generalized bundle methods we study a class of generalized bundle methods for which the stabilizing term can be any closed convex function satisfying certain properties this setting covers several algorithms from the literature that have been so far regarded as distinct under a different hypothesis on the stabilizing term and or the function to be minimized we prove finite termination asymptotic convergence and finite convergence to an optimal point with or without limits on the number of serious steps and or requiring the proximal parameter to go to infinity the convergence proofs leave a high degree of freedom in the crucial implementative features of the algorithm i e the management of the bundle of subgradients beta strategy and of the proximal parameter t strategy we extensively exploit a dual view of bundle methods which are shown to be a dual ascent approach to one nonlinear problem in an appropriate dual space where nonlinear subproblems are approximately solved at each step with an inner linearization approach this allows us to precisely characterize the changes in the subproblems during the serious steps since the dual problem is not tied to the local concept of varepsilon subdifferential for some of the proofs a generalization of inf compactness called compactness is required this concept is related to that of asymptotically well behaved functions

the numerical solution of laplace s equation in three dimensions we consider the dirichlet problem for laplace s equation on a simply connected three dimensional region with a smooth boundary this problem is easily converted to the solution of a fredholm integral equation of the second kind based on representing the harmonic solution as a double layer potential function we solve this integral equation formulation by using galerkin s method with spherical harmonics as the basis functions this approach leads to small linear systems and once the galerkin coefficients for the region have been calculated the computation time is small for any particular boundary function the major disadvantage of the method is the calculation of the galerkin coefficients each of which is a four fold integral with a singular integrand theoretical and computational details of the method are presented

a subband adaptive iterative shrinkage thresholding algorithm we investigate a subband adaptive version of the popular iterative shrinkage thresholding algorithm that takes different update steps and thresholds for each subband in particular we provide a condition that ensures convergence and discuss why making the algorithm subband adaptive accelerates the convergence we also give an algorithm to select appropriate update steps and thresholds for when the distortion operator is linear and time invariant the results in this paper may be regarded as extensions of the recent work by vonesch and unser

a fundamental solution method for the reduced wave problem in a domain exterior to a disc the aim of the present paper is to establish unique solvability theorems for approximate problems derived by fundamental solution method applied to the dirichlet boundary value problem of the two dimensional reduced wave equation in the exterior region of a disc

a new twist two step iterative shrinkage thresholding algorithms for image restoration iterative shrinkage thresholding 1st algorithms have been recently proposed to handle a class of convex unconstrained optimization problems arising in image restoration and other linear inverse problems this class of problems results from combining a linear observation model with a nonquadratic regularizer e g total variation or wavelet based regularization it happens that the convergence rate of these 1st algorithms depends heavily on the linear observation operator becoming very slow when this operator is ill conditioned or ill posed in this paper we introduce two step 1st twist algorithms exhibiting much faster convergence rate than 1st for ill conditioned problems for a vast class of nonquadratic convex regularizers lscrp norms some besov norms and total variation we show that twist converges to a minimizer of the objective function for a given range of values of its parameters for noninvertible observation operators we introduce a monotonic version of twist mtwist although the convergence proof does not apply to this scenario we give experimental evidence that mtwist exhibits similar speed gains over ist the effectiveness of the new methods are experimentally confirmed on problems of image deconvolution and of restoration with missing samples

tikhonov type regularization method for the cauchy problem of the modified helmholtz equation abstract in this paper we propose a numerical method for solving the cauchy problem of the modified helmholtz equation by using green s formula the cauchy problem is transformed to a moment problem then we propose a tikhonov type regularization algorithm for obtaining an approximate solution to the neumann date on the unspecified boundary convergence analysis and error estimation are also discussed for numerical verification several numerical examples in the two dimensional case are presented

a generalization of column generation to accelerate convergence this paper proposes a generalization of column generation reformulating the master problem with fewer variables at the expense of adding more constraints the sub problem structure does not change it shows both analytically and computationally that the reformulation promotes faster convergence to an optimal solution in application to a linear program and to the relaxation of an integer program at each node in the branch and bound tree further it shows that this reformulation subsumes and generalizes prior approaches that have been shown to improve the rate of convergence in special cases

exponential decay of errors of a fundamental solution method applied to a reduced wave problem in the exterior region of a disc this paper concerns a fundamental solution method fsm in abbreviation applied to a reduced wave problem in the exterior region of a disc the convergent rate of approximate solutions to the exact one is proven to be asymptotically exponentially decreasing with respect to the number n of collocation points employed in an approximate problem using obtained fsm solutions we add two numerical tests numerical estimate of errors including cases of high wave numbers and visualization of total waves appeared in the scattering phenomena around a circular obstacle in the cases of k 50 and k 100 where k is a normalized wave number defined through k length of wave number vector x radius of the disc in the second test the total waves almost vanish behind the disc seemingly corresponding to the phenomenon of shadowing in the classical literature of physics

