seismic data restoration via data driven tight frame restoration interpolation of missing traces plays a crucial role in the seismic data processing pipeline efficient restoration methods have been proposed based on sparse signal representation in a transform domain such as fourier wavelet curvelet and shearlet transforms most existing methods are based on transforms with a fixed basis we considered an adaptive sparse transform for restoration of data with complex structures in particular we evaluated a data driven tight frame based sparse regularization method for seismic data restoration the main idea of the data driven tight frame tf is to adaptively learn a set of framelet filters from the currently interpolated data under which the data can be more sparsely represented hence the sparsity promoting l1 norm spl1 minimization methods can produce better restoration results by using the learned filters a split inexact uzawa algorithm which can be viewed as a generalization of the alternating direction of multiplier method admm was applied to solve the presented spl1 model numerical tests were performed on synthetic and real seismic data for restoration of randomly missing traces over a regular data grid our experiments showed that our proposed method obtains the state of the art restoration results in comparison with the traditional fourier based projection onto convex sets the tight frame based method and the recent shearlet regularization admm method

new developments in the primal dual column generation technique abstract the optimal solutions of the restricted master problems typically leads to an unstable behavior of the standard column generation technique and consequently originates an unnecessarily large number of iterations of the method to overcome this drawback variations of the standard approach use interior points of the dual feasible set instead of optimal solutions in this paper we focus on a variation known as the primal dual column generation technique which uses a primal dual interior point method to obtain well centered non optimal solutions of the restricted master problems we show that the method converges to an optimal solution of the master problem even though non optimal solutions are used in the course of the procedure also computational experiments are presented using linear relaxed reformulations of three classical integer programming problems the cutting stock problem the vehicle routing problem with time windows and the capacitated lot sizing problem with setup times the numerical results indicate that the appropriate use of a primal dual interior point method within the column generation technique contributes to a reduction of the number of iterations as well as the running times on average furthermore the results show that the larger the instance the better the relative performance of the primal dual column generation technique

a warm start approach for large scale stochastic linear programs we describe a way of generating a warm start point for interior point methods in the context of stochastic programming our approach exploits the structural information of the stochastic problem so that it can be seen as a structure exploiting initial point generator we solve a small scale version of the problem corresponding to a reduced event tree and use the solution to generate an advanced starting point for the complete problem the way we produce a reduced tree tries to capture the important information in the scenario space while keeping the dimension of the corresponding reduced deterministic equivalent small we derive conditions which should be satisfied by the reduced tree to guarantee a successful warm start of the complete problem the implementation within the hopdm and oops interior point solvers shows remarkable advantages

construction of parseval wavelets from redundant filter systems we consider wavelets in l2 rd which have generalized multiresolutions this means that the initial resolution subspace v0 in l2 rd is not singly generated as a result the representation of the integer lattice zd restricted to v0 has a nontrivial multiplicity function we show how the corresponding analysis and synthesis for these wavelets can be understood in terms of unitary matrix valued functions on a torus acting on a certain vector bundle specifically we show how the wavelet functions on rd can be constructed directly from the generalized wavelet filters

pointwise rates of convergence for the oliker prussner method for the monge ampere equation we study the oliker prussner method exploiting its geometric nature we derive discrete stability and continuous dependence estimates in the max norm by using a discrete alexandroff estimate and the brunn minkowski inequality we show that the method is exact for all convex quadratic polynomials provided the underlying set of nodes is translation invariant within the domain nodes still conform to the domain boundary this gives a suitable notion of operator consistency which combined with stability leads to pointwise rates of convergence for classical and non classical solutions of the monge ampere equation

exponential bases paley wiener spaces and applications abstract we investigate the connection between translation bases for paley wiener spaces and exponential fourier bases for a domain we apply these results to the characterization of vector valued time frequency translations of a paley wiener window signal

blind motion deblurring using multiple images recovery of degraded images due to motion blurring is a challenging problem in digital imaging most existing techniques on blind deblurring are not capable of removing complex motion blurring from the blurred images of complex structures one promising approach is to recover the clear image using multiple images captured for the scene however in practice it is observed that such a multi frame approach can recover a high quality clear image of the scene only after multiple blurred image frames are accurately aligned during pre processing which is a very challenging task even with user interactions in this paper by exploring the sparsity of the motion blur kernel and the clear image under certain domains we propose an alternative iteration approach to simultaneously identify the blur kernels of given blurred images and restore a clear image our proposed approach is not only robust to image formation noises but is also robust to the alignment errors among multiple images a modified version of linearized bregman iteration is then developed to efficiently solve the resulting minimization problem experiments show that our proposed algorithm is capable of accurately estimating the blur kernels of complex camera motions with minimal requirements on the accuracy of image alignment as a result our method is capable of automatically recovering a high quality clear image from multiple blurred images

from frame like wavelets to wavelet frames keeping approximation properties and symmetry abstract for a given symmetric refinable mask obeying the sum rule of order n an explicit method is suggested for the construction of mutually symmetric almost frame like wavelet system providing approximation order n a transformation based on the lifting scheme is described that allows to improve almost frame like wavelets up to dual wavelet frames and preserve other properties a direct method for the construction of dual wavelet frames providing approximation order n and mutual symmetry properties is also discussed for an abelian symmetry group h a technique providing the h symmetry property for each wavelet function is given for the above three methods

