arXiv subjects

Per-Gunnar Martinsson

Publications and source records attributed to Per-Gunnar Martinsson.

54 records · Page 3Linked to original sources

arXiv2016

Householder QR Factorization with Randomization for Column Pivoting (HQRRP). FLAME Working Note #78

A fundamental problem when adding column pivoting to the Householder QR factorization is that only about half of the computation can be cast in terms of high performing matrix-matrix multiplications, which greatly limits the benefits that can be derived from so-called blocking of algorithms. This paper describes a technique for selecting groups of pivot vectors by means of randomized projections. It is demonstrated that the asymptotic flop count for the proposed method is $2mn^2 - (2/3)n^3$ for an $m\times n$ matrix, identical to that of the best classical unblocked Householder QR factorization algorithm (with or without pivoting). Experiments demonstrate acceleration in speed of close to an order of magnitude relative to the {\sc geqp3} function in LAPACK, when executed on a modern CPU with multiple cores. Further, experiments demonstrate that the quality of the randomized pivot selection strategy is roughly the same as that of classical column pivoting. The described algorithm is made available under Open Source license and can be used with LAPACK or libflame.

math.NA↗

arXiv2016

Efficient Algorithms for CUR and Interpolative Matrix Decompositions

The manuscript describes efficient algorithms for the computation of the CUR and ID decompositions. The methods used are based on simple modifications to the classical truncated pivoted QR decomposition, which means that highly optimized library codes can be utilized for implementation. For certain applications, further acceleration can be attained by incorporating techniques based on randomized projections. Numerical experiments demonstrate advantageous performance compared to existing techniques for computing CUR factorizations.

math.NA↗

arXiv2016

RSVDPACK: An implementation of randomized algorithms for computing the singular value, interpolative, and CUR decompositions of matrices on multi-core and GPU architectures

RSVDPACK is a library of functions for computing low rank approximations of matrices. The library includes functions for computing standard (partial) factorizations such as the Singular Value Decomposition (SVD), and also so called "structure preserving" factorizations such as the Interpolative Decomposition (ID) and the CUR decomposition. The ID and CUR factorizations pick subsets of the rows/columns of a matrix to use as bases for its row/column space. Such factorizations preserve properties of the matrix such as sparsity or non-negativity, are helpful in data interpretation, and require in certain contexts less memory than a partial SVD. The package implements highly efficient computational algorithms based on randomized sampling, as described and analyzed in [N. Halko, P.G. Martinsson, J. Tropp, "Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions," SIAM Review, 53(2), 2011], and subsequent papers. This manuscript presents some modifications to the basic algorithms that improve performance and ease of use. The library is written in C and supports both multi-core CPU and GPU architectures.

math.NA↗

arXiv2015

A randomized blocked algorithm for efficiently computing rank-revealing factorizations of matrices

This manuscript describes a technique for computing partial rank-revealing factorizations, such as, e.g, a partial QR factorization or a partial singular value decomposition. The method takes as input a tolerance $\varepsilon$ and an $m\times n$ matrix $A$, and returns an approximate low rank factorization of $A$ that is accurate to within precision $\varepsilon$ in the Frobenius norm (or some other easily computed norm). The rank $k$ of the computed factorization (which is an output of the algorithm) is in all examples we examined very close to the theoretically optimal $\varepsilon$-rank. The proposed method is inspired by the Gram-Schmidt algorithm, and has the same $O(mnk)$ asymptotic flop count. However, the method relies on randomized sampling to avoid column pivoting, which allows it to be blocked, and hence accelerates practical computations by reducing communication. Numerical experiments demonstrate that the accuracy of the scheme is for every matrix that was tried at least as good as column-pivoted QR, and is sometimes much better. Computational speed is also improved substantially, in particular on GPU architectures.