convergence of the method of fundamental solutions for solving the boundary value problem of modified helmholtz equation in this paper we derive the rate of convergence for the method of fundamental methods in solving the boundary value problem of a homogeneous modified helmholtz equation by using the fourier sine expansion to find particular solutions we establish the convergence results for solving the boundary value problem of a non homogeneous equation

generating alternative mixed integer programming models using variable redefinition dropping the complicating constraints in a mixed integer linear program often yields a special structure subproblem that can be reformulated using a different set of decision variables once the new variables have been identified the entire problem can be reformulated in terms of the new variables we develop a theory of variable redefinition based on relating the two sets of decision variables by a linear transformation and describe methods for reformulating the special structure problem the reformulated models have a more accurate linear relaxation than the problems from which they were derived an important property within the context of linear programming based branch and bound modeling approaches

a meshless method for solving the cauchy problem in three dimensional elastostatics the application of the method of fundamental solutions to the cauchy problem in three dimensional isotropic linear elasticity is investigated the resulting system of linear algebraic equations is ill conditioned and therefore its solution is regularized by employing the first order tikhonov functional while the choice of the regularization parameter is based on the l curve method numerical results are presented for both under and equally determined cauchy problems in a piece wise smooth geometry the convergence accuracy and stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain and decreasing the amount of noise added into the input data respectively are analysed

a parallel proximal algorithm for anisotropic total variation minimization total variation tv is a one of the most popular regularizers for stabilizing the solution of ill posed inverse problems this paper proposes a novel proximal gradient algorithm for minimizing tv regularized least squares cost functionals unlike traditional methods that require nested iterations for computing the proximal step of tv our algorithm approximates the latter with several simple proximals that have closed form solutions we theoretically prove that the proposed parallel proximal method achieves the tv solution with arbitrarily high precision at a global rate of converge that is equivalent to the fast proximal gradient methods the results in this paper have the potential to enhance the applicability of tv for solving very large scale imaging inverse problems

diffusion problems using transient discrete source superposition discrete source superposition techniques seek sets of discrete singular source strengths which satisfy the boundary conditions of a particular problem and necessarily the governing differential equation the sum of the fields produced by such sources thus provides an approximation to the required solution in many respects these techniques are a special case of conventional boundary integral equation methods where density distributions of smeared sources are sought in this paper an attempt is made to extend the discrete source superposition technique to transient problems the example addressed is the transient diffusion equation the method developed does seem to have some promise it is found that even coarse spatial discretization can yield quite accurate results and that surprisingly large timesteps can be employed and accuracy maintained further although no direct comparisons are made the absence of spatial integrations indicates that the computational effort per timestep is likely to be less than for conventional bie

a new optimization algorithm for the vehicle routing problem with time windows the vehicle routing problem with time windows vrptw is a generalization of the vehicle routing problem where the service of a customer can begin within the time window defined by the earliest and the latest times when the customer will permit the start of service in this paper we present the development of a new optimization algorithm for its solution the lp relaxation of the set partitioning formulation of the vrptw is solved by column generation feasible columns are added as needed by solving a shortest path problem with time windows and capacity constraints using dynamic programming the lp solution obtained generally provides an excellent lower bound that is used in a branch and bound algorithm to solve the integer set partitioning formulation our results indicate that this algorithm proved to be successful on a variety of practical sized benchmark vrptw test problems the algorithm was capable of optimally solving 100 customer problems this problem size is six times larger than any reported to date by other published research

toeplitz based iterative image reconstruction for mri with correction for magnetic field inhomogeneity in some types of magnetic resonance mr imaging particularly functional brain scans the conventional fourier model for the measurements is inaccurate magnetic field inhomogeneities which are caused by imperfect main fields and by magnetic susceptibility variations induce distortions in images that are reconstructed by conventional fourier methods these artifacts hamper the use of functional mr imaging fmri in brain regions near air tissue interfaces recently iterative methods that combine the conjugate gradient cg algorithm with nonuniform fft nufft operations have been shown to provide considerably improved image quality relative to the conjugate phase method however for non cartesian k space trajectories each cg nufft iteration requires numerous k space interpolations these are operations that are computationally expensive and poorly suited to fast hardware implementations this paper proposes a faster iterative approach to field corrected mr image reconstruction based on the cg algorithm and certain toeplitz matrices this cg toeplitz approach requires k space interpolations only for the initial iteration thereafter only fast fourier transforms ffts are required simulation results show that the proposed cg toeplitz approach produces equivalent image quality as the cg nufft method with significantly reduced computation time