the morley element for fourth order elliptic equations in any dimensions in this paper the well known nonconforming morley element for biharmonic equations in two spatial dimensions is extended to any higher dimensions in a canonical fashion the general n dimensional morley element consists of all quadratic polynomials defined on each n simplex with degrees of freedom given by the integral average of the normal derivative on each n 1 subsimplex and the integral average of the function value on each n 2 subsimplex explicit expressions of nodal basis functions are also obtained for this element on general n simplicial grids convergence analysis is given for this element when it is applied as a nonconforming finite element discretization for the biharmonic equation

a fast optimization transfer algorithm for image inpainting in wavelet domains a wavelet inpainting problem refers to the problem of filling in missing wavelet coefficients in an image a variational approach was used by chan et al the resulting functional was minimized by the gradient descent method in this paper we use an optimization transfer technique which involves replacing their univariate functional by a bivariate functional by adding an auxiliary variable our bivariate functional can be minimized easily by alternating minimization for the auxiliary variable the minimum has a closed form solution and for the original variable the minimization problem can be formulated as a classical total variation tv denoising problem and hence can be solved efficiently using a dual formulation we show that our bivariate functional is equivalent to the original univariate functional we also show that our alternating minimization is convergent numerical results show that the proposed algorithm is very efficient and outperforms that of chan et al

coupled variational image decomposition and restoration model for blurred cartoon plus texture images with missing pixels in this paper we develop a decomposition model to restore blurred images with missing pixels our assumption is that the underlying image is the superposition of cartoon and texture components we use the total variation norm and its dual norm to regularize the cartoon and texture respectively we recommend an efficient numerical algorithm based on the splitting versions of augmented lagrangian method to solve the problem theoretically the existence of a minimizer to the energy function and the convergence of the algorithm are guaranteed in contrast to recently developed methods for deblurring images the proposed algorithm not only gives the restored image but also gives a decomposition of cartoon and texture parts these two parts can be further used in segmentation and inpainting problems numerical comparisons between this algorithm and some state of the art methods are also reported

on uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde s we prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations one of them extends some recent uniqueness results by jensen some establish the uniqueness of solutions for second order isaacs equations and hence include the uniqueness results for bellman equations by p l lions our comparison results apply even for discontinuous solutions and so perron s method readily yields the existence of continuous solutions

matricial filters and crystallographic composite dilation wavelets in 7 8 guo labate lim weiss and wilson introduced the theory of mra composite dilation wavelets we continue their work by studying the filter properties of such wavelets and present several important examples

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

the structure of shift invariant subspaces of l2 rn using the range function approach to shift invariant spaces in l2 rn we give a simple characterization of frames and riesz families generated by shifts of a countable set of generators in terms of their behavior on subspaces of l2 zn this in turn gives a simplified approach to the analysis of frames and riesz families done by gramians and dual gramians we prove a decomposition of a shift invariant space into the orthogonal sum of spaces each of which is generated by a quasi orthogonal generator as an application of this fact we characterize shift preserving operators in terms of range operators and prove some facts about the dimension function

construction and reconstruction of tight framelets and wavelets via matrix mask functions abstract the paper develops construction procedures for tight framelets and wavelets using matrix mask functions in the setting of a generalized multiresolution analysis gmra we show the existence of a scaling vector of a gmra such that its first component exhausts the spectrum of the core space near the origin the corresponding low pass matrix mask has an especially advantageous form enabling an effective reconstruction procedure of the original scaling vector we also prove a generalization of the unitary extension principle for an infinite number of generators this results in the construction scheme for tight framelets using low pass and high pass matrix masks generalizing the classical mra constructions we prove that our scheme is flexible enough to reconstruct all possible orthonormal wavelets as an illustration we exhibit a pathwise connected class of non msf non mra wavelets sharing the same wavelet dimension function

the lifting factorization and construction of wavelet bi frames with arbitrary generators and scaling in this paper we present the lifting factorization and construction of wavelet bi frames with arbitrary generators and scaling we show that an arbitrary polyphase matrix of a wavelet bi frame can be factorized into a series of lifting steps based on the proposed factorization we present a general construction of bi frames especially we give an explicit formula to construct the bi frames of two scaling and two generators with symmetry and one vanishing moment this paper does not involve many theories of wavelet frames in mathematics but focuses on the algebraic issues related to laurent polynomials which are efficient expressions in redundant filter banks associated with wavelet frames as an extension of the classical two channel filter bank the redundant filter bank is more complicated but also more flexible furthermore we present an algorithm to increase the number of vanishing moments to arbitrary order by lifting which is iterated and is straightforward in implementation

properties of the solutions of the linearized monge ampere equation let r n r be a function strictly convex and smooth and det d 2 is the monge ampere generated by given x r n and t 0 let s x t y r n y x x y x t the purpose of this paper is to study the properties of the solutions of the linearized monge ampere equation given by a ij x d ij u 0 where the coefficients a ij x are the cofactors of the matrix d 2 x it is assumed that satisfies a doubling condition on the sets s x t and a uniform continuity condition at every scale with respect to lebesgue measure we establish that the distribution functions of nonnegative solutions u at altitude t decay like a negative power of t and prove an invariant harnack s inequality on the sections s x t all the estimates are independent of the regularity of and depend only on the constants in the hypotheses made on the measure

