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A fast MEMORY EFFICIENT CONSTRUCTION ALGORITHM FOR HIERARCHICALLY SEMI-SEPARABLE REPRESENTATIONS

SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2016年第1期37卷 338-353页

作者： Lessel, K. Hartman, M. Chandrasekaran, S. Univ Calif Santa Barbara Dept Elect & Comp Engn Santa Barbara CA 93106 USA

Existing hierarchically semi-separable construction algorithms for dense n x n matrices require as much as O(n(2)) peak workspace memory, at a cost of O(n(2)) flops. An algorithm is presented which requires O(n(1.5)) ... 详细信息

关键词： fast multipole method hierarchically semi-separable representations fast algorithms orthogonal factorizations low-rank structures memory efficient

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A fast directional algorithm for high-frequency electromagnetic scattering

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JOURNAL OF COMPUTATIONAL PHYSICS 2011年第14期230卷 5471-5487页

作者： Tsuji, Paul Ying, Lexing Univ Texas Austin ICES Austin TX 78712 USA Univ Texas Austin Dept Math Austin TX 78712 USA

This paper is concerned with the fast solution of high-frequency electromagnetic scattering problems using the boundary integral formulation. We extend the O(N log N) directional multilevel algorithm previously proposed for the acoustic scattering case to the vector electromagnetic case. We also detail how to incorporate the curl operator of the magnetic field integral equation into the algorithm. When combined with a standard iterative method, this results in an almost linear complexity solver for the combined field integral equations. In addition, the butterfly algorithm is utilized to compute the far field pattern and radar cross section with O(N log N) complexity. (C) 2011 Elsevier Inc. All rights reserved.

关键词： Electromagnetic scattering Boundary integral equations fast algorithms fast multipole methods Sparse Fourier transforms

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A fast Fourier transform accelerated marching-on-in-time algorithm for electromagnetic analysis

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ELECTROMAGNETICS 2001年第3期21卷 181-197页

作者： Yilmaz, AE Jin, JM Weile, DS Univ Illinois Dept Elect & Comp Engn Ctr Computat Electromagnet Urbana IL 61801 USA Univ Delaware Dept Elect & Comp Engn Newark DE USA

A fast algorithm is presented for solving a time-domain electric field integral equation (EFIE) pertinent to the analysis of scattering from uniformly meshed, perfectly conducting structures. The marching-on-in-time (MOT) scheme that results from discretizing this EFIE is accelerated by using the fast Fourier transform to perform spatial convolutions. The computational cost and storage requirements of this algorithm scale as O(NtNs1.5) and O(N-s(1.5)), respectively, as opposed to O(NtNs2) and O(N-s(2)) for classical MOT methods. Simulation results demonstrate the accuracy and efficiency of the approach and suggestions for extending the technique are proffered.

关键词： time-domain integral equations fast Fourier transform fast algorithms

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fast Grid-Based Refining Segmentation Method in Video-Based Point Cloud Compression

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IEEE ACCESS 2021年 9卷 80088-80099页

作者： Kim, Jieon Kim, Yong-Hwan Korea Elect Technol Inst Seongnam Si 13488 South Korea

The video-based point cloud compression (V-PCC, ISO/IEC 23090-5) is the state-of-the-art international standard for compressing dynamic point clouds developed by the moving picture experts group (MPEG). It has been achieved good rate-distortion (RD) performance by employing the 2D-based dynamic point cloud compression. As a brief look, V-PCC first converts the 3D input point cloud into a set of 2D patches followed by a packing process. The packing process then maps the patches into a 2D grid. Such a way allows compressing the patches utilizing the existing video coding standards. Besides the RD performance, complexity is another vital factor to consider in performance evaluations. In the V-PCC encoder, the self-time accounts for on average 15.9% and a maximum of 48.2% of the total-time, which can be a hindrance to realizing real-time V-PCC applications. One of the most computationally intensive modules of V-PCC is the grid-based refining segmentation (G-RS). Thus this paper proposes a fast G-RS method that can adaptively select voxels that need the refining segmentation. More concretely, the proposed method classifies the voxels based on the projection plane indices of 3D points and only applies the refining process to the selected voxels. Experimental results demonstrate that the proposed method reduces the complexity of the refining steps in G-RS, on average, by 60.7% and 62.5% without coding efficiency loss compared to the test model for category 2 (TMC2) version 12.0 reference software under the random access (RA) and all-intra (AI) configurations, respectively.