multicommodity network flows the impact of formulation on decomposition this paper investigates the impact of problem formulation on dantzig wolfe decomposition for the multicommodity network flow problem these problems are formulated in three ways origin destination specific destination specific and product specific the path based origin destination specific formulation is equivalent to the tree based destination specific formulation by a simple transformation supersupply and superdemand nodes are appended to the tree based product specific formulation to create an equivalent path based product specific formulation we show that solving the path based problem formulations by decomposition results in substantially fewer master problem iterations and lower cpu times than by using decomposition on the equivalent tree based formulations computational results on a series of multicommodity network flow problems are presented

selected topics in column generation dantzig wolfe decomposition and column generation devised for linear programs is a success story in large scale integer programming we outline and relate the approaches and survey mainly recent contributions not yet found in textbooks we emphasize the growing understanding of the dual point of view which has brought considerable progress to the column generation theory and practice it stimulated careful initializations sophisticated solution techniques for the restricted master problem and subproblem as well as better overall performance thus the dual perspective is an ever recurring concept in our selected topics

fast iterative image reconstruction for mri in the presence of field inhomogeneities in magnetic resonance imaging magnetic field inhomogeneities cause distortions in images that are reconstructed by conventional fast fourier transform fft methods several noniterative image reconstruction methods are used currently to compensate for field inhomogeneities but these methods assume that the field map that characterizes the off resonance frequencies is spatially smooth recently iterative methods have been proposed that can circumvent this assumption and provide improved compensation for off resonance effects however straightforward implementations of such iterative methods suffer from inconveniently long computation times this paper describes a tool for accelerating iterative reconstruction of field corrected mr images a novel time segmented approximation to the mr signal equation we use a min max formulation to derive the temporal interpolator speedups of around 60 were achieved by combining this temporal interpolator with a nonuniform fast fourier transform with normalized root mean squared approximation errors of 0 07 the proposed method provides fast accurate field corrected image reconstruction even when the field map is not smooth

nonlinear wavelet image processing variational problems compression and noise removal through wavelet shrinkage this paper examines the relationship between wavelet based image processing algorithms and variational problems algorithms are derived as exact or approximate minimizers of variational problems in particular we show that wavelet shrinkage can be considered the exact minimizer of the following problem given an image f defined on a square i minimize over all g in the besov space b11 l1 i the functional f g l2 i 2 g b11 l1 i we use the theory of nonlinear wavelet image compression in l2 i to derive accurate error bounds for noise removal through wavelet shrinkage applied to images corrupted with i i d mean zero gaussian noise a new signal to noise ratio snr which we claim more accurately reflects the visual perception of noise in images arises in this derivation we present extensive computations that support the hypothesis that near optimal shrinkage parameters can be derived if one knows or can estimate only two parameters about an image f the largest for which f bq lq i 1 q 2 1 2 and the norm f bq lq i both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than donoho and johnstone s visushrink procedure an example suggests however that donoho and johnstone s 1994 1995 1996 sureshrink method which uses a different shrinkage parameter for each dyadic level achieves a lower error than our procedure

dual optimal inequalities for stabilized column generation column generation is one of the most successful approaches for solving large scale linear programming problems however degeneracy difficulties and long tail effects are known to occur as problems become larger in recent years several stabilization techniques of the dual variables have proven to be effective we study the use of two types of dual optimal inequalities to accelerate and stabilize the whole convergence process added to the dual formulation these constraints are satisfied by all or a subset of the dual optimal solutions therefore the optimal objective function value of the augmented dual problem is identical to the original one adding constraints to the dual problem leads to adding columns to the primal problem and feasibility of the solution may be lost we propose two methods for recovering primal feasibility and optimality depending on the type of inequalities that are used our computational experiments on the binary and the classical cutting stock problems and more specifically on the so called triplet instances show that the use of relevant dual information has a tremendous effect on the reduction of the number of column generation iterations

nonuniform fast fourier transforms using min max interpolation the fast fourier transform fft is used widely in signal processing for efficient computation of the ft of finite length signals over a set of uniformly spaced frequency locations however in many applications one requires nonuniform sampling in the frequency domain i e a nonuniform ft several papers have described fast approximations for the nonuniform ft based on interpolating an oversampled fft this paper presents an interpolation method for the nonuniform ft that is optimal in the min max sense of minimizing the worst case approximation error over all signals of unit norm the proposed method easily generalizes to multidimensional signals numerical results show that the min max approach provides substantially lower approximation errors than conventional interpolation methods the min max criterion is also useful for optimizing the parameters of interpolation kernels such as the kaiser bessel function

a minimal algorithm for the multiple choice knapsack problem abstract the multiple choice knapsack problem is defined as a 0 1 knapsack problem with the addition of disjoined multiple choice constraints as for other knapsack problems most of the computational effort in the solution of these problems is used for sorting and reduction but although o n algorithms which solve the linear multiple choice knapsack problem without sorting have been known for more than a decade such techniques have not been used in enumerative algorithms in this paper we present a simple o n partitioning algorithm for deriving the optimal linear solution and show how it may be incorporated in a dynamic programming algorithm such that a minimal number of classes are enumerated sorted and reduced computational experiments indicate that this approach leads to a very efficient algorithm which outperforms any known algorithm for the problem