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

framelet based blind motion deblurring from a single image how to recover a clear image from a single motion blurred image has long been a challenging open problem in digital imaging in this paper we focus on how to recover a motion blurred image due to camera shake a regularization based approach is proposed to remove motion blurring from the image by regularizing the sparsity of both the original image and the motion blur kernel under tight wavelet frame systems furthermore an adapted version of the split bregman method is proposed to efficiently solve the resulting minimization problem the experiments on both synthesized images and real images show that our algorithm can effectively remove complex motion blurring from natural images without requiring any prior information of the motion blur kernel

a new unblocking technique to warmstart interior point methods based on sensitivity analysis one of the main drawbacks associated with interior point methods ipms is the perceived lack of an efficient warmstarting scheme which would enable the use of information from a previous solution of a similar problem recently there has been renewed interest in the subject a common problem with warmstarting for ipm is that an advanced starting point which is close to the boundary of the feasible region as is typical might lead to blocking of the search direction several techniques have been proposed to address this issue most of these aim to lead the iterate back into the interior of the feasible region we classify them as either modification steps or unblocking steps depending on whether the modification is taking place before solving the modified problem to prevent future problems or during the solution if and when problems become apparent a new unblocking strategy is suggested which attempts to directly address the issue of blocking by performing sensitivity analysis on the newton step with the aim of increasing the size of the step that can be taken this analysis is used in a new technique to warmstart interior point methods we identify components of the starting point that are responsible for blocking and aim to improve these by using our sensitivity analysis the relative performance of a selection of different warmstarting techniques suggested in the literature and the new proposed unblocking by sensitivity analysis is evaluated on the warmstarting test set based on a selection of netlib problems proposed by benson and shanno comput optim appl 38 2007 pp 371 399 warmstarting techniques are also applied in the context of solving nonlinear programming problems as a sequence of quadratic programs solved by interior point methods we also apply the warmstarting technique to the problem of finding the complete efficient frontier in portfolio management problems a problem with 192 million variables to our knowledge the largest problem to date solved by a warmstarted ipm we find that the resulting best combined warmstarting strategy manages to save between 50 and 60 of interior point iterations consistently outperforming similar approaches reported in current optimization literature

on the design of multi dimensional compactly supported parseval framelets with directional characteristics abstract in this paper we propose a new method for the construction of multi dimensional wavelet like families of affine frames commonly referred to as framelets with specific directional characteristics small and compact support in space directional vanishing moments dvm and axial symmetries or anti symmetries the framelets we construct arise from readily available refinable functions the filters defining these framelets have few non zero coefficients custom selected orientations and can act as finite difference operators

wavelet bi frames with uniform symmetry for curve multiresolution processing this paper is about the construction of univariate wavelet bi frames with each framelet being symmetric as bivariate filter banks are used for surface multiresolution processing it is required that the corresponding decomposition and reconstruction algorithms have high symmetry so that it is possible to design the corresponding multiresolution algorithms for extraordinary vertices for open surfaces special multiresolution algorithms are designed to process boundary vertices when the multiresolution algorithms derived from univariate wavelet bi frames are used as the boundary algorithms it is desired that not only the scaling functions but also all framelets be symmetric in addition the algorithms for curve surface multiresolution processing should be given by templates so that they can be easily implemented in this paper first by appropriately associating the lowpass and highpass outputs to the nodes of z we show that both biorthogonal wavelet multiresolution algorithms and bi frame multiresolution algorithms can be represented by templates then using the idea of the lifting scheme we provide frame algorithms given by several iterative steps with each step represented by a symmetric template finally with the given templates of algorithms we obtain the corresponding filter banks and construct bi frames based on their smoothness and vanishing moments two types of symmetric bi frames are studied in this paper in order to provide a clearer picture on the template based procedure for bi frame construction in this paper we also consider the template based construction of biorthogonal wavelets the approach of the template based bi frame construction introduced in this paper can be extended easily to the construction of bivariate bi frames with high symmetry for surface multiresolution processing

nonstationary iterated tikhonov regularization in banach spaces with uniformly convex penalty terms we consider the nonstationary iterated tikhonov regularization in banach spaces which defines the iterates via minimization problems with uniformly convex penalty term the penalty term is allowed to be non smooth to include l 1 l 1 and total variation tv like penalty functionals which are significant in reconstructing special features of solutions such as sparsity and discontinuities in practical applications we present the detailed convergence analysis and obtain the regularization property when the method is terminated by the discrepancy principle in particular we establish the strong convergence and the convergence in bregman distance which sharply contrast with the known results that only provide weak convergence for a subsequence of the iterative solutions some numerical experiments on linear integral equations of first kind and parameter identification in differential equations are reported

matrix splitting with symmetry and dyadic framelet filter banks over algebraic number fields abstract algebraic number fields are of particular interest and play an important role in both mathematics and engineering since an algebraic number field can be viewed as a finite dimensional linear space over the rational number field q algorithms using algebraic number fields can be efficiently implemented involving only integer arithmetics in this paper we properly formulate the matrix splitting problem over any general subfield of it c including an algebraic number field as a special case and provide a simple necessary and sufficient condition for a 2 2 matrix of laurent polynomials with symmetry to be able to be factorized by a 2 2 matrix of laurent polynomials with certain symmetry structure we propose an effective algorithm on how to obtain the factorization matrix step by step as an application we obtain a satisfactory algorithm for constructing dyadic framelet filter banks with the perfect reconstruction property and with symmetry over algebraic number fields several examples are provided to illustrate the algorithms proposed in this paper

