Fast Direct Algorithm Research Articles

A fast semi-analytical meshless method in two-dimensions

This paper introduces a fast direct algorithm for the singular boundary method (SBM) in two-dimensional (2D) problems, utilizing the hierarchical off-diagonal low-rank (HODLR) matrix concept as the foundation of the fast direct solver. The HODLR matrix is constructed by hierarchically partitioning the coefficient matrix into blocks using a binary tree, with all off-diagonal blocks at each level being represented as low-rank factors. Furthermore, The Sherman-Morrison-Woodbury formula can be used to efficiently compute the inverse of the HODLR matrix. The numerical experiment results demonstrate that the new fast solver can significantly improve the computational efficiency of the SBM while maintaining the same level of precision.

Enhanced Global Optimization With Parallel Global and Local Structures for Real-Time Control Systems

In practice, objective functions of real-time control systems can have multiple local minimums or can dramatically change over the function space, making them hard to optimize. To efficiently optimize such systems, in this paper, we develop a parallel global optimization framework that combines direct search methods with parallel Bayesian optimization. It consists of an iterative global and local search that searches broadly through the entire global space for promising regions and then efficiently exploits each local promising region. We prove the asymptotic convergence properties of the proposed framework and conduct several numerical experiments to illustrate its empirical performance. We also provide a real-time control problem to illustrate the efficiency of our proposed algorithm. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This work is motivated by a collision avoidance problem of vessels aided with onboard agent-based simulations. The simulation on one vessel can predict potential conflicts with other vessels on a pre-defined trajectory. In heavy congestion regions, the environment is highly dynamic and thus it is difficult to find a much safer alternative trajectory if collision is predicted on the current one. Moreover, for such real-time decisions, the control system should be quick in response to improve safety. The proposed metamodel based algorithm is designed for quick decision in such highly dynamic systems. The algorithm employs a decomposition of the response surface to better handle the multi-modal surface resulting from the highly dynamic environment. Specifically, it first looks at the large-scale trend globally (filter out the many local fluctuations that may otherwise trap the algorithm) to locate potential promising regions and then proceeds to this local regions for more detailed local search. To make quick decisions, it uses fast direct search algorithms in the local search phase and applies a parallel search scheme to enjoy the abundant computing power. Both the theoretical analysis and the simulation studies demonstrate that the proposed algorithm can provide better decisions quickly. We also note that this algorithm is not limited to real-time control or simulation-based system. In the case where each run of the experiment is expensive and the budget is limited for the final decision and when the response function is multi-modal, this algorithm can hopefully become a quite efficient and competitive approach. The multi-modal responses have broad applications in the area of control, planning and operations research, such as robot navigating and reinforcement learning.

Markov chains improve the significance computation of overlapping genome annotations.

MotivationGenome annotations are a common way to represent genomic features such as genes, regulatory elements or epigenetic modifications. The amount of overlap between two annotations is often used to ascertain if there is an underlying biological connection between them. In order to distinguish between true biological association and overlap by pure chance, a robust measure of significance is required. One common way to do this is to determine if the number of intervals in the reference annotation that intersect the query annotation is statistically significant. However, currently employed statistical frameworks are often either inefficient or inaccurate when computing P-values on the scale of the whole human genome.ResultsWe show that finding the P-values under the typically used ‘gold’ null hypothesis is -hard. This motivates us to reformulate the null hypothesis using Markov chains. To be able to measure the fidelity of our Markovian null hypothesis, we develop a fast direct sampling algorithm to estimate the P-value under the gold null hypothesis. We then present an open-source software tool MCDP that computes the P-values under the Markovian null hypothesis in time and memory, where m and n are the numbers of intervals in the reference and query annotations, respectively. Notably, MCDP runtime and memory usage are independent from the genome length, allowing it to outperform previous approaches in runtime and memory usage by orders of magnitude on human genome annotations, while maintaining the same level of accuracy.Availability and implementationThe software is available at https://github.com/fmfi-compbio/mc-overlaps. All data for reproducibility are available at https://github.com/fmfi-compbio/mc-overlaps-reproducibility.Supplementary information Supplementary data are available at Bioinformatics online.

A fast direct singular boundary method for three-dimensional potential problems

The main purpose of the study is to develop a fast singular boundary method for solving the three-dimensional (3D) potential problems. It is a difficult issue to simulate the large-scale problems by the traditional SBM, because of the fully populated coefficient matrix. A fast direct algorithm based on recursive skeletonization factorization (RSF) is combined in the SBM to reduce the CPU time and memory requirements in this study. For the dense coefficient matrix in the SBM, a hierarchically generalized LU decomposition is constructed to invert it fast and accurately. The numerical results demonstrate the great performance of the RSF-SBM for solving large-scale 3D potential problems.

