the multifrontal method for sparse matrix solution theory and practice this paper presents an overview of the multifrontal method for the solution of large sparse symmetric positive definite linear systems the method is formulated in terms of frontal matrices update matrices and an assembly tree formal definitions of these notions are given based on the sparse matrix structure various advances to the basic method are surveyed they include the role of matrix reorderings the use of supernodes and other implementatjon techniques the use of the method in different computational environments is also described

wavelet like bases for the fast solutions of second kind integral equations a class of vector space bases is introduced for the sparse representation of discretizations of integral operators an operator with a smooth nonoscillatory kernel possessing a finite number of singularities in each row or column is represented in these bases as a sparse matrix to high precision a method is presented that employs these bases for the numerical solution of second kind integral equations in time bounded by o n log 2 n where n is the number of points in the discretization numerical results are given which demonstrate the effectiveness of the approach and several generalizations and applications of the method are discussed

efficient inversion of the galerkin matrix of general second order elliptic operators with nonsmooth coefficients this article deals with the efficient approximate inversion of finite element stiffness matrices of general second order elliptic operators with l coefficients it will be shown that the inverse stiffness matrix can be approximated by hierarchical matrices h matrices furthermore numerical results will demonstrate that it is possible to compute an approximate inverse with almost linear complexity

on a property of random walk polynomials involving christoffel functions abstract discrete time birth death processes may or may not have certain properties known as asymptotic aperiodicity and the strong ratio limit property in all cases known to us a suitably normalized process having one property also possesses the other suggesting equivalence of the two properties for a normalized process we show that equivalence may be translated into a property involving christoffel functions for a type of orthogonal polynomials known as random walk polynomials the prevalence of this property and thus the equivalence of asymptotic aperiodicity and the strong ratio limit property for a normalized birth death process is proven under mild regularity conditions

random walk polynomials and random walk measures random walk polynomials and random walk measures play a prominent role in the analysis of a class of markov chains called random walks without any reference to random walks however a random walk polynomial sequence can be defined and will be defined in this paper as a polynomial sequence pn x which is orthogonal with respect to a measure on 1 1 and which is such that the parameters alfa n in the recurrence relations pn 1 x x alfa n pn x snpn 1 x are nonnegative any measure with respect to which a random walk polynomial sequence is orthogonal is a random walk measure we collect some properties of random walk measures and polynomials and use these findings to obtain a limit theorem for random walk measures which is of interest in the study of random walks we conclude with a conjecture on random walk measures involving

a missing family of classical orthogonal polynomials we study a family of classical orthogonal polynomials which satisfy apart from a three term recurrence relation an eigenvalue problem with a differential operator of dunkl type these polynomials can be obtained from the little q jacobi polynomials in the limit q 1 we also show that these polynomials provide a nontrivial realization of the askey wilson algebra for q 1

efficient structured multifrontal factorization for general large sparse matrices rank structures provide an opportunity to develop new efficient numerical methods for practical problems when the off diagonal blocks of certain dense intermediate matrices have small numerical

a fast approximation of the bilateral filter using a signal processing approach the bilateral filter is a nonlinear filter that smoothes a signal while preserving strong edges it has demonstrated great effectiveness for a variety of problems in computer vision and computer graphics and fast versions have been proposed unfortunately little is known about the accuracy of such accelerations in this paper we propose a new signal processing analysis of the bilateral filter which complements the recent studies that analyzed it as a pde or as a robust statistical estimator the key to our analysis is to express the filter in a higher dimensional space where the signal intensity is added to the original domain dimensions importantly this signal processing perspective allows us to develop a novel bilateral filtering acceleration using downsampling in space and intensity this affords a principled expression of accuracy in terms of bandwidth and sampling the bilateral filter can be expressed as linear convolutions in this augmented space followed by two simple nonlinearities this allows us to derive criteria for downsampling the key operations and achieving important acceleration of the bilateral filter we show that for the same running time our method is more accurate than previous acceleration techniques typically we are able to process a 2 megapixel image using our acceleration technique in less than a second and have the result be visually similar to the exact computation that takes several tens of minutes the acceleration is most effective with large spatial kernels furthermore this approach extends naturally to color images and cross bilateral filtering