framing pyramids burt and adelson 1983 introduced the laplacian pyramid lp as a multiresolution representation for images we study the lp using the frame theory and this reveals that the usual reconstruction is suboptimal we show that the lp with orthogonal filters is a tight frame and thus the optimal linear reconstruction using the dual frame operator has a simple structure that is symmetric with the forward transform in more general cases we propose an efficient filterbank fb for the reconstruction of the lp using projection that leads to a proved improvement over the usual method in the presence of noise setting up the lp as an oversampled fb we offer a complete parameterization of all synthesis fbs that provide perfect reconstruction for the lp finally we consider the situation where the lp scheme is iterated and derive the continuous domain frames associated with the lp

the structure of shift invariant spaces on a locally compact abelian group abstract we investigate shift invariant subspaces of l 2 g where g is a locally compact abelian group we show among other things that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a parseval frame

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

alternating direction method of multipliers for linear inverse problems in this paper we propose an iterative method using alternating direction method of multipliers admm strategy to solve linear inverse problems in hilbert spaces with a general convex penalty term when the data is given exactly we give a convergence analysis of our admm algorithm without assuming the existence of a lagrange multiplier in case the data contains noise we show that our method is a regularization method as long as it is terminated by a suitable stopping rule various numerical simulations are performed to test the efficiency of the method

the structure of finitely generated shift invariant spaces in l2 rd a simple characterization is given of finitely generated subspaces of l2 rd which are invariant under translation by any multi integer and is used to give conditions under which such a space has a particularly nice generating set namely a basis and more than that a basis with desirable properties such as stability orthogonality or linear independence the last property makes sense only for local spaces i e shift invariant spaces generated by finitely many compactly supported functions and special attention is paid to such spaces as an application we prove that the approximation order provided by a given local space is already provided by the shift invariant space generated by just one function with this function constructible as a finite linear combination of the finite generating set for the whole space hence compactly supported this settles a question of some 20 years standing

wavelet frame based scene reconstruction from range data abstract how to reconstruct the scene a visible surface from a set of scattered noisy and possibly sparse range data is a challenging problem in robotic navigation and computer graphics as most real scenes can be modeled by piecewise smooth surfaces traditional surface fitting techniques e g smoothing spline generally can not preserve sharp discontinuities of surfaces based on sparse approximation of piecewise smooth functions in frame domain we propose a new tight frame based formulation for reconstructing a piecewise smooth surface from a sparse range data set which is robust to both additive noise and outliers furthermore the resulting minimization problem from our formulation can be efficiently solved by the split bregman method 1 2 the numerical experiments show that the proposed approach is capable of reconstructing a piecewise smooth surface with sharp edges from sparse range data corrupted with noise and outliers

a strictly contractive peaceman rachford splitting method for convex programming in this paper we focus on the application of the peaceman rachford splitting method prsm to a convex minimization model with linear constraints and a separable objective function compared to the douglas rachford splitting method drsm another splitting method from which the alternating direction method of multipliers originates prsm requires more restrictive assumptions to ensure its convergence while it is always faster whenever it is convergent we first illustrate that the reason for this difference is that the iterative sequence generated by drsm is strictly contractive while that generated by prsm is only contractive with respect to the solution set of the model with only the convexity assumption on the objective function of the model under consideration the convergence of prsm is not guaranteed but for this case we show that the first t iterations of prsm still enable us to find an approximate solution with an accuracy of o 1 t a worst case o 1 t convergence rate of prsm in t

weak existence for the semigeostrophic equations formulated as a coupled monge ampere transport problem hoskins s semigeostrophic equations are reformulated as a coupled monge ampere transport problem b j hoskins quart j royal met soc 97 1971 pp 139 153 existence of global weak solutions is obtained for this formulation

warm start of the primal dual method applied in the cutting plane scheme a practical warm start procedure is described for the infeasible primal dual interior point method ipm employed to solve the restricted master problem within the cutting plane method in contrast to the theoretical developments in this field the approach presented in this paper does not make the unrealistic assumption that the new cuts are shallow moreover it treats systematically the case when a large number of cuts are added at one time the technique proposed in this paper has been implemented in the context of hopdm the state of the art yet public domain interior point code numerical results confirm a high degree of efficiency of this approach regardless of the number of cuts added at one time can be thousands in the largest examples and regardless of the depth of the new cuts reoptimizations are usually done with a few additional iterations

compactly supported tight affine spline frames in l 2 r d the theory of fiberization is applied to yield compactly supported tight affine frames wavelets in l 2 r d from box splines the wavelets obtained are smooth piecewise polynomials on a simple mesh furthermore they exhibit a wealth of symmetries and have a relatively small support the number of mother wavelets however increases with the increase of the required smoothness two bivariate constructions of potential practical value are highlighted in both the wavelets are derived from four direction mesh box splines that are refinable with respect to the dilation matrix 1 1 1 1

on the o 1 n convergence rate of the douglas rachford alternating direction method alternating direction methods adms have been well studied in the literature and they have found many efficient applications in various fields in this note we focus on the douglas rachford adm scheme proposed by glowinski and marrocco and we aim at providing a simple approach to estimating its convergence rate in terms of the iteration number the linearized version of this adm scheme which is known as the split inexact uzawa method in the image processing literature is also discussed