a note on a lp based heuristic for a time constrained routing problem in their paper avella et al 2006 investigate a time constrained routing problem the core of the proposed solution approach is a large scale linear program that grows both row and column wise when new variables are introduced thus a column and row generation algorithm is proposed to solve this linear program optimally and an optimality condition is presented to terminate the column and row generation algorithm we demonstrate by using lagrangian duality that this optimality condition is incorrect and may lead to a suboptimal solution at termination

non cartesian mri reconstruction with automatic regularization via monte carlo sure magnetic resonance image mri reconstruction from undersampled k space data requires regularization to reduce noise and aliasing artifacts proper application of regularization however requires appropriate selection of associated regularization parameters in this work we develop a data driven regularization parameter adjustment scheme that minimizes an estimate based on the principle of stein s unbiased risk estimate sure of a suitable weighted squared error measure in k space to compute this sure type estimate we propose a monte carlo scheme that extends our previous approach to inverse problems e g mri reconstruction involving complex valued images our approach depends only on the output of a given reconstruction algorithm and does not require knowledge of its internal workings so it is capable of tackling a wide variety of reconstruction algorithms and nonquadratic regularizers including total variation and those based on the l1 norm experiments with simulated and real mr data indicate that the proposed approach is capable of providing near mean squared error optimal regularization parameters for single coil undersampled non cartesian mri reconstruction

computation of the scattering amplitude for a scattering wave produced by a disc approach by a fundamental solution method our fundamental solution method gives an analytic representation of the approximate solution for the reduced wave problem in the exterior region of a disc the asymptotic behavior of this representation yields an approximate formula for the scattering amplitude an error estimate for this formula is given we add two numerical tests the numerical estimate of errors and profiles of scattering cross sections and the far field coefficient both tests include cases of high wave numbers

comparable upper and lower bounds for boundary values of neumann eigenfunctions and tight inclusion of eigenvalues for smooth bounded domains in rn we prove upper and lower l2 bounds on the boundary data of neumann eigenfunctions and we prove quasiorthogonality of this boundary data in a spectral window the bounds are tight in the sense that both are independent of the eigenvalues this is achieved by working with an appropriate norm for boundary functions which includes a spectral weight that is a function of the boundary laplacian this spectral weight is chosen to cancel concentration at the boundary that can happen for whispering gallery type eigenfunctions these bounds are closely related to wave equation estimates due to tataru using this we bound the distance from an arbitrary helmholtz parameter e 0 to the nearest neumann eigenvalue in terms of boundary normal derivative data of a trial function u solving the helmholtz equation e u 0 this inclusion bound improves over previously known bounds by a factor of e5 6 analogously to a recently improved inclusion bound in the dirichlet case due to the first two authors finally we apply our theory to present an improved numerical implementation of the method of particular solutions for computation of neumann eigenpairs on smooth planar domains we show that the new inclusion bound improves the relative accuracy in a computed neumann eigenvalue around the 42000th from nine to fourteen digits with negligible extra numerical effort

mathematical and numerical studies on meshless methods for exterior unbounded domain problems the method of fundamental solutions mfs is an efficient meshless method for solving boundary value problems in an exterior unbounded domain the numerical solution obtained by the mfs is accurate while the corresponding matrix equation is ill conditioned a modified mfs mmfs with the proper basis functions is proposed by the introduction of the modified trefftz method mtm the concrete expressions of the corresponding condition numbers are given in mathematical forms and the solvability by these methods is mathematically proven thereby the optimal parameter minimizing the condition number is also mathematically given numerical experiments show that the condition numbers of the matrices corresponding to the mtm and the mmfs are reduced and that the numerical solution by the mmfs is more accurate than the one by the conventional method

analysis of the method of fundamental solutions for the modified helmholtz equation the method of fundamental solutions mfs was first proposed by kupradze in 1963 since then the mfs has been extensively applied for solving various kinds of problems in science and engineering however few theoretical works have been reported in the literature in this paper we devote our work to the error analysis and stability of the mfs for the case of the modified helmholtz equation for disk domains a convergence analysis of the mfs was provided by li j comput appl math 159 113125 2004 for solving the modified helmholtz equation in this paper the error bounds of the mfs for bounded and simply connected domains are derived for smooth solutions of the modified helmholtz equation the exponential convergence rates can be achieved for analytic solutions the bounds of condition numbers of the mfs are derived for both disk domains and the bounded and simply connected domains to give the exponential growth via the number of fundamental solutions used numerical experiments are carried out to support the theoretical analysis moreover the analysis of this paper is applied to parabolic equations and some reviews of proof techniques and the analytical characteristics of the mfs are addressed