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

efficient preconditioning of hp fem matrix sequences with slowly varying coefficients an application to topology optimization abstract we previously introduced a preconditioner that has proven effective for h p fem discretizations of various challenging elliptic and hyperbolic problems the construction is inspired by standard nested dissection and relies on the assumption that the schur complements can be approximated to high precision by hierarchically semi separable matrices the preconditioner is built as an approximate l d m t factorization through a divide and conquer approach this implies an enhanced flexibility which allows to handle unstructured geometric meshes anisotropies and discontinuities we build on our previous numerical experiments and develop a preconditioner update strategy that allows us to handle matrix sequences arising from problems with slowly varying coefficients we investigate the performance of the preconditioner along with the update strategy in context of topology optimization of an acoustic cavity

emulation of higher order tensors in manifold monte carlo methods for bayesian inverse problems the bayesian approach to inverse problems relies predominantly on markov chain monte carlo methods for posterior inference the typical nonlinear concentration of posterior measure observed in many such inverse problems presents severe challenges to existing simulation based inference methods motivated by these challenges the exploitation of local geometric information in the form of covariant gradients metric tensors levi civita connections and local geodesic flows have been introduced to more effectively locally explore the configuration space of the posterior measure however obtaining such geometric quantities usually requires extensive computational effort and despite their effectiveness affects the applicability of these geometrically based monte carlo methods in this paper we explore one way to address this issue by the construction of an emulator of the model from which all geometric objects can be obtained in a much more computationally feasible manner the main concept is to approximate the geometric quantities using a gaussian process emulator which is conditioned on a carefully chosen design set of configuration points which also determines the quality of the emulator to this end we propose the use of statistical experiment design methods to refine a potentially arbitrarily initialized design online without destroying the convergence of the resulting markov chain to the desired invariant measure the practical examples considered in this paper provide a demonstration of the significant improvement possible in terms of computational loading suggesting this is a promising avenue of further development

improving multifrontal methods by means of block low rank representations matrices coming from elliptic partial differential equations pdes have been shown to have a low rank property well defined off diagonal blocks of their schur complements can be approximated by low rank products given a suitable ordering of the matrix which gives to the blocks a geometrical meaning such approximations can be computed using an svd or a rank revealing qr factorization the resulting representation offers a substantial reduction of the memory requirement and gives efficient ways to perform many of the basic dense algebra operations several strategies have been proposed to exploit this property we propose a low rank format called block low rank blr and explain how it can be used to reduce the memory footprint and the complexity of direct solvers for sparse matrices based on the multifrontal method we present experimental results that show how the blr format delivers gains that are comparable to those obtained with hierarchical formats such as hierarchical matrices h matrices and hierarchically semi separable hss matrices but provides much greater flexibility and ease of use which are essential in the context of a general purpose algebraic solver

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 robustness and or speed these claims are supported by extensive numerical experiments and a detailed error analysis the specific benefits of randomized techniques depend on the computational environment consider the model problem of finding the k dominant components of the singular value decomposition of an m times n matrix i for a dense input matrix randomized algorithms require bigo mn log k floating point operations flops in contrast to bigo mnk for classical algorithms ii for a sparse input matrix the flop count matches classical krylov subspace methods but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures iii for a matrix that is too large to fit in fast memory the randomized techniques require only a constant number of passes over the data as opposed to bigo k passes for classical algorithms in fact it is sometimes possible to perform matrix approximation with a single pass over the data

diffusion tensors for processing sheared and rotated rectangles image restoration and simplification methods that respect important features such as edges play a fundamental role in digital image processing however known edge preserving methods like common nonlinear diffusion methods tend to round vertices for large diffusion times in this paper we adapt the diffusion tensor for anisotropic diffusion to avoid this effects in images containing rotated and sheared rectangles respectively in this context we propose a new method for estimating rotation angles and shear parameters based on the so called structure tensor further we show how the knowledge of appropriate diffusion tensors can be used in variational models numerical examples including orientation estimation denoising and segmentation demonstrate the good performance of our methods