a nonconforming morley finite element method for the fully nonlinear monge ampere equation in this paper we study finite element approximations of the viscosity solution of the fully nonlinear monge ampere equation det d 2 u f 0 using the well known nonconforming morley element our approach is based on the vanishing moment method which was recently proposed as a constructive way to approximate fully nonlinear second order equations by the author and feng j sci comput 38 1 74 98 2009 the vanishing moment method approximates the monge ampere equation by the fourth order quasilinear equation epsilon delta 2u epsilon rm det d 2u epsilon f with appropriate boundary conditions we develop a finite element scheme using the n dimensional morley element introduced in wang and xu numer math 103 155 169 2006 to approximate the regularized fourth order problem in two and three dimensions and then derive optimal order error estimates

approximation theory and functional analysis part 1 research and survey articles on bernstein durrineyer polynomials with jacob weights h berens and y xu an overview of wavelets c chui a note on weak inequalities in orlicz and lorentz spaces g a edgar and l sucheston bivariate birkhoff interpolation a survey r a lorentz linear approximations of functions with several restricted derivatives y makovoz some characterizations theorems for measures associated with orthogonal polynomials on the unit circle k pan and e b saff box splines cardinal series and wavelets s d reimenschneider and z shen some aspects of the subspace structure of infinite dimensional banach space h rosenthal fairness and monotone curvature j roulier et al projections on 2 dimensional spaces n tomczak jacgermann real versus complex best rational approximation r s varga and a ruttan

a modified characteristic finite element method for a fully nonlinear formulation of the semigeostrophic flow equations this paper develops a fully discrete modified characteristic finite element method for a coupled system consisting of the fully nonlinear monge ampere equation and a transport equation the system is the eulerian formulation in the dual space for b j hoskins semigeostrophic flow equations which are widely used in meteorology to model frontogenesis to overcome the difficulty caused by the strong nonlinearity we first formulate at the differential level a vanishing moment approximation of the semigeostrophic flow equations a methodology recently proposed by the authors x feng and m neilan j sci comput 38 2009 pp 74 98 which involves approximating the fully nonlinear monge ampere equation by a family of fourth order quasilinear equations we then construct a fully discrete modified characteristic finite element method for the regularized problem it is shown that under certain mesh constraint the proposed numerical method converges with an optimal order rate of convergence in particular the obtained error bounds show explicit dependence on the regularization parameter varepsilon numerical tests are also presented to validate the theoretical results and to gauge the efficiency of the proposed fully discrete modified characteristic finite element method

a nonstationary accelerating alternating direction method for frame based poissonian image deblurring abstract poissonian image deblurring problem has been an attractive research subject in many areas of imaging science and engineering in order to obtain accurate restorations one has to run several times to manually select a good value of a regularization parameter in many models in this paper we first propose an optimization model by combining a new data fitting term with a frame based analysis regularizer and a nonnegative constraint a discrepancy principle based nonstationary accelerating alternating direction method called dp naadm is then proposed to solve the optimization model without estimating the regularization parameter numerical results show that dp naadm provides competitive restoration results compared to the well known poisson image deconvolution by augmented lagrangian method called pidal also dp naadm provides better restorations in terms of both accuracy and visual quality than the discrepancy principle based pidal method

denoising for 3 d photon limited imaging data using nonseparable filterbanks in this paper we present a novel frame based denoising algorithm for photon limited 3d images we first construct a new 3d nonseparable filterbank by adding elements to an existing frame in a structurally stable way in contrast with the traditional 3d separable wavelet system the new filterbank is capable of using edge information in multiple directions we then propose a data adaptive hysteresis thresholding algorithm based on this new 3d nonseparable filterbank in addition we develop a new validation strategy for denoising of photon limited images containing sparse structures such as neurons the structure of interest is less than 5 of total volume the validation method based on tubular neighborhoods around the structure is used to determine the optimal threshold of the proposed denoising algorithm we compare our method with other state of the art methods and report very encouraging results on applications utilizing both synthetic and real data

finite element approximations of general fully nonlinear second order elliptic partial differential equations based on the vanishing moment method the vanishing moment method was introduced by the authors in feng and neilan 2009 as a reliable methodology for computing viscosity solutions of fully nonlinear second order partial differential equations pdes it is based on the simple idea of approximating a hard to handle fully nonlinear second order pde by a family parameterized by a small parameter e of easy to handle quasilinear fourth order pdes the primary objective of this article is to present a comprehensive finite element analysis for the vanishing moment approximation of general fully nonlinear second order elliptic pdes which fulfill some structure conditions abstract methodological and convergence analysis frameworks of conforming finite element methods are first developed for fully nonlinear second order pdes in a general setting the abstract framework is then applied to three prototypical nonlinear equations namely the monge ampere equation the equation of prescribed gauss curvature and the infinity laplacian equation numerical experiments are presented for each problem to validate the theoretical error estimate results and to gauge the efficiency of the proposed numerical methods and the vanishing moment methodology

