Adaptive Cross Approximation Method Research Articles

A Hybrid Method of Adaptive Cross Approximation Algorithm and Chebyshev Approximation Technique for Fast Broadband BCS Prediction Applicable to Passive Radar Detection

A hybrid method combining the adaptive cross approximation method (ACA) and the Chebyshev approximation technique (CAT) is presented for fast wideband BCS prediction of arbitrary-shaped 3D targets based on non-cooperative radiation sources. The incident and scattering angles can be computed by using their longitudes, latitudes and altitudes according to the relative positions of the satellite, the target and the passive bistatic radar. The ACA technique can be employed to reduce the memory requirement and computation time by compressing the low-rank matrix blocks. By exploiting the CAT into ACA, it is only required to calculate the currents at several Chebyshev–Gauss frequency sampling points instead of direct point-by-point simulations. Moreover, a wider frequency band can be obtained by using the Maehly approximation. Three numerical examples are presented to validate the accuracy and efficiency of the hybrid ACA-CAT method.

Multilevel Kernel Degeneration–Adaptive Cross Approximation Method to Model Eddy Current NDE Problems

Multilevel Kernel Degeneration–Adaptive Cross Approximation Method to Model Eddy Current NDE Problems

Hybrid Method of ACA and Interpolation-based Algorithms for Analysis of Eddy Current Nondestructive Evaluations

ABSTRACT In this article, the hybridization of adaptive cross approximation (ACA) algorithm and interpolation-based separation of the kernel function is proposed to accelerate solving the matrix equations resulted in the boundary element method (BEM) for 3D arbitrary-shaped eddy current nondestructive evaluation problems. The hybrid method combines the advantages of both the ACA algorithm and the interpolation-based methods, and resolves the shortcoming of pure ACA method, when modeling the planar eddy current nondestructive evaluation problems, that it cannot compress the null entries the BEM generated when the testing and basis patches are co-planar. In the proposed method, the submatrices associated with the null entries are compressed by the interpolation-based method, while the others are compressed by the ACA algorithm. Several benchmarks are shown to demonstrate both the robustness and efficiency of the proposed fast and general solver.

Analysis of multiple inclusion potential problems by the Adaptive Cross Approximation method

Analysis of multiple inclusion potential problems by the Adaptive Cross Approximation method

Skin effect and proximity effect analysis of stranded conductor based on mixed order MoM with adaptive cross approximation algorithm

In this paper, an efficient adaptive cross approximation (ACA) method is employed for the skin effect and proximity effect analysis. Multi-layer stranded cables with multiple strands and different materials are used for analysis to approximate the effect. The skin and proximity matrices are computed by the ACA low-rank matrix approximation technique. The ACA method can provide better solutions through different iterations. The ACA algorithm is very stable and useful for calculating a low rank matrix approximation, because the adaptive mesh refinement for generating optimal computation. The ACA approach provides the less computational memory and less CPU time. Compare the calculated results with standard work results (such as reference) to verify the effectiveness of the adaptive cross-approximation algorithm.

A fast solver for spectral elements applied to fractional differential equations using hierarchical matrix approximation

We develop a fast solver for the spectral element method (SEM) applied to the two-sided fractional diffusion equation on uniform, geometric and graded meshes. By approximating the singular kernel with a degenerate kernel we construct a hierarchical matrix (H-matrix) to represent the stiffness matrix of the SEM and provide error estimates, which we verify numerically. We can solve efficiently the H-matrix approximation problem using a hierarchical LU (H-LU) decomposition method, which reduces the computational cost to O(r2Ndlog2N)+O(r3NdlogN), where r is the rank of the H-LU decomposition, Nd is the total number of degrees of freedom and N is the number of elements. However, we lose the high accuracy of the SEM. To prevent this, we solve the corresponding preconditioned system by using the H-matrix approximation problem as a preconditioner, hence, recovering the high-order accuracy of the SEM. Numerical results show that the condition number of the preconditioned system is independent of the polynomial degree P and grows with the number of elements, but at modest values of the rank R (here R is the rank of submatrices of the H-matrix approximation) it is below order 10 in our experiments, which represents a reduction of more than 11 orders of magnitude from the unpreconditioned system. This reduction is higher in the two-sided fractional derivative compared to one-sided fractional derivative. The corresponding cost is O(r2Ndlog2N)+O(r3NdlogN)+O(Nd2). Moreover, by using a structured mesh (uniform or geometric mesh), we can further reduce the computational cost to O(r2Ndlog2N)+O(r3NdlogN)+O(P2NlogN) for the preconditioned system. Furthermore, we compare the present method with the adaptive cross approximation (ACA), which is kernel independent and relies on the coefficient matrix. The comparison shows that the accuracy using the present algorithm is better than the one using ACA for the H-matrix approximation problem, and furthermore, we have observed that the ACA method may fail to converge in some cases.