application of the inverse fast multipole method as a preconditioner in a 3d helmholtz boundary element method abstract we investigate an efficient preconditioning of iterative methods such as gmres for solving dense linear systems a x b that follow from a boundary element method bem for the 3d helmholtz equation focusing on the low frequency regime while matrix vector products in gmres can be accelerated through the low frequency fast multipole method lffmm the bem often remains computationally expensive due to the large number of gmres iterations we propose the application of the inverse fast multipole method ifmm as a preconditioner to accelerate the convergence of gmres the ifmm is in essence an approximate direct solver that uses a multilevel hierarchical decomposition and low rank approximations the proposed ifmm based preconditioning has a tunable parameter e that balances the cost to construct a preconditioner m which is an approximation of a 1 and the cost to perform the iterative process by means of m namely using a small respectively large value of e takes a long respectively short time to construct m while the number of iterations can be small respectively large a comprehensive set of numerical examples involving various boundary value problems with complicated geometries and mixed boundary conditions is presented to validate the efficiency of the proposed method we show that the ifmm preconditioner with a nearly optimal e of 10 2 clearly outperforms some common preconditioners for the bem achieving 1 2 10 8 times speed up of the computations in particular when the scale of the underlying scatterer is about five wavelengths or more in addition the ifmm preconditioner is capable of solving complicated problems in a reasonable amount of time that bd preconditioner can not

on the fast matrix multiplication in the boundary element method by panel clustering the boundary element method bem leads to a system of linear equations with a full matrix while fem yields sparse matrices this fact seems to require much computational work for the definition of the matrix for the solution of the system and in particular for the matrix vector multiplication which always occurs as an elementary in this paper a method for the approximate matrix vector multiplication is described which requires much less arithmetical work in addition the storage requirements are strongly reduced

mathcal h h matrix approximability of the inverses of fem matrices we study the question of approximability for the inverse of the fem stiffness matrix for scalar second order elliptic boundary value problems by blockwise low rank matrices such as those given by the mathcal h h matrix format introduced by hackbusch computing 62 2 89 108 1999 we show that exponential convergence in the local block rank r r can be achieved we also show that exponentially accurate lu lu decompositions in the mathcal h h matrix format are possible for the stiffness matrices arising in the fem our analysis avoids any coupling of the block rank r r to the mesh width h h we also cover fairly general boundary conditions of mixed dirichlet neumann robin boundary conditions

domain decomposition based ℌ lu preconditioners hierarchical matrices provide a data sparse way to approximate fully populated matrices the two basic steps in the construction of an h matrix are a the hierarchical construction of a matrix block partition and b the blockwise approxi mation of matrix data by low rank matrices in this paper we develop a new approach to construct the necessary partition based on domain decomposition compared to standard geometric bisection based h matrices this new approach yields h lu fac torizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements these rigorously proven and numerically verified improvements result from an h matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an lu factorization we provide numerical results in which a domain decom position based h lu factorization is used as a preconditioner in the iterative solution of the discrete three dimensional convection diffusion equation

a tensor train accelerated solver for integral equations in complex geometries abstract we present a framework using the quantized tensor train qtt decomposition to accurately and efficiently solve volume and boundary integral equations in three dimensions we describe how the qtt decomposition can be used as a hierarchical compression and inversion scheme for matrices arising from the discretization of integral equations for a broad range of problems computational and storage costs of the inversion scheme are extremely modest o log n and once the inverse is computed it can be applied in o n log n we analyze the qtt ranks for hierarchically low rank matrices and discuss its relationship to commonly used hierarchical compression techniques such as fmm and hss we prove that the qtt ranks are bounded for translation invariant systems and argue that this behavior extends to non translation invariant volume and boundary integrals for volume integrals the qtt decomposition provides an efficient direct solver requiring significantly less memory compared to other fast direct solvers we present results demonstrating the remarkable performance of the qtt based solver when applied to both translation and non translation invariant volume integrals in 3d for boundary integral equations we demonstrate that using a qtt decomposition to construct preconditioners for a krylov subspace method leads to an efficient and robust solver with a small memory footprint we test the qtt preconditioners in the iterative solution of an exterior elliptic boundary value problem laplace formulated as a boundary integral equation in complex multiply connected geometries