math.NA↗

arXiv2015

Compressing rank-structured matrices via randomized sampling

Randomized sampling has recently been proven a highly efficient technique for computing approximate factorizations of matrices that have low numerical rank. This paper describes an extension of such techniques to a wider class of matrices that are not themselves rank-deficient, but have off-diagonal blocks that are; specifically, the classes of so called \textit{Hierarchically Off-Diagonal Low Rank (HODLR)} matrices and \textit{Hierarchically Block Separable (HBS)} matrices. Such matrices arise frequently in numerical analysis and signal processing, in particular in the construction of fast methods for solving differential and integral equations numerically. These structures admit algebraic operations (matrix-vector multiplications, matrix factorizations, matrix inversion, etc.) to be performed very rapidly; but only once a data-sparse representation of the matrix has been constructed. How to rapidly compute this representation in the first place is much less well understood. The present paper demonstrates that if an $N\times N$ matrix can be applied to a vector in $O(N)$ time, and if the ranks of the off-diagonal blocks are bounded by an integer $k$, then the cost for constructing a HODLR representation is $O(k^{2}\,N\,(\log N)^{2})$, and the cost for constructing an HBS representation is $O(k^{2}\,N\,\log N)$. The point is that when legacy codes (based on, e.g., the Fast Multipole Method) can be used for the fast matrix-vector multiply, the proposed algorithm can be used to obtain the data-sparse representation of the matrix, and then well-established techniques for HODLR/HBS matrices can be used to invert or factor the matrix. The proposed scheme is also useful in simplifying the implementation of certain operations on rank-structured matrices such as the matrix-matrix multiplication, low-rank update, addition, etc.

math.NA↗

arXiv2014

An efficient and highly accurate solver for multi-body acoustic scattering problems involving rotationally symmetric scatterers

A numerical method for solving the equations modeling acoustic scattering is presented. The method is capable of handling several dozen scatterers, each of which is several wave-lengths long, on a personal work station. Even for geometries involving cavities, solutions accurate to seven digits or better were obtained. The method relies on a Boundary Integral Equation formulation of the scattering problem, discretized using a high-order accurate Nyström method. A hybrid iterative/direct solver is used in which a local scattering matrix for each body is computed, and then GMRES, accelerated by the Fast Multipole Method, is used to handle reflections between the scatterers. The main limitation of the method described is that it currently applies only to scattering bodies that are rotationally symmetric.

math.NA↗

arXiv2013

A high-order accurate accelerated direct solver for acoustic scattering from surfaces

We describe an accelerated direct solver for the integral equations which model acoustic scattering from curved surfaces. Surfaces are specified via a collection of smooth parameterizations given on triangles, a setting which generalizes the typical one of triangulated surfaces, and the integral equations are discretized via a high-order Nystrom method. This allows for rapid convergence in cases in which high-order surface information is available. The high-order discretization technique is coupled with a direct solver based on the recursive construction of scattering matrices. The result is a solver which often attains $O(N^{1.5})$ complexity in the number of discretization nodes $N$ and which is resistant to many of the pathologies which stymie iterative solvers in the numerical simulation of scattering. The performance of the algorithm is illustrated with numerical experiments which involve the simulation of scattering from a variety of domains, including one consisting of a collection of 1000 ellipsoids with randomly oriented semiaxes arranged in a grid, and a domain whose boundary has 12 curved edges and 8 corner points.

math.NA↗

arXiv2013

A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media

This paper presents a direct solution technique for the scattering of time-harmonic waves from a bounded region of the plane in which the wavenumber varies smoothly in space.The method constructs the interior Dirichlet-to-Neumann (DtN) map for the bounded region via bottom-up recursive merges of (discretization of) certain boundary operators on a quadtree of boxes.These operators take the form of impedance-to-impedance (ItI) maps. Since ItI maps are unitary, this formulation is inherently numerically stable, and is immune to problems of artificial internal resonances. The ItI maps on the smallest (leaf) boxes are built by spectral collocation on tensor-product grids of Chebyshev nodes. At the top level the DtN map is recovered from the ItI map and coupled to a boundary integral formulation of the free space exterior problem, to give a provably second kind equation.Numerical results indicate that the scheme can solve challenging problems 70 wavelengths on a side to 9-digit accuracy with 4 million unknowns, in under 5 minutes on a desktop workstation. Each additional solve corresponding to a different incident wave (right-hand side) then requires only 0.04 seconds.

