Results 1  10
of
47
Hierarchical interpolative factorization for elliptic operators: differential equations
 Comm. Pure Appl. Math
"... This paper introduces the hierarchical interpolative factorization for integral equations (HIFIE) associated with elliptic problems in two and three dimensions. This factorization takes the form of an approximate generalized LU decomposition that permits the efficient application of the discretiz ..."
Abstract

Cited by 16 (5 self)
 Add to MetaCart
(Show Context)
This paper introduces the hierarchical interpolative factorization for integral equations (HIFIE) associated with elliptic problems in two and three dimensions. This factorization takes the form of an approximate generalized LU decomposition that permits the efficient application of the discretized operator and its inverse. HIFIE is based on the recursive skeletonization algorithm but incorporates a novel combination of two key features: (1) a matrix factorization framework for sparsifying structured dense matrices and (2) a recursive dimensional reduction strategy to decrease the cost. Thus, higherdimensional problems are effectively mapped to one dimension, and we conjecture that constructing, applying, and inverting the factorization all have linear or quasilinear complexity. Numerical experiments support this claim and further demonstrate the performance of our algorithm as a generalized fast multipole method, direct solver, and preconditioner. HIFIE is compatible with geometric adaptivity and can handle both boundary and volume problems. MATLAB ® codes are freely available.
A superfast structured solver for Toeplitz linear systems via randomized sampling
 SIAM J. Matrix Anal. Appl
"... Abstract. We propose a superfast solver for Toeplitz linear systems based on rank structured matrix methods and randomized sampling. The solver uses displacement equations to transform a Toeplitz matrix T into a Cauchylike matrix C, which is known to have lownumericalrank offdiagonal blocks. Thu ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We propose a superfast solver for Toeplitz linear systems based on rank structured matrix methods and randomized sampling. The solver uses displacement equations to transform a Toeplitz matrix T into a Cauchylike matrix C, which is known to have lownumericalrank offdiagonal blocks. Thus, we design a fast scheme for constructing a hierarchically semiseparable (HSS) matrix approximation to C, where the HSS generators have internal structures. Unlike classical HSS methods, our solver employs randomized sampling techniques together with fast Toeplitz matrixvector multiplications, and thus converts the direct compression of the offdiagonal blocks of C into the compression of much smaller blocks. A strong rankrevealing QR factorization method is used to generate/preserve certain special structures, and also to ensure stability. A fast ULV HSS factorization scheme is provided to take advantage of the special structures. We also propose a precomputation procedure for the HSS construction so as to further improve the efficiency. The complexity of these methods is significantly lower than some similar Toeplitz solvers for large matrix size n. Detailed flop counts are given, with the aid of a rank relaxation technique. The total cost of our methods includes O(n) flops for HSS operations and O(n log2 n) flops for matrix multiplications via FFTs, where n is the order of T. Various numerical tests on classical examples, including illconditioned ones, demonstrate the efficiency, and also indicate that the methods are stable in practice. This work shows a practical way of using randomized sampling in the development of fast rank structured methods.
A (2011) Fast direct solvers for elliptic partial differential equations
"... The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii Gillman, A. (Ph.D., Applied Mathematics) Fast direct solvers for elliptic partial differe ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii Gillman, A. (Ph.D., Applied Mathematics) Fast direct solvers for elliptic partial differential equations Thesis directed by Prof. PerGunnar Martinsson The dissertation describes fast, robust, and highly accurate numerical methods for solving boundary value problems associated with elliptic PDEs such as Laplace’s and Helmholtz ’ equations, the equations of elasticity, and timeharmonic Maxwell’s equation. In many areas of science and engineering, the cost of solving such problems determines what can and cannot be modeled computationally. Elliptic boundary value problems may be solved either via discretization of the PDE (e.g., finite element methods) or by first reformulating the equation as an integral equation, and then discretizing the integral equation. In either case, one is left with the task of solving a system of
A fast randomized algorithm for computing a hierarchically semiseparable representation of a matrix
 SIAM J Matrix Anal Appl
"... Abstract. 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 rankdeficient but have off ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
(Show Context)
Abstract. 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 rankdeficient but have offdiagonal blocks that are; specifically, the class of socalled hierarchically semiseparable (HSS) matrices. HSS matrices arise frequently in numerical analysis and signal processing, particularly in the construction of fast methods for solving differential and integral equations numerically. The HSS structure admits algebraic operations (matrixvector multiplications, matrix factorizations, matrix inversion, etc.) to be performed very rapidly, but only once the HSS 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 × N matrix can be applied to a vector in O(N) time, and if individual entries of the matrix can be computed rapidly, then provided that an HSS representation of the matrix exists, it can be constructed in O(N k2) operations, where k is an upper bound for the numerical rank of the offdiagonal blocks. The point is that when legacy codes (based on, e.g., the fast multipole method) can be used for the fast matrixvector multiply, the proposed algorithm can be used to obtain the HSS representation of the matrix, and then wellestablished techniques for HSS matrices can be used to invert or factor the matrix.