decomposition of images by the anisotropic rudin osher fatemi model the total variation based image denoising model of rudin osher and fatemi can be generalized in a natural way to favor certain edge directions we consider the resulting anisotropic energies and study properties of their minimizers 2004 wiley periodicals inc

fast directional multilevel algorithms for oscillatory kernels this paper introduces a new directional multilevel algorithm for solving n body or n point problems with highly oscillatory kernels these systems often result from the boundary integral formulations of scattering problems and are difficult due to the oscillatory nature of the kernel and the non uniformity of the particle distribution we address the problem by first proving that the interaction between a ball of radius r and a well separated region has an approximate low rank representation as long as the well separated region belongs to a cone with a spanning angle of o 1 r and is at a distance which is at least o r 2 away from from the ball we then propose an efficient and accurate procedure which utilizes random sampling to generate such a separated low rank representation based on the resulting representations our new algorithm organizes the high frequency far field computation by a multidirectional and multiscale strategy to achieve maximum efficiency the algorithm performs well on a large group of highly oscillatory kernels our algorithm is proved to have o n log n computational complexity for any given accuracy when the points are sampled from a two dimensional surface we also provide numerical results to demonstrate these properties

analysis of superimposed oriented patterns estimation of local orientation in images may be posed as the problem of finding the minimum gray level variance axis in a local neighborhood in bivariate images the solution is given by the eigenvector corresponding to the smaller eigenvalue of a 2times2 tensor for an ideal single orientation the tensor is rank deficient i e the smaller eigenvalue vanishes a large minimal eigenvalue signals the presence of more than one local orientation what may be caused by non opaque additive or opaque occluding objects crossings bifurcations or corners we describe a framework for estimating such superimposed orientations our analysis is based on the eigensystem analysis of suitably extended tensors for both additive and occluding superpositions unlike in the single orientation case the eigensystem analysis does not directly yield the orientations rather it provides so called mixed orientation parameters mops we therefore show how to decompose the mops into the individual orientations we also show how to use tensor invariants to increase efficiency and derive a new feature for describing local neighborhoods which is invariant to rigid transformations applications are e g in texture analysis directional filtering and interpolation feature extraction for corners and crossings tracking and signal separation

approximation of solution operators of elliptic partial differential equations by mathcal h and mathcal h 2 matrices we investigate the problem of computing the inverses of stiffness matrices resulting from the finite element discretization of elliptic partial differential equations since the solution operators are non local the inverse matrices will in general be dense therefore representing them by standard techniques will require prohibitively large amounts of storage in the field of integral equations a successful technique for handling dense matrices efficiently is to use a data sparse representation like the popular multipole method in this paper we prove that this approach can be generalized to cover inverse matrices corresponding to partial differential equations by switching to data sparse mathcal h and mathcal h 2 matrices the key results are existence proofs for local low rank approximations of the solution operator and its discrete counterpart which give rise to error estimates for mathcal h and mathcal h 2 matrix approximations of the entire matrices

an analysis of the perona malik scheme the perona malik scheme is a numerical procedure for de noising digital pictures in this work we investigate how it evolves piecewise smooth initial data in one space dimension by scaling a natural parameter that appears in the scheme in an appropriate way with respect to the grid size we obtain a meaningful continuum limit the resulting evolution is unusual it involves solving a system of heat equations coupled to each other through nonlinear boundary conditions that become singular at special times leading to gradient blow up for the solutions however the scheme suggests a natural continuation beyond each of the singular times that involves a change in the p d e system we give a convergence proof that applies under some technical restrictions on any bounded interval of time which might include singular times furthermore the proposed continuum limit has some of the essential features observed in applications of the numerical scheme

some generalized jacobi polynomials abstract we give explicitly the recurrence coefficients in the three term recurrence relation of some generalized jacobi polynomials defined by the positive weight p x 1 x 2 1 x p on 1 1 the case p 0 can be found in chihara s book the case p 1 is treated by the first author and we consider here the cases p 2 3 4 these recurrence coefficients can be used to prove that their corresponding polynomials are random walk and then are of interest in birth and death process 1 2 for instance the cases p 1 2 are treated