a practical guide to the recovery of wavelet coefficients from fourier measurements in a series of recent papers b adcock a c hansen and c poon appl comput harmon anal 36 2014 pp 387 415 b adcock m gataric and a c hansen siam j imaging sci 7 2014 pp 1690 1723 adcock et al siam j math anal 47 2015 pp 1196 1233 it was shown that one can optimally recover the wavelet coefficients of an unknown compactly supported function from pointwise evaluations of its fourier transform via the method of generalized sampling while these papers focused on the optimality of generalized sampling in terms of its stability and error bounds the current paper explains how this optimal method can be implemented to yield a computationally efficient algorithm in particular we show that generalized sampling has a computational complexity of mathcal o left m n log n right when recovering the first n boundary corrected wavelet coefficients of an unknown compactly supported function from m n fourier samples therefore due to the linear correspondences bet

on solving boundary value problems of modified helmholtz equations by plane wave functions plane wave functions e x w in r2 where 0 x x y w cos sin and x w x cos y sin are used as basis functions to solve boundary value problems of modified helmholtz equations u x 2u x 0 x u x h x x where is the laplace operator and a bounded and simply connected domain in r2 approximations of the exact solution of the above problem by plane wave functions are explicitly constructed for the case that is a disc and the order of approximations is derived a computational algorithm by collocation methods based on a simple singular decomposition of circular matrices is proposed and numerical examples are shown to demonstrate the efficiency of the methods

a fast multilevel algorithm for wavelet regularized image restoration we present a multilevel extension of the popular ldquothresholded landweberrdquo algorithm for wavelet regularized image restoration that yields an order of magnitude speed improvement over the standard fixed scale implementation the method is generic and targeted towards large scale linear inverse problems such as 3 d deconvolution microscopy the algorithm is derived within the framework of bound optimization the key idea is to successively update the coefficients in the various wavelet channels using fixed subband adapted iteration parameters step sizes and threshold levels the optimization problem is solved efficiently via a proper chaining of basic iteration modules the higher level description of the algorithm is similar to that of a multigrid solver for pdes but there is one fundamental difference the latter iterates though a sequence of multiresolution versions of the original problem while in our case we cycle through the wavelet subspaces corresponding to the difference between successive approximations this strategy is motivated by the special structure of the problem and the preconditioning properties of the wavelet representation we establish that the solution of the restoration problem corresponds to a fixed point of our multilevel optimizer we also provide experimental evidence that the improvement in convergence rate is essentially determined by the unconstrained linear part of the algorithm irrespective of the type of wavelet finally we illustrate the technique with some image deconvolution examples including some real 3 d fluorescence microscopy data

numerical implementation of the edem for modified helmholtz bvps on annular domains in a recent paper by the current authors a new methodology called the extended domain eigenfunction method edem was proposed for solving elliptic boundary value problems on annular like domains in this paper we present and investigate one possible numerical algorithm to implement the edem this algorithm is used to solve modified helmholtz bvps on annular like domains two examples of annular like domains are studied the results and performance are compared with those of the well known boundary element method bem the high accuracy of the edem solutions and the superior efficiency of the edem over the bem make edem an excellent alternate candidate to use in the animation industry where speed is a predominant requirement and by the scientific community where accuracy is the paramount objective

fundamental solutions method for elliptic boundary value problems we consider a procedure for solving boundary value problems for elliptic homogeneous equations known as the fundamental solutions method we prove its applicability for some second order operators as well as for fourth order ones the boundary conditions of an elliptic problem are approximated by using fundamental solutions of the corresponding operator with singularities located outside the domain of interest in a specific case of the laplacian an estimate shows that the method discussed possesses convergence properties as good as those of any method using harmonic polynomials as trial functions numerical examples are given among which are simple signorini s problems for harmonic and lame operators

approximation of surfaces by modified helmholtz splines abstract we present a smoothing method of a set of points to be approximated from a given boundary value problem for the modified helmholtz equation a solution of such problem will be called mh spline two convergence results are carefully established some numerical and graphic examples are shown in order to test the validity of the method

an augmented rbf method for solving fractional sturm liouville eigenvalue problems in this work we construct numerical schemes using radial basis functions rbfs to solve sturm liouville eigenvalue problems in which the so called fractional derivatives are involved we take into account the fact that the solutions to such problems might be nonsmooth and consequently we augment the space generated by rbf base functions with some fractional polynomials to overcome this problem we show through some examples the effectiveness of our method