math.NA↗

arXiv2013

An O(N) Direct Solver for Integral Equations on the Plane

An efficient direct solver for volume integral equations with O(N) complexity for a broad range of problems is presented. The solver relies on hierarchical compression of the discretized integral operator, and exploits that off-diagonal blocks of certain dense matrices have numerically low rank. Technically, the solver is inspired by previously developed direct solvers for integral equations based on "recursive skeletonization" and "Hierarchically Semi-Separable" (HSS) matrices, but it improves on the asymptotic complexity of existing solvers by incorporating an additional level of compression. The resulting solver has optimal O(N) complexity for all stages of the computation, as demonstrated by both theoretical analysis and numerical examples. The computational examples further display good practical performance in terms of both speed and memory usage. In particular, it is demonstrated that even problems involving 10^{7} unknowns can be solved to precision 10^{-10} using a simple Matlab implementation of the algorithm executed on a single core.

math.NA↗

arXiv2013

An O(N) algorithm for constructing the solution operator to 2D elliptic boundary value problems in the absence of body loads

The large sparse linear systems arising from the finite element or finite difference discretization of elliptic PDEs can be solved directly via, e.g., nested dissection or multifrontal methods. Such techniques reorder the nodes in the grid to reduce the asymptotic complexity of Gaussian elimination from $O(N^{2})$ to $O(N^{1.5})$ for typical problems in two dimensions. It has recently been demonstrated that the complexity can be further reduced to O(N) by exploiting structure in the dense matrices that arise in such computations (using, e.g., $\mathcal{H}$-matrix arithmetic). This paper demonstrates that such \textit{accelerated} nested dissection techniques become particularly effective for boundary value problems without body loads when the solution is sought for several different sets of boundary data, and the solution is required only near the boundary (as happens, e.g., in the computational modeling of scattering problems, or in engineering design of linearly elastic solids.

math.NA↗

arXiv2011

A direct solver with O(N) complexity for integral equations on one-dimensional domains

An algorithm for the direct inversion of the linear systems arising from Nystrom discretization of integral equations on one-dimensional domains is described. The method typically has O(N) complexity when applied to boundary integral equations (BIEs) in the plane with non-oscillatory kernels such as those associated with the Laplace and Stokes' equations. The scaling coefficient suppressed by the "big-O" notation depends logarithmically on the requested accuracy. The method can also be applied to BIEs with oscillatory kernels such as those associated with the Helmholtz and Maxwell equations; it is efficient at long and intermediate wave-lengths, but will eventually become prohibitively slow as the wave-length decreases. To achieve linear complexity, rank deficiencies in the off-diagonal blocks of the coefficient matrix are exploited. The technique is conceptually related to the H and H^2 matrix arithmetic of Hackbusch and co-workers, and is closely related to previous work on Hierarchically Semi-Separable matrices.

math.NA↗

arXiv2011

An algorithm for the principal component analysis of large data sets

Recently popularized randomized methods for principal component analysis (PCA) efficiently and reliably produce nearly optimal accuracy --- even on parallel processors --- unlike the classical (deterministic) alternatives. We adapt one of these randomized methods for use with data sets that are too large to be stored in random-access memory (RAM). (The traditional terminology is that our procedure works efficiently "out-of-core.") We illustrate the performance of the algorithm via several numerical examples. For example, we report on the PCA of a data set stored on disk that is so large that less than a hundredth of it can fit in our computer's RAM.

stat.CO↗

arXiv2011

A high-order accurate discretization scheme for variable coefficient elliptic PDEs in the plane with smooth solutions

A discretization scheme for variable coefficient elliptic PDEs in the plane is presented. The scheme is based on high-order Gaussian quadratures and is designed for problems with smooth solutions, such as scattering problems involving soft scatterers. The resulting system of linear equations is very well suited to efficient direct solvers such as nested dissection and the more recently proposed accelerated nested dissection schemes with O(N) complexity.