关键词： Three-dimensional displays Complexity theory Software Encoding Standards Geometry Transform coding Video-based point cloud compression dynamic point cloud fast algorithms low-complex video encoders voxel-based refining segmentation encoder optimization

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fast regression methods in a Lanczos (or PLS-1) basis. Theory and applications

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CHEMOMETRICS AND INTELLIGENT LABORATORY SYSTEMS 2000年第2期51卷 145-161页

作者： Wu, W Manne, R Univ Bergen Dept Chem NO-5007 Bergen Norway Free Univ Brussels Inst Pharmaceut ChemoAC B-1090 Brussels Belgium

In order to improve the calibration speed for very large data sets, novel algorithms for principal component regression (PCR) and partial-least-squares (PLS) regression are presented. They use the Lanczos or PLS-1 transformation to reduce the data matrix X to a small bidiagonal matrix (R), after which the small tridiagonal matrix (R'R) is diagonalized and inverted. The complexity of the PCR model may be optimized by cross-validation (PCRL) but also using simpler and faster recipes based upon sound-off monitoring and model fit (PCRF). A similar fast PLS procedure (PLSF) is also presented. Calculations are made for five near infrared spectroscopy (NIR) data sets and compared with PCR with feature selection (PCRS) based on correlation and with de Jong's simple partial least squares (SIMPLS). The Lanczos-based methods have comparable prediction performance and similar model complexity to PCRS and SIMPLS but are considerably faster. From a detailed comparison of the methods, some insight is gained into the performance of the PLS method. (C) 2000 Elsevier Science B.V. All rights reserved.

关键词： PCR PLS lanczos fast algorithms fast optimization calibration NIR

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A fast ALGORITHM FOR EXTRAPOLATION OF DISCRETE-TIME PERIODIC BAND-LIMITED SIGNALS

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SIGNAL PROCESSING 1993年第2期33卷 183-196页

作者： SOLTANIANZADEH, H YAGLE, AE UNIV MICHIGAN DEPT ELECT ENGN & COMP SCIANN ARBORMI 48109

We present a fast algorithm for exact extrapolation of a discrete-time and periodic band-limited signal from its values in an interval having the same length as the band-width of the signal. Here band-width is the number of non-zero values in one period of the discrete Fourier transform of the signal. Applications to problems in which the number of given values is unequal to the band-width are also given. The procedure is a simple autoregression on the time-domain values of the signal, and is much simpler than previous algorithms for discrete-discrete extrapolation. which required computation of a large pseudo-inverse. The procedure is highly parallelizable, and the computational savings are especially significant for multidimensional signal extrapolation. Numerical examples for 1-D and 2-D extrapolation demonstrate (1) that the procedure works perfectly in the absence of noise, and (2) that it works well in the presence of band-limited noise. In the presence of wide-band noise the procedure breaks down, due to ill-posedness of the problem;some regularization techniques are proposed.

关键词： BAND-LIMITED EXTRAPOLATION SIGNAL RECONSTRUCTION MULTIDIMENSIONAL SIGNAL RECONSTRUCTION fast algorithms

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A fast DIRECT SOLVER FOR STRUCTURED LINEAR SYSTEMS BY RECURSIVE SKELETONIZATION

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2012年第5期34卷 A2507-A2532页

作者： Ho, Kenneth L. Greengard, Leslie NYU Courant Inst Math Sci New York NY 10012 USA NYU Program Computat Biol New York NY USA

We present a fast direct solver for structured linear systems based on multilevel matrix compression. Using the recently developed interpolative decomposition of a low-rank matrix in a recursive manner, we embed an approximation of the original matrix into a larger but highly structured sparse one that allows fast factorization and application of the inverse. The algorithm extends the Martinsson-Rokhlin method developed for 2D boundary integral equations and proceeds in two phases: a precomputation phase, consisting of matrix compression and factorization, followed by a solution phase to apply the matrix inverse. For boundary integral equations which are not too oscillatory, e. g., based on the Green functions for the Laplace or low-frequency Helmholtz equations, both phases typically have complexity O(N) in two dimensions, where N is the number of discretization points. In our current implementation, the corresponding costs in three dimensions are O(N-3/2) and O(N log N) for precomputation and solution, respectively. Extensive numerical experiments show a speedup of similar to 100 for the solution phase over modern fast multipole methods;however, the cost of precomputation remains high. Thus, the solver is particularly suited to problems where large numbers of iterations would be required. Such is the case with ill-conditioned linear systems or when the same system is to be solved with multiple right-hand sides. Our algorithm is implemented in Fortran and freely available.