the black box fast multipole method a new o n fast multipole formulation is proposed for non oscillatory kernels this algorithm is applicable to kernels k x y which are only known numerically that is their numerical value can be obtained for any x y this is quite different from many fast multipole methods which depend on analytical expansions of the far field behavior of k for x y large other black box or kernel independent fast multipole methods have been devised our approach has the advantage of requiring a small pre computation time even for very large systems and uses the minimal number of coefficients to represent the far field for a given l 2 tolerance error in the approximation this technique can be very useful for problems where the kernel is known analytically but is quite complicated or for kernels which are defined purely numerically

existence of ℋ matrix approximants to the inverse fe matrix of elliptic operators with l coefficients this article deals with the existence of blockwise low rank approximants so called matrices to inverses of fem matrices in the case of uniformly elliptic operators with l coefficients unlike operators arising from boundary element methods for which the matrix theory has been extensively developed the inverses of these operators do not benefit from the smoothness of the kernel function however it will be shown that the corresponding green functions can be approximated by degenerate functions giving rise to the existence of blockwise low rank approximants of fem inverses numerical examples confirm the correctness of our estimates as a side product we analyse the matrix property of the inverse of the fe mass matrix

second order cone programming second order cone programming socp problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the cartesian product of second order lorentz cones linear programs convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as socp problems as can many other problems that do not fall into these three categories these latter problems model applications from a broad range of fields from engineering control and finance to robust optimization and combinatorial optimization on the other hand semidefinite programming sdp that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices includes socp as a special case therefore socp falls between linear lp and quadratic qp programming and sdp like lp qp and sdp problems socp problems can be solved in polynomial time by interior point methods the computational effort per iteration required by these methods to solve socp problems is greater than that required to solve lp and qp problems but less than that required to solve sdp s of similar size and structure because the set of feasible solutions for an socp problem is not polyhedral as it is for lp and qp problems it is not readily apparent how to develop a simplex or simplex like method for socp while socp problems can be solved as sdp problems doing so is not advisable both on numerical grounds and computational complexity concerns for instance many of the problems presented in the survey paper of vandenberghe and boyd vb96 as examples of sdps can in fact be formulated as socps and should be solved as such in 2 3 below we give socp formulations for four of these examples the convex quadratically constrained quadratic programming qcqp problem problems involving fractional quadratic functions rutcor rutgers university e mail alizadeh rutcor rutgers edu research supported in part by the u s national science foundation grant ccr 9901991 ieor columbia university e mail gold ieor columbia edu research supported in part by the department of energy grant de fg02 92er25126 national science foundation grants dms 94 14438 cda 97 26385 and dms 01 04282

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

adaptive finite element method for fractional differential equations using hierarchical matrices abstract a robust and fast solver for the fractional differential equation fdes involving the riesz fractional derivative is developed using an adaptive finite element method it is based on the utilization of hierarchical matrices h matrices for the representation of the stiffness matrix resulting from the finite element discretization of the fdes we employ a geometric multigrid method for the solution of the algebraic system of equations we combine it with an adaptive algorithm based on a posteriori error estimation a posteriori error estimation based adaptive algorithm is used to deal with general type singularities arising in the solution of the fdes through various test examples we demonstrate the efficiency of the method and the high accuracy of the numerical solution even in the presence of singularities the proposed technique has been verified effectively through fundamental examples including riesz left right riemann liouville fractional derivative and furthermore it can be readily extended to more general fractional differential equations with different boundary conditions and low order terms

a fast summation tree code for matern kernel the matern family of functions is a widely used covariance kernel in spatial statistics for gaussian process modeling which in many instances requires calculations with a covariance matrix in this paper we design a fast summation algorithm for the matern kernel in order to efficiently perform matrix vector multiplications this algorithm is based on the barnes hut tree code framework and addresses several practical issues the anisotropy of the kernel the nonuniform distribution of the point set and a tight error estimate of the approximation even though the algorithmic details differ from the standard tree code in several aspects empirically the computational cost of our algorithm scales as o n log n f orn points comprehensive numerical experiments are shown to demonstrate the practicality of the design

a fast direct solver for structured linear systems by recursive skeletonization 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 mathcal o n in two dimensions where n is the number of discretization points in our current implementation the corresponding costs in three dimensions are mathcal o n 3 2 and mathcal o

