proximal thresholding algorithm for minimization over orthonormal bases the notion of soft thresholding plays a central role in problems from various areas of applied mathematics in which the ideal solution is known to possess a sparse decomposition in some orthonormal basis using convex analytical tools we extend this notion to that of proximal thresholding and investigate its properties providing in particular several characterizations of such thresholders we then propose a versatile convex variational formulation for optimization over orthonormal bases that covers a wide range of problems and we establish the strong convergence of a proximal thresholding algorithm to solve it numerical applications to signal recovery are demonstrated

half linear differential equations linearization technique and its application abstract oscillatory properties of the half linear second order differential equation r t x c t x 0 x x p 2 x p 1 are investigated where is viewed as a perturbation of an equation of the same form and these properties are compared with the oscillatory behavior of a certain associated linear second order differential equation as a corollary a conjecture posed by the first author in the paper o dosl perturbations of the half linear euler weber differential equations j math anal appl 323 2006 426 440 is proved and some new research problems are formulated

integral conditions for nonoscillation of second order nonlinear differential equations an extension of the fubini theorem is presented and applied in studying the possible coexistence of nonoscillatory solutions for the differential equation r t x sgnx q t x sgnx some discrepancies among the cases and are illustrated and compared with the linear case in addition when an application to the integral characterization of principal solutions which answers an open problem stated in o dosl a elbert integral characterization of the principal solution of half linear second order differential equations studia sci math hungar 36 2000 455 469 is given

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

deconvolution by thresholding in mirror wavelet bases the deconvolution of signals is studied with thresholding estimators that decompose signals in an orthonormal basis and threshold the resulting coefficients a general criterion is established to choose the orthonormal basis in order to minimize the estimation risk wavelet bases are highly sub optimal to restore signals and images blurred by a low pass filter whose transfer function vanishes at high frequencies a new orthonormal basis called mirror wavelet basis is constructed to minimize the risk for such deconvolutions an application to the restoration of satellite images is shown

linearized bregman iterations for compressed sensing finding a solution of a linear equation au f with various minimization properties arises from many applications one such application is compressed sensing where an efficient and robust to noise algorithm to find a minimal l 1 norm solution is needed this means that the algorithm should be tailored for large scale and completely dense matrices a while au and a t u can be computed by fast transforms and the solution we seek is sparse recently a simple and fast algorithm based on linearized bregman iteration was proposed in 28 32 for this purpose this paper is to analyze the convergence of linearized bregman iterations and the minimization properties of their limit based on our analysis here we derive also a new algorithm that is proven to be convergent with a rate furthermore the new algorithm is simple and fast in approximating a minimal l 1 norm solution of au f as shown by numerical simulations hence it can be used as another choice of an efficient tool in compressed sensing

analysis of online composite mirror descent algorithm we study the convergence of the online composite mirror descent algorithm which involves a mirror map to reflect the geometry of the data and a convex objective function consisting of a loss and a

split bregman iteration algorithm for total bounded variation regularization based image deblurring many existing algorithms taking the seminorm in bv for regularization have achieved great success in image processing however this paper considers the total bounded variation regularization based approach to perform image deblurring based on this novel model we introduce an extended split bregman iteration to obtain the optimum solution quickly we also provide the rigorous convergence analysis of the iterative algorithm here compared with the results of the rof method numerical simulations illustrate the more excellent reconstruction performance of the proposed algorithm

a chaotic iterative algorithm based on linearized bregman iteration for image deblurring in this paper we propose a new chaotic iterative algorithm for image deblurring the algorithm is based on generalized inverse iteration and linearized bregman iteration for the basis pursuit problem min u r n u 1 au f numerical experiments show that the chaotic algorithm for image restoration is effective and efficient

local strong homogeneity of a regularized estimator this paper deals with regularized pointwise estimation of discrete signals which contain large strongly homogeneous zones where typically they are constant or linear or more generally satisfy a linear equation the estimate is defined as the minimizer of an objective function combining a quadratic data fidelity term and a regularization prior term the latter term is the sum of the values obtained by applying a potential function pf to each component called a difference of a linear transform of the signal minimizers of functions of this form arise in various settings in statistics and optimization the features exhibited by such an estimate are closely related to the shape of the pf our goal is to determine estimators providing solutions which involve large strongly homogeneous zones where more precisely the differences are null in spite of the noise corrupting the data to this end we require that the strongly homogeneous zones recovered by the estimator be insensitive to any variation of th

asymptotic properties of kneser solutions to nonlinear second order odes with regularly varying coefficients in this work we investigate properties of a class of solutions to the second order ode p t u t q t f u t 0 on the interval a a 0 where p and q are functions regularly varying at infinity and f satisfies f l 0 f 0 f l 0 with l0 0