AN O(N) DIRECT SOLVER FOR INTEGRAL EQUATIONS ON THE PLANE
"... Abstract. 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 offdiagonal blocks of certain dense matrices have numerically low ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
(Show Context)
Abstract. 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 offdiagonal 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 SemiSeparable” (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 107 unknowns can be solved to precision 10−10 using a simple Matlab implementation of the algorithm executed on a single core.
Compression and Direct Manipulation of Complex Blendshape Models
"... Figure 1: Left: This facial model has 348 MB of uncompressed blendshape data, and runs at 8 frames per second on 8 CPUs. Our compression method reduces the storage to 25.4 MB and achieves 300 frames per second with a GPU implementation. No artifacts are visible. Right: The user can interactively man ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
Figure 1: Left: This facial model has 348 MB of uncompressed blendshape data, and runs at 8 frames per second on 8 CPUs. Our compression method reduces the storage to 25.4 MB and achieves 300 frames per second with a GPU implementation. No artifacts are visible. Right: The user can interactively manipulate the blendshape puppet by dragging any vertex on the model. The deforming region is colored red for visual feedback. We present a method to compress complex blendshape models and thereby enable interactive, hardwareaccelerated animation of these models. Facial blendshape models in production are typically large in terms of both the resolution of the model and the number of target shapes. They are represented by a single huge blendshape matrix, whose size presents a storage burden and prevents realtime processing. To address this problem, we present a new matrix compression scheme based on a hierarchically semiseparable (HSS) representation with matrix block reordering. The compressed data are also suitable for parallel processing. An efficient GPU implementation provides very fast feedback of the resulting animation. Compared with the original data, our technique leads to a huge improvement in both storage and processing efficiency without incurring any visual artifacts. As an application, we introduce an extended version of the direct manipulation method to control a large number of facial blendshapes efficiently and intuitively.
A HIGHORDER ACCURATE ACCELERATED DIRECT SOLVER FOR ACOUSTIC SCATTERING FROM SURFACES
"... Abstract. 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 i ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
Abstract. 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 highorder Nyström method. This allows for rapid convergence in cases in which highorder surface information is available. The highorder 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(N1.5) 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. 1.
Fast and accurate numerical methods for solving elliptic difference equations defined on lattices
 J. Comput. Phys
"... Techniques for rapidly computing approximate solutions to elliptic PDEs such as Laplace’s equation are well established. For problems involving general domains, and operators with constant coefficients, a highly efficient approach is to rewrite the boundary value problem as a boundary integral equat ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
Techniques for rapidly computing approximate solutions to elliptic PDEs such as Laplace’s equation are well established. For problems involving general domains, and operators with constant coefficients, a highly efficient approach is to rewrite the boundary value problem as a boundary integral equation (BIE), and then solve the BIE using fast methods such as, e.g., the Fast Multipole Method (FMM). The current paper demonstrates that this procedure can be extended to elliptic difference equations defined on infinite lattices, or on finite lattice with boundary conditions of either Dirichlet or Neumann type. As a representative model problem, a lattice equivalent of Laplace’s equation on a square lattice in two dimensions is considered: discrete analogs of BIEs are derived and fast solvers analogous to the FMM are constructed. Fast techniques are also constructed for problems involving lattices with inclusions and local deviations from perfect periodicity. The complexity of the methods described is O(Nboundary +Nsource +Ninc) where Nboundary is the number of nodes on the boundary of the domain, Nsource is the number of nodes subjected to body loads, and Ninc is the number of nodes that deviate from perfect periodicity. This estimate should be compared to the O(Ndomain logNdomain) estimate for FFT based methods, where Ndomain is the total number of nodes in the lattice (so that in two dimensions, Nboundary ∼ N
A FAST SEMIDIRECT LEAST SQUARES ALGORITHM FOR HIERARCHICALLY BLOCK SEPARABLE MATRICES∗
"... Abstract. 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 ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
Abstract. 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, equalityconstrained, 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 illconditioned. For an M×N HBS matrix with M ≥ N having bounded offdiagonal 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 +N3/2) in two dimensions, and O(M + N2) 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.
Conditioning analysis of incomplete Cholesky factorizations with
, 2012
"... The analysis of preconditioners based on incomplete Cholesky factorization in which the neglected (dropped) components are orthogonal to the approximations being kept is presented. General estimate for the condition number of the preconditioned system is given which only depends on the accuracy of i ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
The analysis of preconditioners based on incomplete Cholesky factorization in which the neglected (dropped) components are orthogonal to the approximations being kept is presented. General estimate for the condition number of the preconditioned system is given which only depends on the accuracy of individual approximations. The estimate is further improved if, for instance, only the newly computed rows of the factor are modified during each approximation step. In this latter case it is further shown to be sharp. The analysis is illustrated with some existing factorizations in the context of discretized elliptic partial differential equations.