Certified and fast computations with shallow covariance kernels

Many techniques for data science and uncertainty quantification demand efficient tools to handle Gaussian random fields, which are defined in terms of their mean functions and covariance operators. Recently, parameterized Gaussian random fields have gained increased attention, due to their higher degree of flexibility. However, especially if the random field is parameterized through its covariance operator, classical random field discretization techniques fail or become inefficient. In this work we introduce and analyze a new and certified algorithm for the low-rank approximation of a parameterized family of covariance operators which represents an extension of the adaptive cross approximation method for symmetric positive definite matrices. The algorithm relies on an affine linear expansion of the covariance operator with respect to the parameters, which needs to be computed in a preprocessing step using, e.g., the empirical interpolation method. We discuss and test our new approach for isotropic covariance kernels, such as Mat\'ern kernels. The numerical results demonstrate the advantages of our approach in terms of computational time and confirm that the proposed algorithm provides the basis of a fast sampling procedure for parameter dependent Gaussian random fields.

Distributed fast boundary element methods for Helmholtz problems

We present an approach for a distributed memory parallelization of the boundary element method. The given mesh is decomposed into submeshes and the respective matrix blocks are distributed among computational nodes (processes). The distribution which takes care of the load balancing during the system matrix assembly and matrix-vector multiplication is based on the cyclic graph decomposition. Moreover, since the individual matrix blocks are approximated using the adaptive cross approximation method, we describe its modification capable of dealing with zero blocks in the double-layer operator matrix since these are usually problematic when using the original adaptive cross approximation algorithm. Convergence and scalability of the method are demonstrated on the half- and full-space sound scattering problems modeled by the Helmholtz equation.

Iterative FEBI-PO Hybrid Method for Analyzing Scattering Problems of Complex Structures Having Junctions With Large-Scale Platform

An iterative hybrid of finite element boundary integral (FEBI) method and physical optics (PO) is presented for scattering problems of complex structures having junctions with electrically large-scale platform. The treatment of basis functions on the junctions between complex structures and platform is discussed in detail. Moreover, the multilevel fast multipole algorithm and the adaptive cross approximation method are employed to accelerate the solution of FEBI and the computation of interaction between FEBI and PO regions, respectively. Compared with FEBI method, the hybrid method can efficiently reduce the number of unknowns and the computational complexity while preserving acceptable accuracy. In addition, the alternative implementation of FEBI and PO can also maintain good independence of the two methods. Some numerical results are discussed to demonstrate the efficiency and accuracy of the approach.

An Adaptive Cross Approximation Method for the Hamilton-Jacobi-Bellman Equation * *This work is supported by the National Natural Science Foundation of China (No. 61473233)

Abstract The Hamilton-Jacobi-Bellman (HJB) equation can provide closed-loop optimal control feedback, but numerical methods for the HJB equation usually suffer from heavy computational burden. In this paper, an adaptive cross approximation (ACA) based method is proposed to alleviate the heavy computational cost. Firstly, when the HJB equation is discretized and solved on uniform grids, it is shown that under certain assumptions the value function tensor can be obtained elementwise by solving the corresponding two-point boundary value problem (TPBVP). Secondly, the value function tensor of the HJB equation is reshaped into a matrix to incorporate the adaptive cross approximation method. Given an error tolerance, the adaptive cross approximation method can reconstruct the value function tensor with only a small subset of the entries. In the simulation for the four-dimensional spacecraft attitude stabilization problem, compared with calculating the full value function tensor, the proposed method achieves about three orders of magnitude improvement in both computation time and memory requirement. Real-time closed-loop receding horizon control is implemented based on the obtained value function, and it is also compared with the open-loop trajectory. Numerical results show the effectiveness and efficiency of the proposed method.

Matrix valued adaptive cross approximation

A new variant of the Adaptive Cross Approximation (ACA) for approximation of dense block matrices is presented. This algorithm can be applied to matrices arising from the Boundary Element Methods (BEM) for elliptic or Maxwell systems of partial differential equations. The usual interpolation property of the ACA is generalised for the matrix valued case. Some numerical examples demonstrate the efficiency of the new method. The main example will be the electromagnetic scattering problem, that is, the exterior boundary value problem for the Maxwell system. Here, we will show that the matrix valued ACA method works well for high order BEM, and the corresponding high rate of convergence is preserved. Another example shows the efficiency of the new method in comparison with the standard technique, whilst approximating the smoothed version of the matrix valued fundamental solution of the time harmonic Maxwell system. Copyright © 2016 John Wiley & Sons, Ltd.

Wideband nested cross approximation for Helmholtz problems

In this article, the construction of nested bases approximations to discretizations of integral operators with oscillatory kernels is presented. The new method has log-linear complexity and generalizes the adaptive cross approximation method to high-frequency problems. It allows for a continuous and numerically stable transition from low to high frequencies.

Constructing nested bases approximations from the entries of non-local operators

In this article, a method for constructing nested bases approximations to large-scale fully populated discretizations of integral operators is introduced. The scheme uses only few of the matrix entries for approximating the whole matrix. In this sense, it is similar to the adaptive cross approximation method. However, its computational complexity is improved.