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

a fast block low rank dense solver with applications to finite element matrices this article presents a fast solver for the dense frontal matrices that arise from the multifrontal sparse elimination process of 3d elliptic pdes the solver relies on the fact that these matrices can be efficiently represented as a hierarchically off diagonal low rank hodlr matrix to construct the low rank approximation of the off diagonal blocks we propose a new pseudo skeleton scheme the boundary distance low rank approximation that picks rows and columns based on the location of their corresponding vertices in the sparse matrix graph we compare this new low rank approximation method to the adaptive cross approximation aca algorithm and show that it achieves better speedup specially for unstructured meshes using the hodlr direct solver as a preconditioner with a low tolerance to the gmres iterative scheme we can reach machine accuracy much faster than a conventional lu solver numerical benchmarks are provided for frontal matrices arising from 3d finite element problems corresponding to a wide range of applications

domain decomposition based mathcal h lu preconditioning hierarchical matrices provide a data sparse way to approximate fully populated matrices the two basic steps in the construction of an mathcal h matrix are a the hierarchical construction of a matrix block partition and b the blockwise approximation of matrix data by low rank matrices in this paper we develop a new approach to construct the necessary partition based on domain decomposition compared to standard geometric bisection based mathcal h matrices this new approach yields mathcal h lu factorizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements these rigorously proven and numerically verified improvements result from an mathcal h matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an lu factorization we provide numerical results in which a domain decomposition based mathcal h lu factorization is used as a preconditioner in the iterative solution of the discrete three dimensional convection diffusion equation

efficient algorithms for computing a strong rank revealing qr factorization given anm n matrixm withm n it is shown that there exists a permutation fi and an integer k such that the qr factorization myi q ak ckbk reveals the numerical rank of m the k k upper triangular matrix ak is well conditioned ilckll2 is small and bk is linearly dependent on ak with coefficients bounded by a low degree polynomial in n existing rank revealing qr rrqr algorithms are related to such factorizations and two algorithms are presented for computing them the new algorithms are nearly as efficient as qr with column pivoting for most problems and take o ran2 floating point operations in the worst case

efficient mesh deformation based on radial basis function interpolation by means of the inverse fast multipole method abstract radial basis function interpolation is often employed in mesh deformation algorithms for unstructured meshes for example in fluid structure interaction or design optimization problems this is known to be a robust methodology that results in high quality deformed meshes the applicability of this method to large problems is currently hampered by its prohibitive computational cost however which is caused by the need to solve a dense system of equations the computation time grows as o n b 3 if a direct solver is employed for the solution where n b denotes the number of boundary nodes in the mesh while alternative iterative solvers often suffer from an unfavorable convergence behavior in this paper we present the inverse fast multipole method as a novel fast approximate direct solver with a computational cost scaling as o n b the linear complexity is achieved by transforming the dense system into an extended sparse system along with the compression of certain matrix blocks into low rank factorizations the solver is inexact although the error can be controlled and made as small as needed a low accuracy solver can hence be used as an efficient preconditioner in an iterative scheme numerical benchmarks are presented demonstrating that the proposed approach enhances the computational efficiency of mesh deformation algorithms based on radial basis function interpolation significantly without jeopardizing their robustness and quality

stability properties of the perona malik scheme the perona malik scheme is a numerical technique for denoising digital images without blurring object boundaries edges in general solutions generated by this scheme do not satisfy a comparison principle we identify conditions under which two solutions initially ordered remain ordered and we state restricted comparison principles these allow us to study stability properties of the scheme we also explore what these results say in the limit as the discretization size goes to 0

