Pub Date : 2021-06-01DOI: 10.4208/csiam-am.so-2020-0007
A. Theljani
{"title":"Combined Second and Fourth-Order PDEs Model and Associated Variational Problems for Geometric Images Inpainting and Denoising","authors":"A. Theljani","doi":"10.4208/csiam-am.so-2020-0007","DOIUrl":"https://doi.org/10.4208/csiam-am.so-2020-0007","url":null,"abstract":"","PeriodicalId":29749,"journal":{"name":"CSIAM Transactions on Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46370311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-01DOI: 10.4208/csiam-am.2020-0029
Yiming Gao & ChunlinWu
. Multi-channel/modality image joint reconstruction has gained much re-search interest in recent years. In this paper, we propose to use a nonconvex and non-Lipschitz joint regularizer in a general variational model for joint reconstruction un-der additive measurement noise. This framework has good ability in edge-preserving by sharing common edge features of individual images. We study the lower bound theory for the non-Lipschitz joint reconstruction model in two important cases with Gaussian and impulsive measurement noise, respectively. In addition, we extend pre-vious works to propose an inexact iterative support shrinking algorithm with prox-imal linearization for multi-channel image reconstruction (InISSAPL-MC) and prove that the iterative sequence converges globally to a critical point of the original objective function. In a special case of single channel image restoration, the convergence result improves those in the literature. For numerical implementation, we adopt primal dual method to the inner subproblem. Numerical experiments in color image restoration and two-modality undersampled magnetic resonance imaging (MRI) reconstruction show that the proposed non-Lipschitz joint reconstruction method achieves consider-able improvements in terms of edge preservation for piecewise constant images com-pared to existing methods.
{"title":"A General Non-Lipschitz Joint Regularized Model for Multi-Channel/Modality Image Reconstruction","authors":"Yiming Gao & ChunlinWu","doi":"10.4208/csiam-am.2020-0029","DOIUrl":"https://doi.org/10.4208/csiam-am.2020-0029","url":null,"abstract":". Multi-channel/modality image joint reconstruction has gained much re-search interest in recent years. In this paper, we propose to use a nonconvex and non-Lipschitz joint regularizer in a general variational model for joint reconstruction un-der additive measurement noise. This framework has good ability in edge-preserving by sharing common edge features of individual images. We study the lower bound theory for the non-Lipschitz joint reconstruction model in two important cases with Gaussian and impulsive measurement noise, respectively. In addition, we extend pre-vious works to propose an inexact iterative support shrinking algorithm with prox-imal linearization for multi-channel image reconstruction (InISSAPL-MC) and prove that the iterative sequence converges globally to a critical point of the original objective function. In a special case of single channel image restoration, the convergence result improves those in the literature. For numerical implementation, we adopt primal dual method to the inner subproblem. Numerical experiments in color image restoration and two-modality undersampled magnetic resonance imaging (MRI) reconstruction show that the proposed non-Lipschitz joint reconstruction method achieves consider-able improvements in terms of edge preservation for piecewise constant images com-pared to existing methods.","PeriodicalId":29749,"journal":{"name":"CSIAM Transactions on Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42485514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-01DOI: 10.4208/CSIAM-AM.2021.NLA.02
J. Xia
In structured matrix computations, existing rank structures such as hierarchically semiseparable (HSS) forms admit fast and stable factorizations. However, for discretized problems, such forms are restricted to 1D cases. In this work, we propose a framework to break such a 1D barrier. We study the feasibility of designing multilayer hierarchically semiseparable (MHS) structures for the approximation of dense matrices arising from multi-dimensional discretized problems such as certain integral operators. The MHS framework extends HSS forms to higher dimensions via the integration of multiple layers of structures, i.e., structures within the dense generator representations of HSS forms. Specifically, in the 2D case, we lay theoretical foundations and justify the existence of MHS structures based on the fast multipole method (FMM) and algebraic techniques such as representative subset selection. Rigorous numerical rank bounds and conditions for the structures are given. Representative subsets of points and a multi-layer tree are used to intuitively illustrate the structures. The MHS framework makes it convenient to explore multidimensional FMM structures. MHS representations are suitable for stable direct factorizations and can take advantage of existing methods and analysis well developed for simple HSS methods. Numerical tests for some discretized operators show that the appropriate inner-layer numerical ranks are significantly smaller than the off-diagonal numerical ranks used in standard HSS approximations. AMS subject classifications: 15A23, 65F05, 65F30