deconvolution a wavelet frame approach this paper devotes to analyzing deconvolution algorithms based on wavelet frame approaches which has already appeared in chan et al siam j sci comput 24 4 1408 1432 2003 appl comput hormon anal 17 91 115 2004a int j imaging syst technol 14 91 104 2004b as wavelet frame based high resolution image reconstruction methods we first give a complete formulation of deconvolution in terms of multiresolution analysis and its approximation which completes the formulation given in chan et al siam j sci comput 24 4 1408 1432 2003 appl comput hormon anal 17 91 115 2004a int j imaging syst technol 14 91 104 2004b this formulation converts deconvolution to a problem of filling the missing coefficients of wavelet frames which satisfy certain minimization properties these missing coefficients are recovered iteratively together with a built in denoising scheme that removes noise in the data set such that noise in the data will not blow up while iterating this approach has already been proven to be efficient in solving various problems in high resolution image reconstructions as shown by the simulation results given in chan et al siam j sci comput 24 4 1408 1432 2003 appl comput hormon anal 17 91 115 2004a int j imaging syst technol 14 91 104 2004b however an analysis of convergence as well as the stability of algorithms and the minimization properties of solutions were absent in those papers this paper is to establish the theoretical foundation of this wavelet frame approach in particular a proof of convergence an analysis of the stability of algorithms and a study of the minimization property of solutions are given

operator splittings bregman methods and frame shrinkage in image processing we examine the underlying structure of popular algorithms for variational methods used in image processing we focus here on operator splittings and bregman methods based on a unified approach via fixed point iterations and averaged operators in particular the recently proposed alternating split bregman method can be interpreted from different points of view as a bregman as an augmented lagrangian and as a douglas rachford splitting algorithm which is a classical operator splitting method we also study similarities between this method and the forward backward splitting method when applied to two frequently used models for image denoising which employ a besov norm and a total variation regularization term respectively in the first setting we show that for a discretization based on parseval frames the gradient descent reprojection and the alternating split bregman algorithm are equivalent and turn out to be a frame shrinkage method for the total variation regularizer we also present a numerical comparison with multistep methods

numerical methods for fully nonlinear elliptic equations of the monge ampere type abstract in this article we discuss the numerical solution of the dirichlet problem for the monge ampere equation in two dimensions the solution of closely related problems is also discussed these include a family of pucci s equations the equation prescribing the harmonic mean of the eigenvalues of the hessian of a smooth function of two variables and a minimization problem from nonlinear elasticity where the cost functional involves the determinant of the gradient of vector valued functions to solve the monge ampere equation we consider two methods the first one reduces the monge ampere equation to a saddle point problem for a well chosen augmented lagrangian to solve this saddle point problem we advocate an uzawa douglas rachford algorithm the second method combines nonlinear least squares and operator splitting this second method being simpler to implements we apply variants of it to the solution of the other problems for the space discretization we use mixed finite element approximations closely related to methods already used for the solution of linear and nonlinear bi harmonic problems through these approximations the solution of the above problems is essentially reduced to the solution of discrete poisson problems the methods discussed in this article are validated by the results of numerical experiments

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

cine cone beam ct reconstruction using low rank matrix factorization algorithm and a proof of principle study respiration correlated cbct commonly called 4dcbct provides respiratory phase resolved cbct images a typical 4dcbct represents averaged patient images over one breathing cycle and the fourth dimension is actually breathing phase instead of time in many clinical applications it is desirable to obtain true 4dcbct with the fourth dimension being time i e each constituent cbct image corresponds to an instantaneous projection theoretically it is impossible to reconstruct a cbct image from a single projection however if all the constituent cbct images of a 4dcbct scan share a lot of redundant information it might be possible to make a good reconstruction of these images by exploring their sparsity and coherence redundancy though these cbct images are not completely time resolved they can exploit both local and global temporal coherence of the patient anatomy automatically and contain much more temporal variation information of the patient geometry than the conventional 4dcbct we propose in this work a computational model and algorithms for the reconstruction of this type of semi time resolved cbct called cine cbct based on low rank approximation that can utilize the underlying temporal coherence both locally and globally such as slow variation periodicity or repetition in those cine cbct images

new error estimates of the morley element for the plate bending problems in this paper we derive some new error estimates of a nonconforming morley element for the plate bending problems our aim is fourfold first it shows that the consistency error of the morley element can be controlled by its corresponding approximation error up to an arbitrarily high order term especially which can be of arbitrary order provided that f on the right hand side is piecewise smooth enough thus the total energy norm is dominated alone by the approximation error second as a byproduct we derive the error estimate under the regularity assumption u h 2 s with s 0 1 which fills the gap in the a priori error estimate of the morley element with low regularity for s 0 1 2 third based on this new error bound a robust convergence is proved even if the solution is exactly in h 2 finally by a slight modification of the variational formulation a new error estimate is also derived for the case f h 1 the key role is played by some quasi interpolation operators and the weak continuous properties of morley finite element space the main ideas and results presented here are motivated by mao and shi 2010 and can be viewed as its extension to fourth order problems

a new warmstarting strategy for the primal dual column generation method this paper presents a new warmstarting technique in the context of a primal dual column generation method applied to solve a particular class of combinatorial optimization problems the technique relies on calculating an initial point and on solving auxiliary linear optimization problems to determine the step direction needed to fully restore primal and dual feasibilities after new columns arrive conditions on the maximum size of the cuts and on a suitable initial point are discussed additionally the strategy ensures that the duality gap of the warmstart is bounded by the old duality gap multiplied with a small constant which depends on the relation between the old and modified problems computational experiments demonstrate the gains achieved when compared to a coldstart approach