Recent Advances and Emerging Applications of the Boundary Element Method

Sponsored by the U.S. National Science Foundation, a workshop on the boundary element method (BEM) was held on the campus of the University of Akron during September 1–3, 2010 (NSF, 2010, “Workshop on the Emerging Applications and Future Directions of the Boundary Element Method,” University of Akron, Ohio, September 1–3). This paper was prepared after this workshop by the organizers and participants based on the presentations and discussions at the workshop. The paper aims to review the major research achievements in the last decade, the current status, and the future directions of the BEM in the next decade. The review starts with a brief introduction to the BEM. Then, new developments in Green's functions, symmetric Galerkin formulations, boundary meshfree methods, and variationally based BEM formulations are reviewed. Next, fast solution methods for efficiently solving the BEM systems of equations, namely, the fast multipole method, the pre-corrected fast Fourier transformation method, and the adaptive cross approximation method are presented. Emerging applications of the BEM in solving microelectromechanical systems, composites, functionally graded materials, fracture mechanics, acoustic, elastic and electromagnetic waves, time-domain problems, and coupled methods are reviewed. Finally, future directions of the BEM as envisioned by the authors for the next five to ten years are discussed. This paper is intended for students, researchers, and engineers who are new in BEM research and wish to have an overview of the field. Technical details of the BEM and related approaches discussed in the review can be found in the Reference section with more than 400 papers cited in this review.

A fast integral equation method for simulating high-field radio frequency coil arrays in magnetic resonance imaging

A fast full-wave numerical approach was developed for simulating high-field multi-channel radio-frequency (RF) receive coil arrays in magnetic resonance imaging. To improve the efficiency, the impedance matrix was compressed by a multilevel adaptive cross approximation method. Furthermore, careful organization of multiple coil simulations was applied so that the impedance matrix associated with biological subjects is constructed and pre-conditioned only once. Numerical examples demonstrate the efficacy of the proposed approach for RF coil simulations.

Randomised pseudo-skeleton approximation and its application in electromagnetics

A very simple and efficient algorithm called randomised pseudo-skeleton approximation is proposed for the decomposition of a low-rank matrix. The complexity of the algorithm is O(r3), where r is the rank of the matrix. Comparison with the adaptive cross-approximation method is presented to show its validity and efficiency.

Recompression techniques for adaptive cross approximation

The adaptive cross approximation method (ACA) generates low-rank approximations to suitable m× n sub-blocks of discrete integral formulations of elliptic boundary value problems. A characteristic property is that the approximation, which requires k(m+ n), k ∼ | log e|∗, units of storage, is generated in an adaptive and purely algebraic manner using only few of the matrix entries. In this article we present further recompression techniques which are based on ACA and bring the required amount of storage down to sublinear order kk′, where k′ depends logarithmically on the accuracy of the approximation but is independent of the matrix size. The additional compression is due to a certain smoothness of the vectors generated by ACA.

Efficient Sampling of Electromagnetic Fields via the Adaptive Cross Approximation

The adaptive cross approximation method is applied to the problem of efficiently sampling an electromagnetic field across two dimensions. The algorithm uses previously collected samples to determine which field samples to take next and reduces the number of samples required to accurately represent the field. In this way the field may be sampled faster than it would have been if it had been raster scanned. This is achieved without significant changes to the sensor hardware.

Efficient crack growth analyzes by combining fast methods for BEM

AbstractThe combination of fast methods for the boundary element method (BEM) for efficient crack growth analyzes is presented. Due to the nonlinearity of fatigue crack growth an incremental procedure has to be applied. Within each increment a stress analysis is needed. Based on the asymptotic stress field the stress intensity factors (SIFs) are calculated by an extrapolation method. Then, a new crack front is determined by a reliable 3D crack growth criterion. Finally, the numerical model has to be updated for the next increment. The time dominant factor in each increment is the computation of the stress field. Due to the stress concentration problem the BEM is utilized. To speed‐up the calculation several independent fast methods are exploited. An algebraic technique is the adaptive cross approximation (ACA) method which is acting on the system matrix itself. The application of the substructure technique leads to a blockwise band matrix and therefore to reduced memory requirements. Further savings in memory and computation time are reached by modelling cracks with the dual discontinuity method (DDM) and using the ACA method in each substructure. The efficiency of the combined methods is shown by a complex industrial example. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Two efficient methods for a multifrequency solution of the Helmholtz equation

In this paper, two efficient methods for the multifrequency analysis for the Helmholtz equation are compared. The first approach is based on the direct collocation method with a subsequent approximation of the matrix by the Adaptive Cross Approximation method. Using the so-called Fourier scheme, the elements of the matrices for a series of frequencies can be computed efficiently. In the second approach an indirect Galerkin type method with piecewise linear ansatz and test functions is combined with a special source simulation technique. This combination allows a rather accurate and systematic approximation of acoustical results in major parts of the considered frequency range, leading to a significant reduction of the computer time needed to calculate complete frequency spectra. A representative example demonstrates how the two proposed approaches can be used. Both procedures turn out to be very promising steps towards a more efficient calculation of complex sound radiation problems.