a fast direct solver for the integral equations of scattering theory on planar curves with corners we describe an approach to the numerical solution of the integral equations of scattering theory on planar curves with corners it is rather comprehensive in that it applies to a wide variety of boundary value problems here we treat the neumann and dirichlet problems as well as the boundary value problem arising from acoustic scattering at the interface of two fluids it achieves high accuracy is applicable to large scale problems and perhaps most importantly does not require asymptotic estimates for solutions instead the singularities of solutions are resolved numerically the approach is efficient however only in the low and mid frequency regimes once the scatterer becomes more than several hundred wavelengths in size the performance of the algorithm of this paper deteriorates significantly we illustrate our method with several numerical experiments including the solution of a neumann problem for the helmholtz equation given on a domain with nearly 10000 corner points

a short time beltrami kernel for smoothing images and manifolds we introduce a short time kernel for the beltrami image enhancing flow the flow is implemented by convolving the image with a space dependent kernel in a similar fashion to the solution of the heat equation by a convolution with a gaussian kernel the kernel is appropriate for smoothing regular flat 2 d images for smoothing images painted on manifolds and for simultaneously smoothing images and the manifolds they are painted on the kernel combines the geometry of the image and that of the manifold into one metric tensor thus enabling a natural unified approach for the manipulation of both additionally the derivation of the kernel gives a better geometrical understanding of the beltrami flow and shows that the bilateral filter is a euclidean approximation of it on a practical level the use of the kernel allows arbitrarily large time steps as opposed to the existing explicit numerical schemes for the beltrami flow in addition the kernel works with equal ease on regular 2 d images and on images painted on parametric or triangulated manifolds we demonstrate the denoising properties of the kernel by applying it to various types of images and manifolds

a fast nested dissection solver for cartesian 3d elliptic problems using hierarchical matrices we present a fast algorithm for solutions to linear systems arising from three dimensional elliptic problems on a regular cartesian mesh we follow the approach of schmitz and ying 2012 on combining the nested dissection matrix factorization method with hierarchical matrices in two dimensions and extend it to the three dimensional case a theoretical linear time complexity is derived and a more practical variant with slightly worse scaling is demonstrated

nonlinear regularized reaction diffusion filters for denoising of images with textures denoising is always a challenging problem in natural imaging and geophysical data processing in this paper we consider the denoising of texture images using a nonlinear reaction diffusion equation and directional wavelet frames in our model a curvelet shrinkage is used for regularization of the diffusion process to preserve important features in the diffusion smoothing and a wave atom shrinkage is used as the reaction in order to preserve and enhance interesting oriented textures we derive a digital reaction diffusion filter that lives on graphs and show convergence of the corresponding iteration process experimental results and comparisons show very good performance of the proposed model for texture preserving denoising

estimating extremal eigenvalues and condition numbers of matrices the paper describes probabilistic algorithms which may be used to make rough estimates of the largest and smallest eigenvalues of a positive definite matrix and the condition number of a nonsingular matrix in the 2 norm given varepsilon 0 and a prescribed relative error the algorithms compute estimates which with probability at least 1 varepsilon have relative errors less than that prescribed in particular the method gives a reliable way to estimate the condition number of a matrix of large degree

gamblets for opening the complexity bottleneck of implicit schemes for hyperbolic and parabolic odes pdes with rough coefficients implicit schemes are popular methods for the integration of time dependent pdes such as hyperbolic and parabolic pdes however the necessity to solve corresponding linear systems at each time step constitutes a complexity bottleneck in their application to pdes with rough coefficients we present a generalization of gamblets introduced in 62 enabling the resolution of these implicit systems in near linear complexity and provide rigorous a priori error bounds on the resulting numerical approximations of hyperbolic and parabolic pdes these generalized gamblets induce a multiresolution decomposition of the solution space that is adapted to both the underlying hyperbolic and parabolic pde and the system of odes resulting from space discretization and to the time steps of the numerical scheme

on the compression of low rank matrices a procedure is reported for the compression of rank deficient matrices a matrix a of rank k is represented in the form a u circ b circ v where b is a k times k submatrix of a and u v are well conditioned matrices that each contain a k times k identity submatrix this property enables such compression schemes to be used in certain situations where the singular value decomposition svd cannot be used efficiently numerical examples are presented