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

an adaptive algorithm for tv based model of three norms lq q 12 1 2 in image restoration abstract in this paper we present an adaptive method for the tv based model of three norms l q q 1 2 1 2 for the image restoration problem the algorithm with the l2 norm is used in the smooth regions where the value of u is small the algorithm with the l 1 2 norm is applied for the jumps where the value of u is large when the value of u is moderate the algorithm with the l1 norm is employed thus the three algorithms are applied for different regions of a given image such that the advantages of each algorithm are adopted the numerical experiments demonstrate that our adaptive algorithm can not only keep the original edge and original detailed information but also weaken the staircase phenomenon in the restored images specifically in contrast to the l1 norm as in the rudin osher fatemi model the l2 norm yields better results in the smooth and flat regions and the l 1 2 norm is more suitable in regions with strong discontinuities therefore our adaptive algorithm is efficient and robust even for images with large noises

asymptotic behavior of positive solutions of sublinear differential equations of emden fowler type this paper is concerned with asymptotic analysis of positive solutions of the second order nonlinear differential equation a x t q t f x t 0 where q a 0 is a continuous function which is regularly varying and f 0 0 is a continuous increasing function which is regularly varying of index c 0 1 an application of the theory of regular variation gives the possibility of determining precise information about the asymptotic behavior at infinity of intermediate solutions of eq a

on an extension of the fubini theorem and its applications in odes the paper deals with the extension of the fubini theorem there are given applications for the second order differential nonlinear equations the first application brings comparison results for oscillation of two generalized emden fowler equations the second one shows the similarities and discrepancies for nonoscillatory half linear and linear differential equations

curvature filters efficiently reduce certain variational energies in image processing the rapid approximate solution of variational problems involving generic data fitting terms is often of practical relevance for example in real time applications variational solvers based on diffusion schemes or the euler lagrange equations are too slow and restricted in the types of data fitting terms here we present a filter based approach to reduce variational energies that contain generic data fitting terms but are restricted to specific regularizations our approach is based on reducing the regularization part of the variational energy while guaranteeing non increasing total energy this is applicable to regularization dominated models where the data fitting energy initially increases while the regularization energy initially decreases we present fast discrete filters for regularizers based on gaussian curvature mean curvature and total variation these pixel local filters can be used to rapidly reduce the energy of the full model we prove the convergence of the resulting iterative scheme in a greedy sense and we show several experiments to demonstrate applications in image processing problems involving regularization dominated variational models

partially smooth variational principles and applications we discuss a smooth variational principle for partially smooth viscosity subdifferentials and explore its applications in nonsmooth analysis

full waveform inversion using seislet regularization abstractbecause of inaccurate incomplete and inconsistent waveform records full waveform inversion fwi in the framework of a local optimization approach may not have a unique solution and thus it remains an ill posed inverse problem to improve the robustness of fwi we have developed a new model regularization approach that enforced the sparsity of solutions in the seislet domain the construction of seislet basis functions requires structural information that can be estimated iteratively from migration images we implement fwi with seislet regularization using nonlinear shaping regularization and impose sparseness by applying soft thresholding on the updated model in the seislet domain at each iteration of the data fitting process the main extra computational cost of the method relative to standard fwi is the cost of applying forward and inverse seislet transforms at each iteration this cost is almost negligible compared with the cost of solving wave equations numerical tests using the syntheti

asymptotic formulae for solutions of half linear differential equations we establish asymptotic formulae for regularly varying solutions of the half linear differential equation r t y 1 sgn y p t y 1 sgn y where r p are positive continuous functions on a and 1 the results can be understood in several ways some open problems posed in the literature are solved results for linear differential equations are generalized some of the observations are new even in the linear case a refinement on information about behavior of solutions in standard asymptotic classes is provided a precise description of regularly varying solutions which are known to exist is given regular variation of all positive solutions is proved