deep convolutional neural network for inverse problems in imaging in this paper we propose a novel deep convolutional neural network cnn based algorithm for solving ill posed inverse problems regularized iterative algorithms have emerged as the standard approach to ill posed inverse problems in the past few decades these methods produce excellent results but can be challenging to deploy in practice due to factors including the high computational cost of the forward and adjoint operators and the difficulty of hyperparameter selection the starting point of this paper is the observation that unrolled iterative methods have the form of a cnn filtering followed by pointwise non linearity when the normal operator h h where h is the adjoint of the forward imaging operator h of the forward model is a convolution based on this observation we propose using direct inversion followed by a cnn to solve normal convolutional inverse problems the direct inversion encapsulates the physical model of the system but leads to artifacts when the problem is ill posed the cnn combines multiresolution decomposition and residual learning in order to learn to remove these artifacts while preserving image structure we demonstrate the performance of the proposed network in sparse view reconstruction down to 50 views on parallel beam x ray computed tomography in synthetic phantoms as well as in real experimental sinograms the proposed network outperforms total variation regularized iterative reconstruction for the more realistic phantoms and requires less than a second to reconstruct a 512 times 512 image on the gpu

a meshless method for some inverse problems associated with the helmholtz equation in this paper a new numerical scheme based on the method of fundamental solutions is proposed for the numerical solution of some inverse boundary value problems associated with the helmholtz equation including the cauchy problem since the resulting matrix equation is badly ill conditioned a regularized solution is obtained by employing truncated singular value decomposition while the regularization parameter for the regularization method is provided by the l curve method numerical results are presented for problems on smooth and piecewise smooth domains with both exact and noisy data and the convergence and stability of the scheme are investigated these results show that the proposed scheme is highly accurate computationally efficient stable with respect to the noise in the data and convergent with respect to decreasing the amount of data noise and increasing the distance between the physical and fictitious boundaries and could be considered as a competitive alternative to existing methods for these problems

a partition of unity fem for time dependent diffusion problems using multiple enrichment functions an enriched partition of unity fem is developed to solve time dependent diffusion problems in the present formulation multiple exponential functions describing the spatial and temporal diffusion decay are embedded in the finite element approximation space the resulting enrichment is in the form of a local asymptotic expansion unlike previous works in this area where the enrichment must be updated at each time step here the temporal decay in the solution is embedded in the asymptotic expansion thus the system matrix that is evaluated at the first time step may be decomposed and retained for every next time step by just updating the right hand side of the linear system of equations the advantage is a significant saving in the computational effort where previously the linear system must be reevaluated and resolved at every time step in comparison with the traditional finite element analysis with p version refinements the present approach is much simpler more efficient and yields more accurate solutions for a prescribed number of dofs numerical results are presented for a transient diffusion equation with known analytical solution the performance of the method is analyzed on two applications the transient heat equation with a single source and the transient heat equation with multiple sources the aim of such a method compared with the classical fem is to solve time dependent diffusion applications efficiently and with an appropriate level of accuracy

study on connections of the mfs trefftz method indirect biem and invariant mfs in the three dimensional laplace problems containing spherical boundaries following the success of a study on the method of fundamental solutions using an image concept 13 we extend to solve the three dimensional laplace problems containing spherical boundaries by using the three approaches the case of eccentric sphere for the laplace problem is considered the optimal locations for the source distribution to include the foci in the mfs are also examined by using the image concept in the 3d problems whether a free constant is required or not in the mfs is also studied the error distribution is discussed after comparing with the analytical solution derived by using the bispherical coordinates besides the relationship between the trefftz bases and the singularity in the mfs for the three dimensional laplace problems is also addressed it is found that one source of the mfs contains several interior and exterior trefftz sets through a degenerate kernel on the contrary one single trefftz base can be superimposed by some lumped sources in the mfs through an indirect biem based on this finding the relationship between the fictitious boundary densities of the indirect biem and the singularity strength in the mfs can be constructed due to the fact that the mfs is a lumped version of an indirect biem

improved computational efficiency of locally low rank mri reconstruction using iterative random patch adjustments this paper presents and analyzes an alternative formulation of the locally low rank llr regularization framework for magnetic resonance image mri reconstruction generally llr based mri reconstruction techniques operate by dividing the underlying image into a collection of matrices formed from image patches each of these matrices is assumed to have low rank due to the inherent correlations among the data whether along the coil temporal or multi contrast dimensions the llr regularization has been successful for various mri applications such as parallel imaging and accelerated quantitative parameter mapping however a major limitation of most conventional implementations of the llr regularization is the use of multiple sets of overlapping patches although the use of overlapping patches leads to effective shift invariance it also results in high computational load which limits the practical utility of the llr regularization for mri to circumvent this problem alternative llr based algorithms instead shift a single set of non overlapping patches at each iteration thereby achieving shift invariance and avoiding block artifacts a novel contribution of this paper is to provide a mathematical framework and justification of llr regularization with iterative random patch adjustments llr irpa this method is compared with a state of the art llr regularization algorithm based on overlapping patches and it is shown experimentally that results are similar but with the advantage of much reduced computational load we also present theoretical results demonstrating the effective shift invariance of the llr irpa approach and we show reconstruction examples and comparisons in both retrospectively and prospectively undersampled mri acquisitions and in t1 parameter mapping