math.NA↗

arXiv2010

Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions

Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed---either explicitly or implicitly---to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, speed, and robustness. These claims are supported by extensive numerical experiments and a detailed error analysis.

math.NA↗

arXiv2010

A Direct Solver for the Rapid Solution of Boundary Integral Equations on Axisymmetric Surfaces in Three Dimensions

A scheme for rapidly and accurately computing solutions to boundary integral equations (BIEs) on rotationally symmetric surfaces in three dimensions is presented. The scheme uses the Fourier transform to reduce the original BIE defined on a surface to a sequence of BIEs defined on a generating curve for the surface. It can handle loads that are not necessarily rotationally symmetric. Nystrom discretization is used to discretize the BIEs on the generating curve. The quadrature used is a high-order Gaussian rule that is modified near the diagonal to retain high-order accuracy for singular kernels. The reduction in dimensionality, along with the use of high-order accurate quadratures, leads to small linear systems that can be inverted directly via, e.g., Gaussian elimination. This makes the scheme particularly fast in environments involving multiple right hand sides. It is demonstrated that for BIEs associated with Laplace's equation, the kernel in the reduced equations can be evaluated very rapidly by exploiting recursion relations for Legendre functions. Numerical examples illustrate the performance of the scheme; in particular, it is demonstrated that for a BIE associated with Laplace's equation on a surface discretized using 320 000 points, the set-up phase of the algorithm takes 2 minutes on a standard desktop, and then solves can be executed in 0.5 seconds.

math.NA↗

arXiv2008

Rapid factorization of structured matrices via randomized sampling

Randomized sampling has recently been demonstrated to be an efficient technique for computing approximate low-rank factorizations of matrices for which fast methods for computing matrix vector products are available. This paper describes an extension of such techniques to a wider class of matrices that are not themselves rank-deficient, but have off-diagonal blocks that are. Such matrices arise frequently in numerical analysis and signal processing, and there exist several methods for rapidly performing algebraic operations (matrix-vector multiplications, matrix factorizations, matrix inversion, \textit{etc}) on them once low-rank approximations to all off-diagonal blocks have been constructed. The paper demonstrates that if such a matrix can be applied to a vector in O(N) time, where the matrix is of size $N\times N$, and if individual entries of the matrix can be computed rapidly, then in many cases, the task of constructing approximate low-rank factorizations for all off-diagonal blocks can be performed in $O(N k^{2})$ time, where $k$ is an upper bound for the numerical rank of the off-diagonal blocks.

math.NA↗

arXiv2007

A fast direct solver for network matrices

A fast direct inversion scheme for the large sparse systems of linear equations resulting from the discretization of elliptic partial differential equations in two dimensions is given. The scheme is described for the particular case of a discretization on a uniform square grid, but can be generalized to more general geometries. For a grid containing $N$ points, the scheme requires $O(N \log^{2}N)$ arithmetic operations and $O(N \log N)$ storage to compute an approximate inverse. If only a single solve is required, then the scheme requires only $O(\sqrt{N} \log N)$ storage; the same storage is sufficient for computing the Dirichlet-to-Neumann operator as well as other boundary-to-boundary operators. The scheme is illustrated with several numerical examples. For instance, a matrix of size $10^6 \times 10^6$ is inverted to seven digits accuracy in four minutes on a 2.8GHz P4 desktop PC.

math.NA↗

arXiv2000

Solutions Near Singular Points to the Eikonal and Related First Order Non-linear Partial Differential Equations in Two Independent Variables

A detailed study of solutions to the first order partial differential equation H(x,y,z_x,z_y)=0, with special emphasis on the eikonal equation z_x^2+z_y^2=h(x,y), is made near points where the equation becomes singular in the sense that dH=0, in which case the method of characteristics does not apply. The main results are that there is a strong lack of uniqueness of solutions near such points and that solutions can be less regular than both the function H and the initial data of the problem, but that this loss of regularity only occurs along a pair of curves through the singular point. The main tools are symplectic geometry and the Sternberg normal form for Hamiltonian vector fields.

math.AP↗