filled without computing the complete original coupling matrix. It follows that the matrix filling in the ID‐MLFMA becomes much less expensive. The memory consumed is thus greatly reduced and the MVM is accelerated as well. Several factors that may influence the accuracy, efficiency and reliability of the proposed ID‐MLFMA are investigated by numerical experiments. Complex targets are calculated to demonstrate the capability of the ID‐MLFMA algorithm.