Fast Fourier Solvers for the Tensor Product High-Order FEM for a Poisson Type Equation

Logarithmically optimal in theory and fast in practice, direct algorithms for implementing a tensor product finite element method (FEM) based on tensor products of 1D high-order FEM spaces on multi-dimensional rectangular parallelepipeds are proposed for solving the N-dimensional Poisson-type equation $$ - \Delta u + \alpha u = f$$ ($$N \geqslant 2$$) with Dirichlet boundary conditions. The algorithms are based on well-known Fourier approaches. The key new points are a detailed description of the eigenpairs of the 1D eigenvalue problems for the high-order FEM, as well as fast direct and inverse eigenvector expansion algorithms that simultaneously employ several versions of the fast Fourier transform. Results of numerical experiments in the 2D and 3D cases are presented. The algorithms can be used in numerous applications, in particular, to implement tensor product high-order finite element methods for various time-dependent partial differential equations, including the multidimensional heat, wave, and Schrodinger ones.

A fast directional boundary element method for wideband multi-domain elastodynamic analysis

A fast directional boundary element method for wideband multi-domain elastodynamic analysis is developed. The directional low rank property of the elastodynamic kernel is proved, which serves as the theoretical basis of the fast directional algorithm. The boundary integral evaluation is accelerated domain by domain. For each domain, the octree is divided into low frequency and high frequency regimes according to its S-wave number. The computation in the low frequency regime is the same with the kernel independent fast multipole method, while the high frequency regime is computed efficiently by applying the parabolic low rank property of the kernels on directional wedges. The matrices on only one directional wedge for each level have to be calculated, since that for other directions can be obtained by coordinate frame rotations. Thus the harmonic responses for any frequencies can be computed efficiently. Numerical results show that the computational complexity for wideband multi-domain elastodynamic problems is successfully brought down to O(NlogαN). Its applications to viscoelastic materials and inverse problems are also validated.

Simulation of two-dimensional steady-state heat conduction problems by a fast singular boundary method

The study presents a fast direct algorithm for solutions of systems arising from singular boundary method (SBM) in two-dimensional (2D) steady-state heat conduction problems. As a strong-form boundary discretization collocation technique, the SBM is mathematically simple, easy-to-implement, and free of mesh. Similar to boundary element method (BEM), the SBM generates a dense coefficient matrix, which requires O(N3) operations and O(N2) memory to solve the system with direct solvers. In this study, a fast direct solver based on recursive skeletonization factorization (RSF) is applied to solve the linear systems in the SBM. Due to the fact that the coefficient matrix is hierarchically block separable, we can construct a multilevel generalized LU decomposition that allows fast application of the inverse of the coefficient matrix. Combing the RSF and the SBM can dramatically reduce the operations and the memory to O(N). The accuracy and effectiveness of the RSF-SBM are tested through some 2D heat conduction problems including isotropic and anisotropic cases.

A fast inverse direct aperture optimization algorithm for intensity-modulated radiation therapy.

The goal of this work was to develop and evaluate a fast inverse direct aperture optimization (FIDAO) algorithm for IMRT treatment planning and plan adaptation. A previously proposed fluence map optimization algorithm called fast inverse dose optimization (FIDO) was extended to optimize the aperture shapes and weights of IMRT beams. FIDO is a very fast fluence map optimization algorithm for IMRT that finds the global minimum using direct matrix inversion without unphysical negative beam weights. In this study, an equivalent second-order Taylor series expansion of the FIDO objective function was used, which allowed for the objective function value and gradient vector to be computed very efficiently during direct aperture optimization, resulting in faster optimization. To evaluate the speed gained with FIDAO, a proof-of-concept algorithm was developed in MATLAB using an interior-point optimization method to solve the reformulated aperture-based FIDO problem. The FIDAO algorithm was used to optimize four step-and-shoot IMRT cases: on the AAPM TG-119 phantom as well as a liver, prostate, and head-and-neck clinical cases. Results were compared with a conventional DAO algorithm that uses the same interior-point method but using the standard formulation of the objective function and its gradient vector. A substantial gain in optimization speed was obtained with the prototype FIDAO algorithm compared to the conventional DAO algorithm while producing plans of similar quality. The optimization time (number of iterations) for the prototype FIDAO algorithm vs the conventional DAO algorithm was 0.3s (17) vs 56.7s (50); 2.0s (28) vs 134.1s (57); 2.5s (26) vs 180.6s (107); and 6.7s (20) vs 469.4s (482) in the TG-119 phantom, liver, prostate, and head-and-neck examples, respectively. A new direct aperture optimization algorithm based on FIDO was developed. For the four IMRT test cases examined, this algorithm executed approximately 70-200 times faster without compromising the IMRT plan quality.