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

a kernel independent adaptive fast multipole algorithm in two and three dimensions we present a new fast multipole method for particle simulations the main feature of our algorithm is that it does not require the implementation of multipole expansions of the underlying kernel and it is based only on kernel evaluations instead of using analytic expansions to represent the potential generated by sources inside a box of the hierarchical fmm tree we use a continuous distribution of an equivalent density on a surface enclosing the box to find this equivalent density we match its potential to the potential of the original sources at a surface in the far field by solving local dirichlet type boundary value problems the far field evaluations are sparsified with singular value decomposition in 2d or fast fourier transforms in 3d we have tested the new method on the single and double layer operators for the laplacian the modified laplacian the stokes the modified stokes the navier and the modified navier operators in two and three dimensions our numerical results indicate that our method compares very well with the best known implementations of the analytic fmm method for both the laplacian and modified laplacian kernels its advantage is the relative simplicity of the implementation and its immediate extension to more general kernels

hierarchical interpolative factorization for elliptic operators differential equations this paper introduces the hierarchical interpolative factorization for integral equations hif ie 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 hif ie 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 higher dimensional 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 hif ie is compatible with geometric adaptivity and can handle both boundary and volume problems matlab codes are freely available 2015 wiley periodicals inc

fast adaptive high order accurate discretization of the lippmann schwinger equation in two dimensions we present a fast direct solver for two dimensional scattering problems where an incident wave impinges on a penetrable medium with compact support we represent the scattered field using a volume potential whose kernel is the outgoing green s function for the exterior domain inserting this representation into the governing partial differential equation we obtain an integral equation of lippmann schwinger type the principal contribution here is the development of an automatically adaptive high order accurate discretization based on a quad tree data structure which provides rapid access to arbitrary elements of the discretized system matrix this permits the straightforward application of state of the art algorithms for constructing compressed versions of the solution operator these solvers typically require o n 3 2 work where n denotes the number of degrees of freedom we demonstrate the performance of the method for a variety of problems in both low and high frequency regimes

a direct solver with o n complexity for variable coefficient elliptic pdes discretized via a high order composite spectral collocation method a numerical method for solving elliptic pdes with variable coefficients on two dimensional domains is presented the method is based on high order composite spectral approximations and is designed for problems with smooth solutions the resulting system of linear equations is solved using a direct as opposed to iterative solver that has optimal o n complexity for all stages of the computation when applied to problems with nonoscillatory solutions such as the laplace and the stokes equations numerical examples demonstrate that the scheme is capable of computing solutions with a relative accuracy of 10 10 or better for challenging problems such as highly oscillatory helmholtz problems and convection dominated convection diffusion equations in terms of speed it is demonstrated that a problem with a nonoscillatory solution that was discretized using 10 8 nodes can be solved in 115 minutes on a personal workstation with two quad core 3 3 ghz cpus since the solver is direct and the solution

a fast direct solver for boundary value problems on locally perturbed geometries abstract many applications including optimal design and adaptive discretization techniques involve solving several boundary value problems on geometries that are local perturbations of an original geometry this manuscript presents a fast direct solver for boundary value problems that are recast as boundary integral equations the idea is to write the discretized boundary integral equation on a new geometry as a low rank update to the discretized problem on the original geometry using the sherman morrison formula the inverse can be expressed in terms of the inverse of the original system applied to the low rank factors and the right hand side numerical results illustrate for problems where perturbation is localized the fast direct solver is three times faster than building a new solver from scratch

ratio limits and limiting conditional distributions for discrete time birth death processes we consider discrete time birth death processes with an absorbing state and study the conditional state distribution at time n given that absorption has not occurred by that time but will occur eventually in particular we establish conditions for the convergence of these distributions to a proper distribution as n the problem turns out to be closely related to that of finding conditions for the existence of limits of ratios of n step transition probabilities as n orthogonal polynomials feature in the spectral representation for the n step transition probabilities of a birth death process and consequently play a key role in the analysis