关键词： fast algorithms multilevel matrix compression interpolative decomposition sparse direct solver integral equations fast multipole method

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Geodesic Matting: A Framework for fast Interactive Image and Video Segmentation and Matting

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INTERNATIONAL JOURNAL OF COMPUTER VISION 2009年第2期82卷 113-132页

作者： Bai, Xue Sapiro, Guillermo Univ Minnesota Minneapolis MN 55455 USA

An interactive framework for soft segmentation and matting of natural images and videos is presented in this paper. The proposed technique is based on the optimal, linear time, computation of weighted geodesic distances to user-provided scribbles, from which the whole data is automatically segmented. The weights are based on spatial and/or temporal gradients, considering the statistics of the pixels scribbled by the user, without explicit optical flow or any advanced and often computationally expensive feature detectors. These could be naturally added to the proposed framework as well if desired, in the form of weights in the geodesic distances. An automatic localized refinement step follows this fast segmentation in order to further improve the results and accurately compute the corresponding matte function. Additional constraints into the distance definition permit to efficiently handle occlusions such as people or objects crossing each other in a video sequence. The presentation of the framework is complemented with numerous and diverse examples, including extraction of moving foreground from dynamic background in video, natural and 3D medical images, and comparisons with the recent literature.

关键词： Interactive image and video segmentation Matting Geodesic computations Weighted distance functions fast algorithms User-provided scribbles

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A fast SEMIDIRECT LEAST SQUARES ALGORITHM FOR HIERARCHICALLY BLOCK SEPARABLE MATRICES

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2014年第2期35卷 725-748页

作者： Ho, Kenneth L. Greengard, Leslie NYU Courant Inst Math Sci New York NY 10012 USA NYU Program Computat Biol New York NY 10012 USA

We present a fast algorithm for linear least squares problems governed by hierarchically block separable (HBS) matrices. Such matrices are generally dense but data sparse and can describe many important operators including those derived from asymptotically smooth radial kernels that are not too oscillatory. The algorithm is based on a recursive skeletonization procedure that exposes this sparsity and solves the dense least squares problem as a larger, equality-constrained, sparse one. It relies on a sparse QR factorization coupled with iterative weighted least squares methods. In essence, our scheme consists of a direct component, comprised of matrix compression and factorization, followed by an iterative component to enforce certain equality constraints. At most two iterations are typically required for problems that are not too ill-conditioned. For an M x N HBS matrix with M >= N having bounded off-diagonal block rank, the algorithm has optimal O(M + N) complexity. If the rank increases with the spatial dimension as is common for operators that are singular at the origin, then this becomes O(M + N) in one dimension, O(M + N-3/2) in two dimensions, and O(M + N-2) in three dimensions. We illustrate the performance of the method on both overdetermined and underdetermined systems in a variety of settings, with an emphasis on radial basis function approximation and efficient updating and downdating.

关键词： fast algorithms matrix compression recursive skeletonization sparse QR decomposition weighted least squares deferred correction radial basis functions updating/downdating

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Parallel- and Cyclic-Iterative Variational Bayes for fast Kalman Filtering in Large-Dimensions

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2022年 70卷 5871-5884页

作者： Ait-El-Fquih, Boujemaa Hoteit, Ibrahim King Abdullah Univ Sci & Technol KAUST Div Phys Sci & Engn Thuwal 23955 Saudi Arabia

This paper considers the linear-Gaussian filtering problem in large-dimensions, a framework in which the Kalman filter (KF) can be computationally prohibitive. As a remedy, a hybrid scheme combining KF with variational Bayes (VB), an approach that designs an approximation of the filtering probability density function (pdf) based on a separable form under the Kullback-Leibler divergence minimization criterion, has been proposed. Despite its approximate and cyclic-iterative characters, this VBKF algorithm can provide comparable performances to the KF, at reduced computational requirements. Here, we resort to the memory gradient subspace (MGS) optimization in the space of separable pdfs to derive a new variant of VBKF endowed with a parallel-iterative structure. At a given iteration of the proposed MGS-VBKF, all separate marginals of the approximate filtering pdf are updated in parallel, which could lead to significant computational time savings, especially when the state is split into small enough partitions. We establish the connection between VBKF and MGS-VBKF, showing that the latter can be derived from the former by iterating instead in parallel on its equations and inserting a correction term representing the contribution of the gradient and memory directions. We further extend these algorithms to the case of unsupervised systems with unknown observation noise uncertainties. The performances of the proposed filters are finally studied and compared to those of the KF and a deterministic ensemble Kalman filter through numerical experiments.

关键词： Unsupervised Bayesian filtering Kalman filtering variational Bayes high dimension fast algorithms subspace optimization

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