a branch and price algorithm for the generalized assignment problem the generalized assignment problem examines the maximum profit assignment of jobs to agents such that each job is assigned to precisely one agent subject to capacity restrictions on the agents a new algorithm for the generalized assignment problem is presented that employs both column generation and branch and bound to obtain optimal integer solutions to a set partitioning formulation of the problem

the method of fundamental solutions for solving options pricing models abstract this paper provides a foundation of the method of fundamental solutions mfs for the options pricing models governed by the black scholes equation in which both the european option and american options are considered in the solution procedure no artificial boundary conditions are imposed for both datum and infinite sides of the stock prices in the cases of the european options no time marching procedures are required and numerical results are verified with the exact solutions since the free boundary conditions are considered for the american options boundary update procedure is thus applied at the same time numerical results are compared with the results in the literatures these numerical results indicate the mfs is an effective and robust meshless numerical solution for solving the options pricing models

the method of fundamental solutions applied to boundary value problems on the surface of a sphere abstract in this work we propose using the method of fundamental solutions mfs to solve boundary value problems for the helmholtz beltrami equation on a sphere we prove density and convergence results that justify the proposed mfs approximation several numerical examples are considered to illustrate the good performance of the method

a fast wavelet based reconstruction method for magnetic resonance imaging in this work we exploit the fact that wavelets can represent magnetic resonance images well with relatively few coefficients we use this property to improve magnetic resonance imaging mri reconstructions from undersampled data with arbitrary k space trajectories reconstruction is posed as an optimization problem that could be solved with the iterative shrinkage thresholding algorithm ista which unfortunately converges slowly to make the approach more practical we propose a variant that combines recent improvements in convex optimization and that can be tuned to a given specific k space trajectory we present a mathematical analysis that explains the performance of the algorithms using simulated and in vivo data we show that our nonlinear method is fast as it accelerates ista by almost two orders of magnitude we also show that it remains competitive with tv regularization in terms of image quality

a fast thresholded landweber algorithm for wavelet regularized multidimensional deconvolution we present a fast variational deconvolution algorithm that minimizes a quadratic data term subject to a regularization on the norm of the wavelet coefficients of the solution previously available methods have essentially consisted in alternating between a landweber iteration and a wavelet domain soft thresholding operation while having the advantage of simplicity they are known to converge slowly by expressing the cost functional in a shannon wavelet basis we are able to decompose the problem into a series of subband dependent minimizations in particular this allows for larger subband dependent step sizes and threshold levels than the previous method this improves the convergence properties of the algorithm significantly we demonstrate a speed up of one order of magnitude in practical situations this makes wavelet regularized deconvolution more widely accessible even for applications with a strong limitation on computational complexity we present promising results in 3 d deconvolution microscopy where the size of typical data sets does not permit more than a few tens of iterations

3d steerable wavelets in practice we introduce a systematic and practical design for steerable wavelet frames in 3d our steerable wavelets are obtained by applying a 3d version of the generalized riesz transform to a primary isotropic wavelet frame the novel transform is self reversible tight frame and its elementary constituents riesz wavelets can be efficiently rotated in any 3d direction by forming appropriate linear combinations moreover the basis functions at a given location can be linearly combined to design custom and adaptive steerable wavelets the features of the proposed method are illustrated with the processing and analysis of 3d biomedical data in particular we show how those wavelets can be used to characterize directional patterns and to detect edges by means of a 3d monogenic analysis we also propose a new inverse problem formalism along with an optimization algorithm for reconstructing 3d images from a sparse set of wavelet domain edges the scheme results in high quality image reconstructions which demonstrate the feature reduction ability of the steerable wavelets as well as their potential for solving inverse problems

a new theoretical error estimate of the method of fundamental solutions applied to reduced wave problems in the exterior region of a disk in this paper we present a mathematical study of the method of fundamental solutions mfs applied to reduced wave problems with dirichlet boundary conditions in the exterior domain of a disk a theorem in this paper shows that the mfs with n source points in equi distantly equally phased arrangement with assignment parameter q 0

dynamic programming and strong bounds for the 0 1 knapsack problem two new algorithms recently proved to outperform all previous methods for the exact solution of the 0 1 knapsack problem this paper presents a combination of such approaches where in additi on valid inequalities are generated and surrogate relaxed and a new initial core problem is adopted the algorithm is able to solve all classical test instances with up to 10 000 variables in less than 0 2 seconds on a hp9000 735 99 computer the c language implementation of the algorithm is available on the internet