$\mathcal {H}$-Matrix Accelerated Solution of Surface–Volume–Surface EFIE for Fast Electromagnetic Analysis on 3-D Composite Dielectric Objects

An efficient fast direct algorithm based on the hierarchical ( $\mathcal {H}$ -) matrices is presented for solution of the radiation problems on piecewise homogeneous dielectric objects using Method of Moment (MoM) discretization of the surface–volume–surface electric field integral equation (SVS-EFIE). The SVS-EFIE for the composite objects introduces independent surface electric current density on the boundary of each region. Therefore, different from the traditional Poggio–Miller–Chang–Harrington–Wu–Tsai formulation, in the SVS-EFIE, the object regions can be meshing independently according to their local properties which improves the flexibility and efficiency of the proposed method. It also makes the proposed algorithms appropriate for the analysis of both multiscale and large-scale composite structures. The numerical results from the proposed fast method are provided for the high-loss biological tissues from bioelectromagnetics applications and agree well with the analytical Mie series solution and commercial software. The CPU time and memory cost of the required $\mathcal {H}$ -matrix operations are analyzed in details and verified through several numerical experiments. The new computational framework allows for fast direct solution of 3-D radiation and scattering problems of moderate electrical size with $O(P^{\alpha } \log ^2 P)$ CPU time and $O(P^{\alpha } \log P)$ memory complexity, $P$ being the number of surface unknowns produced by the MoM discretization, and $1\leq \alpha \leq 1.5$ being a geometry-dependent parameter.

A fast direct algorithm for implementing a high-order finite element method on rectangles as applied to boundary value problems for the Poisson equation

Fast direct and inverse algorithms for expansion in terms of eigenvectors of one-dimensional eigenvalue problems for a high-order finite element method (FEM) are proposed based on the fast discrete Fourier transform. They generalize logarithmically optimal Fourier algorithms for solving boundary value problems for Poisson-type equations on rectangular meshes to high-order FEM. The algorithms can be extended to the multidimensional case and can be applied to nonstationary problems.

A fast direct search algorithm for contact detection of convex polygonal or polyhedral particles

A fast direct search (FDS) algorithm is presented to increase the efficiency of contact detection for convex polygonal and polyhedral particles. All contact types are detected using only a small subset of these contact types: vertex-to-edge for polygons while vertex-to-face and edge-to-edge for polyhedra. First, an initial contact list is generated. Then in subsequent steps the contact list is updated by checking only local boundaries of the blocks and their separation. An exclusion algorithm is applied to avoid unnecessary examination for particles that are near but not-in-contact. The benchmark tests show that the FDS produces significant speed-up in various cases.

The FFTRR-based fast direct algorithms for complex inhomogeneous biharmonic problems with applications to incompressible flows

We develop analysis-based fast and accurate direct algorithms for several biharmonic problems in a unit disk derived directly from the Green’s functions of these problems and compare the numerical results with the “decomposition” algorithms (see Ghosh and Daripa, IMA J. Numer. Anal. 36(2), 824–850 [17]) in which the biharmonic problems are first decomposed into lower order problems, most often either into two Poisson problems or into two Poisson problems and a homogeneous biharmonic problem. One of the steps in the “decomposition algorithm” as discussed in Ghosh and Daripa (IMA J. Numer. Anal. 36(2), 824–850 [17]) for solving certain biharmonic problems uses the “direct algorithm” without which the problem can not be solved. Using classical Green’s function approach for these biharmonic problems, solutions of these problems are represented in terms of singular integrals in the complex z−plane (the physical plane) involving explicitly the boundary conditions. Analysis of these singular integrals using FFT and recursive relations (RR) in Fourier space leads to the development of these fast algorithms which are called FFTRR based algorithms. These algorithms do not need to do anything special to overcome coordinate singularity at the origin as often the case when solving these problems using finite difference methods in polar coordinates. These algorithms have some other desirable properties such as the ease of implementation and parallel in nature by construction. Moreover, these algorithms have O(logN) complexity per grid point where N2 is the total number of grid points and have very low constant behind this order estimate of the complexity. Performance of these algorithms is shown on several test problems. These algorithms are applied to solving viscous flow problems at low and moderate Reynolds numbers and numerical results are presented.

A fast direct sampling algorithm for equilateral closed polygons

Sampling equilateral closed polygons is of interest in the statistical study of ring polymers. Over the past 30 years, previous authors have proposed a variety of simple Markov chain algorithms (but have not been able to show that they converge to the correct probability distribution) and complicated direct samplers (which require extended-precision arithmetic to evaluate numerically unstable polynomials). We present a simple direct sampler which is fast and numerically stable, and analyze its runtime using a new formula for the volume of equilateral polygon space as a Dirichlet-type integral.