a customized douglas rachford splitting algorithm for separable convex minimization with linear constraints we consider applying the douglas rachford splitting method drsm to the convex minimization problem with linear constraints and a separable objective function the dual application of drsm has been well studied in the literature resulting in the well known alternating direction method of multipliers admm in this paper we show that the primal application of drsm in combination with an appropriate decomposition can yield an efficient structure exploiting algorithm for the model under consideration whose subproblems could be easier than those of admm both the exact and inexact versions of this customized drsm are studied and their numerical efficiency is demonstrated by some preliminary numerical results

an improved algorithm for basis pursuit problem and its applications abstract we propose an algorithm for solving the basis pursuit problem min u c n u 1 a u f our starting motivation is the algorithm for compressed sensing proposed by qiao li and wu which is based on linearized bregman iteration with generalized inverse qiao li and wu defined new algorithm for solving the basis pursuit problem in compressive sensing using a linearized bregman iteration and the iterative formula of linear convergence for computing the matrix generalized inverse in our proposed approach we combine a partial application of the newton s second order iterative scheme for computing the generalized inverse with the bregman iteration our scheme takes lesser computational time and gives more accurate results in most cases the effectiveness of the proposed scheme is illustrated in two applications signal recovery from noisy data and image deblurring

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

restoration of chopped and nodded images by framelets in infrared astronomy an observed image from a chop and nod process can be considered as the result of passing the original image through a high pass filter here we propose a restoration algorithm which builds up a tight framelet system that has the high pass filter as one of the framelet filters our approach reduces the solution of restoration problem to that of recovering the missing coefficients of the original image in the tight framelet decomposition the framelet approach provides a natural setting to apply various sophisticated framelet denoising schemes to remove the noise without reducing the intensity of major stars in the image a proof of the convergence of the algorithm based on convex analysis is also provided simulated and real images are tested to illustrate the efficiency of our method over the projected landweber method

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

convergence of the linearized bregman iteration for l1 norm minimization one of the key steps in compressed sensing is to solve the basis pursuit problem min u n u au f bregman iteration was very successfully used to solve this problem in 40 also a simple and fast iterative algorithm based on linearized bregman iteration was proposed in 40 which is described in detail with numerical simulations in 35 a convergence analysis of the smoothed version of this algorithm was given in 11 the purpose of this paper is to prove that the linearized bregman iteration proposed in 40 for the basis pursuit problem indeed converges

handbook of differential equations chapter headings definitions and concepts transformations exact analytical methods exact methods for odes exact methods for pdes approximate analytical methods numerical methods concepts numerical methods for odes numerical methods for pdes list of tables list of programs list of figures subject index

projected landweber iteration for matrix completion recovering an unknown low rank or approximately low rank matrix from a sampling set of its entries is known as the matrix completion problem in this paper a nonlinear constrained quadratic program problem concerning the matrix completion is obtained a new algorithm named the projected landweber iteration plw is proposed and the convergence is proved strictly numerical results show that the proposed algorithm can be fast and efficient under suitable prior conditions of the unknown low rank matrix

wavelet algorithms for high resolution image reconstruction high resolution image reconstruction refers to the reconstruction of high resolution images from multiple low resolution shifted degraded samples of a true image in this paper we analyze this problem from the wavelet point of view by expressing the true image as a function in cal l bbb r 2 we derive iterative algorithms which recover the function completely in the cal l sense from the given low resolution functions these algorithms decompose the function obtained from the previous iteration into different frequency components in the wavelet transform domain and add them into the new iterate to improve the approximation we apply wavelet packet thresholding methods to denoise the function obtained in the previous step before adding it into the new iterate our numerical results show that the reconstructed images from our wavelet algorithms are better than that from the tikhonov least squares approach extension to super resolution image reconstruction where some of the low resolution images are missing is also considered