an admm numerical approach to linear parabolic state constrained optimal control problems optimal control problems arising from systems modeled by linear parabolic equations may be difficult for both theoretical analysis and algorithmic design for the case where there are additional constraints on the state variables restrictive regularity assumptions are usually required to guarantee the existence of the associated lagrange multiplier and thus some regularization type methods such as the moreau yosida and lavrentiev methods have been discussed in the literature in this article we study the application of the alternating direction method of multipliers admm to linear parabolic state constrained optimal control problems and propose an admm numerical approach we prove the convergence of the admm algorithm without any existence or regularity assumption on the lagrange multiplier and estimate its worst case convergence rate in both the ergodic and nonergodic senses an important feature of the admm approach is that it decouples the state constraints and the parabolic optimal control problems inside each iteration we show the efficiency of the admm approach by testing some control problems in two space dimensions

tight frames of compactly supported affine wavelets this paper extends the class of orthonormal bases of compactly supported wavelets recently constructed by daubechies commun pure appl math 41 909 1988 for each integer n 1 a family of wavelet functions having support 0 2n 1 is constructed such that jk x 2j 2 2jx k kj k z is a tight frame of l2 r i e for every f l2 r f c jk jk f jk for some c 0 this family is parametrized by an algebraic subset vn of r4n furthermore for n 2 a proper algebraic subset wn of vn is specified such that all points in vn outside of wn yield orthonormal bases the relationship between these tight frames and the theory of group representations and coherent states is discussed

new constructions of nonseparable tight wavelet frames abstract we present two methods for constructing new nonseparable multidimensional tight wavelet frames by combining the ideas of sum of squares representations of nonnegative trigonometric polynomials with the coset sum method of generating nonseparable multidimensional lowpass filters from univariate lowpass filters in effect these methods allow one to select a univariate lowpass filter and generate nonseparable multidimensional tight wavelet frames from it in any dimension n 2 under certain conditions on the input filter which are given explicitly we construct sum of hermitian squares representations for a particular class of trigonometric polynomials f in several variables each related to a coset sum generated lowpass mask in that nonnegativity of f implies the sub qmf condition for in two ways for interpolatory inputs to the coset sum method satisfying the univariate sub qmf condition we find this representation using the fejer riesz lemma and in the general case by writing f x p x where x is a vector of complex exponential functions and p is a constant positive semidefinite matrix that is constructed to reduce the number of generators in this representation the generators of this representation of f may then be used to generate the filters in a tight wavelet frame with lowpass mask several examples of these representations and the corresponding frames are given throughout

time delayed finite difference reaction diffusion systems with nonquasimonotone functions this paper is concerned with finite difference solutions of a system of reaction diffusion equations with coupled nonlinear boundary conditions and time delays the reaction functions and the boundary functions are not necessarily quasimonotone and the time delays may appear in the reaction functions as well as in the boundary functions the investigation is devoted to the finite difference system for both the time dependent problem and its corresponding steady state problem some monotone iteration processes for the finite difference systems are given and the asymptotic behavior of the time dependent solution in relation to the steady state solution is discussed the asymptotic behavior result leads to some local and global attractors of the time dependent problem including the convergence of the time dependent solution to a unique steady state solution an application and some numerical results to an enzyme substrate reaction diffusion problem are given all the results are directly applicable to parabolic ordinary systems and to reaction diffusion systems without time delays

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

construction of nonseparable dual ω wavelet frames in l2 rs suppose j 1 a 1 j 1 a 1 and j 1 2 a 2 1 j 1 2 a 2 1 are two pairs of dual m wavelet frames and n wavelet frames in l 2 r s 1 and l 2 r s 2 respectively where m and n are s 1xs 1 and s 2xs 2 dilation matrices with a 1 det m 1 and 2a 2 1 det n 1 moreover their mask symbols both satisfy mixed extension principle mep based on the mask symbols a family of nonseparable dual w wavelet frames in l 2 r s are constructed where s s 1 s 2 and w m qon with q and m 1 q both being integer matrices then a convolution scheme for improving regularity of wavelet frames is given from the nonseparable dual w wavelet frames nonseparable w wavelet frames with high regularity can be constructed easily we give an algorithm for constructing nonseparable dual symmetric or antisymmetric wavelet frames in l 2 r s from the dual w wavelet frames nonseparable dual w wavelet frames with symmetry can be obtained easily in the end two examples are given

hybrid spline frames using their unitary extension principle ron and shen have constructed a normalized tight frame for l 2 consisting of spline functions with uniform knots this paper constructs a normalized tight frame for l 2 0 oo comprising spline functions with knots on a hybrid of uniform and geometric mesh the construction is motivated by applications in adaptive approximation using spline functions on a hybrid mesh that admits a natural dyadic multiresolution approximation of l 2 0 based on dilation and translation