Total Variation on a Tree

We consider the problem of minimizing the continuous valued total variation subject to different unary terms on trees and propose fast direct algorithms based on dynamic programming to solve these problems. We treat both the convex and the nonconvex case and derive worst-case complexities that are equal to or better than existing methods. We show applications to total variation based two dimensional image processing and computer vision problems based on a Lagrangian decomposition approach. The resulting algorithms are very efficient, offer a high degree of parallelism, and come along with memory requirements which are only in the order of the number of image pixels.

A fast directional BEM for large-scale acoustic problems based on the Burton–Miller formulation

In this paper, a highly efficient fast boundary element method (BEM) for solving large-scale engineering acoustic problems in a broad frequency range is developed and implemented. The acoustic problems are modeled by the Burton–Miller boundary integral equation (BIE), thus the fictitious frequency issue is completely avoided. The BIE is discretized by using the Nyström method based on the curved quadratic elements, leading to simple numerical implementation (no edge or corner problems) and high accuracy in the BEM analysis. The linear systems are solved iteratively and accelerated by using a newly developed kernel-independent wideband fast directional algorithm (FDA) for fast summation of oscillatory kernels. In addition, the computational efficiency of the FDA is further promoted by exploiting the low-rank features of the translation matrices, resulting in two- to three-fold reduction in the computational time of the multipole-to-local translations. The high accuracy and nearly linear computational complexity of the present method are clearly demonstrated by typical examples. An acoustic scattering problem with dimensionless wave number kD (where k is the wave number and D is the typical length of the obstacle) up to 1000 and the degrees of freedom up to 4 million is successfully solved within 10h on a computer with one core and the memory usage is 24GB.

Modified Fast Directional Multilevel Algorithm for Analysis of Scattering of Microstrip Patch

In this paper, the modified fast directional multilevel algorithm (FDMA) is presented for analyzing microstrip patch. The FDMA has been successfully applied to analyze scattering in free space and then it was extended to solve mircrostrip problems. The quad-tree structure is used in this method, which is the same as that in fast multipole method. In our method, the finest level group size can be set smaller than 0.2λ without suffering from subwavelength breakdown. The resulting algorithm has the memory requirement and the CPU time per iteration proportional to O (NLogN), where N refers to the number of unknowns. Numerical results of microstrip structures are presented to show the efficiency of this method

Error Controllable Solutions of Large-Scale Problems in Electromagnetics: MLFMA-Accelerated Locally Corrected Nyström Solutions of CFIE in 3D [Open Problems in CEM

This work provides an overview of a parallel, high-order, error-controllable framework for solving large-scale scattering problems in electromagnetics, as well as open problems pertinent to such solutions. The method is based on the higher-order locally corrected Nyström (LCN) discretization of the combined-field integral equation (CFIE), accelerated with the error-controlled Multi-Level Fast Multipole Algorithm (MLFMA). Mechanisms for controlling the accuracy of calculations are discussed, including geometric representation, stages of the locally corrected Nyström method, and the MLFMA. Also presented are the key attributes of parallelization for the developed numerical framework. Numerical results validate the proposed numerical scheme by demonstrating higher-order error convergence for smooth scatterers. For the problem of scattering from a sphere, the developed numerical solution is shown to have the ability to produce a solution with a maximum relative error of the order 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-9</sup> . Open-ended problems, such as treatment of general scatterers with geometric singularities, construction of well-conditioned operators, and current challenges in development of fast iterative and direct algorithms, are also discussed.

A fast direct solver for elliptic problems on general meshes in 2D

We present a fast direct algorithm for solutions to linear systems arising from 2D elliptic equations. We follow the approach in Xia et al. (2009) on combining the multifrontal method with hierarchical matrices. We present a variant of that approach with additional hierarchical structure, extend it to quasi-uniform meshes, and detail an adaptive decomposition procedure for general meshes. Linear time complexity is shown for a quasi-regular grid and demonstrated via numerical results for the adaptive algorithm.

Fast Directional Multilevel Algorithm for Analyzing Wave Scattering

A new method named fast directional multilevel algorithm is proposed for analyzing wave scattering. Similar to a fast multipole method, an oct-tree structure is used in the new method. Unlike the conventional MLFMA, the kernel is evaluated by a new method, instead of approximating the Green's function by multipole expansion. The fast directional multilevel algorithm is first used to solve the wave scattering problem via combined field integral equation (CFIE) using Rao-Wilton-Glisson (RWG) basis functions. The low rank representations are extended to apply in the low frequency regime which is proved efficient and stable. The numerical results demonstrate that the computational complexity of this new multilevel algorithm can achieve the complexity of O(N log N). This algorithm is robust and performs well on other oscillatory kernels because of its not depending on the explicit form of kernels.

A fast directional algorithm for high-frequency electromagnetic scattering

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.