multiscale methods for polyhedral regularizations in this paper we present the extension and generalization of the adaptive inverse scale space aiss method proposed for ell 1 regularization in math comp 82 2013 pp 269 299 to arbitrary polyhedral functions we will see that the representation of a convex polyhedral function as a finitely generated function allows us to conclude that the inverse scale space flow the time continuous formulation of the augmented lagrangian method can be solved exactly without discretization we present an algorithm for the computation of such a solution for arbitrary polyhedral functions analyze its convergence and interpret the well known forward scale space flow as the inverse scale space flow on the convex conjugate functional thus including this class of flows in our analysis a surprising result is the equivalence of the scale space or gradient flow with a standard variational problem finally we give examples of the applications for the aiss algorithm for polyhedral functions which illustrate th

on the equivalence of some basic principles in variational analysis the primary goal of this paper is to study relationships between certain basic principles of variational analysis and its applications to nonsmooth calculus and optimization considering a broad class of banach spaces admitting smooth renorms with respect to some bornology we establish an equivalence between useful versions of a smooth variational principle for lower semicontinuous functions an extremal principle for nonconvex sets and an enhanced fuzzy sum rule formulated in terms of viscosity normals and subgradients with controlled ranks further refinements of the equivalence result are obtained in the case of a frechet differentiable norm based on the new enhanced sum rule we provide a simplified proof for the refined sequential description of approximate normals and subgradients in smooth spaces

signal and noise separation in prestack seismic data using velocity dependent seislet transform abstractthe seislet transform is a waveletlike transform that analyzes seismic data by following varying slopes of seismic events across different scales and provides a multiscale orthogonal basis for seismic data it generalizes the discrete wavelet transform dwt in the sense that the dwt in the lateral direction is simply the seislet transform with a zero slope our earlier work used plane wave destruction pwd to estimate smoothly varying slopes however the pwd operator can be sensitive to strong noise interference which makes the seislet transform based on pwd pwd seislet transform occasionally fail in providing a sparse multiscale representation for seismic field data we adopted a new velocity dependent vd formulation of the seislet transform in which the normal moveout equation served as a bridge between local slope patterns and conventional moveout parameters in the common midpoint domain the vd slope has better resistance to strong random noise which indicated the potential of vd sei

a dual split bregman method for fast l minimization in this paper we propose a new algorithm for fast l1 minimization as frequently arising in compressed sensing our method is based on a split bregman algorithm applied to the dual of the problem of minimizing u 1 1 2 u 2 such that u solves the under determined linear system au f which was recently investigated in the context of linearized bregman methods furthermore we provide a convergence analysis for split bregman methods in general and show with our compressed sensing example that a split bregman approach to the primal energy can lead to a different type of convergence than split bregman applied to the dual thus making the analysis of different ways to minimize the same energy interesting for a wide variety of optimization problems

asplund sets differentiability and subdifferentiability of functions in banach spaces abstract we show that asplund sets are effective tools to study differentiability of lipschitz functions and e subdifferentiability of lower semicontinuous functions on general banach spaces if a locally lipschitz function defined on an asplund generated space x t y has a minimal clarke subdifferential mapping then it is t b y uniformly strictly differentiable on a dense g subset of x examples are given of locally lipschitz functions that are t b y uniformly strictly differentiable everywhere but nowhere frechet differentiable

on super linear emden fowler type differential equations we study the second order emden fowler type differential equation in the super linear case using a holder type inequality we resolve the open problem on the possible coexistence on three possible types of nononscillatory solutions subdominant intermediate and dominant solutions jointly with this sufficient conditions for the existence of globally positive intermediate solutions are established some of our results are new also for the emden fowler equation

increasing solutions of thomas fermi type differential equations the superlinear case abstract an asymptotic analysis of increasing solutions x t of equations of super linear thomas fermi type is performed in the framework of regular variation leading to the accurate behavior of all such x t for t

adaptive wavelet galerkin methods for linear inverse problems we introduce and analyze numerical methods for the treatment of inverse problems based on an adaptive wavelet galerkin discretization these methods combine the theoretical advantages of the wavelet vaguelette decomposition wvd in terms of optimally adapting to the unknown smoothness of the solution together with the numerical simplicity of galerkin methods in a first step we simply combine a thresholding algorithm on the data with a galerkin inversion on a fixed linear space in a second step a more elaborate method performs the inversion by an adaptive procedure in which a smaller space adapted to the solution is iteratively constructed this leads to a significant reduction of the computational cost