mixed finite element methods for the fully nonlinear monge ampere equation based on the vanishing moment method this paper studies mixed finite element approximations of the viscosity solution to the dirichlet problem for the fully nonlinear monge ampere equation det d 2u 0 f 0 based on the vanishing moment method which was proposed recently by the authors in x feng and m neilan j scient comp doi 10 1007 s10915 008 9221 9 2008 in this approach the second order fully nonlinear monge ampere equation is approximated by the fourth order quasilinear equation varepsilon delta 2 u varepsilon det d 2u varepsilon f it was proved in x feng trans ams submitted that the solution u varepsilon converges to the unique convex viscosity solution u 0 of the dirichlet problem for the monge ampere equation this result then opens a door for constructing convergent finite element methods for the fully nonlinear second order equations a task which has been impracticable before the goal of this paper is threefold first we develop a family of hermann miyoshi type mixed finite element methods for approximating the solution u varepsilon of the regularized fourth order problem which computes simultaneously u varepsilon and the moment tensor sigma varepsilon d 2u varepsilon second we derive error estimates which track explicitly the dependence of the error constants on the parameter varepsilon for the errors u varepsilon u varepsilon h and sigma 0 sigma h varepsilon finally we present a detailed numerical study on the rates of convergence in terms of powers of varepsilon for the error u 0 u h varepsilon and sigma varepsilon sigma h varepsilon and numerically examine what is the best mesh size h in relation to varepsilon in order to achieve these rates due to the strong nonlinearity of the underlying equation the standard perturbation argument for error analysis of finite element approximations of nonlinear problems does not work for the problem to overcome the difficulty we employ a fixed point technique which strongly relies on the stability of the linearized problem and its mixed finite element approximations

l minimization for wavelet frame based image restoration the theory of tight wavelet frames has been extensively studied in the past twenty years and they are currently widely used for image restoration and other image processing and analysis problems the success of wavelet frame based models including balanced approach 18 7 and analysis based approach 11 31 50 is due to their capability of sparsely approximating piecewise smooth functions like images motivated by the balanced approach and analysis based approach we shall propose a wavelet frame based l0 minimization model where the l0 norm of the frame coefficients is penalized we adapt the penalty decomposition pd method of 40 to solve the proposed optimization problem some convergence analysis of the adapted pd method will also be provided numerical results showed that the proposed model solved by the pd method can generate images with better quality than those obtained by either analysis based approach or balanced approach in terms of restoring sharp features as well as maintaining smoothness of the recovered images

affine systems in l2 rd the analysis of the analysis operator abstract discrete affine systems are obtained by applying dilations to a given shift invariant system the complicated structure of the affine system is due first and foremost to the fact that it is not invariant under shifts affine frames carry the additional difficulty that they are global in nature it is the entire interaction between the various dilation levels that determines whether the system is a frame and not the behaviour of the system within one dilation level we completely unravel the structure of the affine system with the aid of two new notions the affine product and a quasi affine system this leads to a characterization of affine frames the induced characterization of tight affine frames is in terms of exact orthogonality relations that the wavelets should satisfy on the fourier domain several results such as a general oversampling theorem follow from these characterizations most importantly the affine product can be factored during a multiresolution analysis construction and this leads to a complete characterization of all tight frames that can be constructed by such methods moreover this characterization suggests very simple sufficient conditions for constructing tight frames from multiresolution of particular importance are the facts that the underlying scaling function does not need to satisfy any a priori conditions and that the freedom offered by redundancy can be fully exploited in these constructions an

a discrete helmholtz decomposition with morley finite element functions and the optimality of adaptive finite element schemes the discrete reliability of a finite element method is a key ingredient to prove optimal convergence of an adaptive mesh refinement strategy and requires the interchange of a coarse triangulation and some arbitrary refinement of it one approach for this is the careful design of an intermediate triangulation with one level refinements and with the remaining difficulty to design some interpolation operator which maps a possibly nonconforming approximation into the finite element space based on the finer triangulation this paper enfolds the second possibility of some novel discrete helmholtz decomposition for the nonconforming morley finite element method this guarantees the optimality of a standard adaptive mesh refining algorithm for the biharmonic equation numerical examples illustrate the crucial dependence of the bulk parameter and the surprisingly short pre asymptotic range of the adaptive morley finite element method

geometric aspects of frame representations of abelian groups hhwe consider frames arising from the action of a unitary representation of a discrete countable abelian group we show that the range of the analysis operator can be determined by computing which characters appear in the representation this allows one to compare the ranges of two such frames which is useful for determining similarity and also for multiplexing schemes our results then partially extend to bessel sequences arising from the action of the group we apply the results to sampling on bandlimited functions and to wavelet and weyl heisenberg frames this yields a sufficient condition for two sampling transforms to have orthogonal ranges and two analysis operators for wavelet and weyl heisenberg frames to have orthogonal ranges the sufficient condition is easy to compute in terms of the periodization of the fourier transform of the frame generators

reoptimization with the primal dual interior point method reoptimization techniques for an interior point method applied to solving a sequence of linear programming problems are discussed conditions are given for problem perturbations that can be absorbed in merely one newton step the analysis is performed for both short step and long step feasible path following methods a practical procedure is then derived for an infeasible path following method it is applied in the context of crash start for several large scale structured linear programs numerical results with oops a new object oriented parallel solver demonstrate the efficiency of the approach for large structured linear programs crash start leads to about 40 reduction in the number of iterations and translates into a 25 reduction of the solution time the crash procedure parallelizes well and speed ups between 3 1 3 8 on four processors are achieved

hopdm version 2 12 a fast lp solver based on a primal dual interior point method hopdm is an implementation of the primal dual interior point method for solving large scale linear programming lp problems hopdm stands for higher order primal dual method its newest version 2 12 of may 1995 is a robust and efficient code that compares favourably with today s commercial lp solvers

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics

forces nlp an efficient implementation of interior point methods for multistage nonlinear nonconvex programs abstractreal time implementation of optimisation based control and trajectory planning can be very challenging for nonlinear systems as a result if an implementation based on a fixed linearisatio