the method of fundamental solutions for helmholtz type equations in composite materials in this paper we introduce and develop the method of fundamental solutions mfs for solving helmholtz type elliptic partial differential equations in composite materials this study builds upon the previous developments and applications of the mfs to linear and nonlinear heat conduction elasticity and functionally graded composite layered materials numerical results are presented and discussed for four examples involving both the modified helmholtz and the helmholtz equations in two dimensional or three dimensional bounded or unbounded smooth or non smooth composite domains it was found that the method produces numerical results which are in good agreement with the analytical solutions where available

error analysis of the method of fundamental solutions for linear elastostatics for linear elastostatics in 2d the trefftz methods i e the boundary methods using the particular solutions and the fundamental solutions satisfying the cauchy navier equation lead to the method of particular solutions mps and the method of fundamental solutions mfs respectively in this paper the mixed types of the displacement and the traction boundary conditions are dealt with and both the direct collocation techniques and the lagrange multiplier are used to couple the boundary conditions the former is just the mfs and the mps and the latter is also called the hybrid trefftz method htm in jirousek 1978 1992 1996 1 3 in bogomolny 1985 4 and li 2009 5 the error analysis of the mfs is given for laplace s equation and in li 2012 6 the error bounds of both mps and htm using particular solutions ps are provided for linear elastostatics in this paper our efforts are devoted to explore the error analysis of the mfs and the htm using fundamental solutions fs the key analysis is to derive the errors between fs and ps of the linear elastostatics where the expansions of the fs in li et al 2011 7 are a basic tool in analysis then the optimal convergence rates can be achieved for the mfs and the htm using fs recently the mfs has been developed with numerous reports in computation the analysis is behind the analysis of the mfs for linear elastostatics in this paper may narrow the existing gap between computation and theory of the mfs

the method of fundamental solutions and quasi monte carlo method for diffusion equations the laplace transform is applied to remove the time dependent variable in the diffusion equation for non harmonic initial conditions this gives rise to a non homogeneous modified helmholtz equation which we solve by the method of fundamental solutions to do this a particular solution must be obtained which we find through a method suggested by atkinson to avoid costly gaussian quadratures we approximate the particular solution using quasi monte carlo integration which has the advantage of ignoring the singularity in the integrand the approximate transformed solution is then inverted numerically using stehfest s algorithm two numerical examples are given to illustrate the simplicity and effectiveness of our approach to solving diffusion equations in 2 d and 3 d 1998 john wiley sons ltd

fmm accelerated source model technique for many scatterer problems motivated by the lack of scalability of the source model technique smt to many scatterer problems a new smt based method for 3 d many scatterer problems is developed the large time complexity of the smt is reduced by deriving an implicit compact representation of the scattering problem via a fast multipole formulation which divides the interactions into far and near ones while grouping and approximating the far interactions this results in significant acceleration of classical smt which is demonstrated in a variety of 3 d many body scattering problems the improvement is notable already at the minimal number of unknowns necessary and grows with the size of the problem

error estimates for a fundamental solution method applied to reduced wave problems in a domain exterior to a disc this paper concerns error estimates for an approximation method for solving the dirichlet boundary value problem of the 2 dimensional reduced wave equation in the exterior region of a disc from theoretical and practical points of view

solution of helmholtz equation by trefftz method the application of the trefftz method for calculating wave forces on offshore structures is presented indirect and direct formulations using complete and non singular systems of trefftz functions for the helmholtz equation are posed in this paper an effective technique using different interpolation functions for the velocity potential and wave force are suggested to improve the computational accuracy of the wave force the numerical examples show that the present method is highly efficient and accurate

exponential wavelet iterative shrinkage thresholding algorithm for compressed sensing magnetic resonance imaging it is beneficial for both hospitals and patients to accelerate mri scanning recently a new fast mri technique based on cs was proposed however the reconstruction quality and computation time of cs mri did not meet the standard of clinical use therefore we proposed a novel algorithm based on three successful components the sparsity of ewt the rapidness of fista and the excellent tuning in sista the proposed method was dubbed exponential wavelet iterative shrinkage threshold algorithm ewista experiments over four kinds of mr images brain ankle knee and adhd indicated that the proposed ewista showed better reconstruction performance than the state of the art algorithms such as fcsa ista fista sista and ewt ista moreover ewista was faster than ista and ewt ista but slightly slower than fcsa fista and sista

wavelet theory demystified we revisit wavelet theory starting from the representation of a scaling function as the convolution of a b spline the regular part of it and a distribution the irregular or residual part this formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory including some new extensions for fractional orders n a self contained accessible fashion in particular we prove that the b spline component is entirely responsible for five key wavelet properties order of approximation reproduction of polynomials vanishing moments multiscale differentiation property and smoothness regularity of the basis functions we also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions this allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal additional results include a new peeling theory of smoothness leading to the extended notion of wavelet differentiability in the l sub p sense and a sharper theorem stating that smoothness implies order