{"title":"Multi-Layer Hierarchical Structures","authors":"J. Xia","doi":"10.4208/CSIAM-AM.2021.NLA.02","DOIUrl":"https://doi.org/10.4208/CSIAM-AM.2021.NLA.02","url":null,"abstract":"In structured matrix computations, existing rank structures such as hierarchically semiseparable (HSS) forms admit fast and stable factorizations. However, for discretized problems, such forms are restricted to 1D cases. In this work, we propose a framework to break such a 1D barrier. We study the feasibility of designing multilayer hierarchically semiseparable (MHS) structures for the approximation of dense matrices arising from multi-dimensional discretized problems such as certain integral operators. The MHS framework extends HSS forms to higher dimensions via the integration of multiple layers of structures, i.e., structures within the dense generator representations of HSS forms. Specifically, in the 2D case, we lay theoretical foundations and justify the existence of MHS structures based on the fast multipole method (FMM) and algebraic techniques such as representative subset selection. Rigorous numerical rank bounds and conditions for the structures are given. Representative subsets of points and a multi-layer tree are used to intuitively illustrate the structures. The MHS framework makes it convenient to explore multidimensional FMM structures. MHS representations are suitable for stable direct factorizations and can take advantage of existing methods and analysis well developed for simple HSS methods. Numerical tests for some discretized operators show that the appropriate inner-layer numerical ranks are significantly smaller than the off-diagonal numerical ranks used in standard HSS approximations. AMS subject classifications: 15A23, 65F05, 65F30","PeriodicalId":29749,"journal":{"name":"CSIAM Transactions on Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47425223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-01DOI: 10.4208/csiam-am.2020-0206
Yingzhou Li, J. Poulson, Lexing Ying
{"title":"Distributed-Memory $mathscr{H}$-Matrix Algebra I: Data Distribution and Matrix-Vector Multiplication","authors":"Yingzhou Li, J. Poulson, Lexing Ying","doi":"10.4208/csiam-am.2020-0206","DOIUrl":"https://doi.org/10.4208/csiam-am.2020-0206","url":null,"abstract":"","PeriodicalId":29749,"journal":{"name":"CSIAM Transactions on Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43697668","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-16DOI: 10.4208/csiam-am.so-2022-0024
Gang-Han Fan, T. Luo, Y. Xiang
We study the 2+1 dimensional continuum model for the evolution of stepped epitaxial surface under long-range elastic interaction proposed by Xu and Xiang (SIAM J. Appl. Math. 69, 1393-1414, 2009). The long-range interaction term and the two length scales in this model makes PDE analysis challenging. Moreover, unlike in the 1+1 dimensional case, there is a nonconvexity contribution in the total energy in the 2+1 dimensional case, and it is not easy to prove that the solution is always in the well-posed regime during the evolution. In this paper, we propose a modified 2+1 dimensional continuum model based on the underlying physics. This modification fixes the problem of possible illposedness due to the nonconvexity of the energy functional. We prove the existence and uniqueness of both the static and dynamic solutions and derive a minimum energy scaling law for them. We show that the minimum energy surface profile is mainly attained by surfaces with step meandering instability. This is essentially different from the energy scaling law for the 1+1 dimensional epitaxial surfaces under elastic effects attained by step bunching surface profiles. We also discuss the transition from the step bunching instability to the step meandering instability in 2+1 dimensions.
本文研究了Xu和Xiang (SIAM J. appll)提出的长时间弹性相互作用下阶梯外延表面演化的2+1维连续体模型。数学。69,1393-1414,2009)。该模型中的远程相互作用项和两个长度尺度使得PDE分析具有挑战性。此外，与1+1维情况不同，2+1维情况下的总能量存在非凸性贡献，且在演化过程中不容易证明解总是处于适定状态。在本文中，我们提出了一个基于基础物理的改进的2+1维连续体模型。这种修正修正了由于能量泛函的非凸性而可能产生的病态问题。我们证明了静态解和动态解的存在唯一性，并推导了它们的最小能量标度律。结果表明，能量最小的表面轮廓主要是由阶梯弯曲不稳定曲面获得的。这与阶跃聚束表面轮廓在弹性效应下得到的1+1维外延表面的能量标度规律有本质区别。我们还讨论了在2+1维中从阶跃聚束不稳定性到阶跃弯曲不稳定性的转变。
{"title":"Existence, Uniqueness and Energy Scaling of (2+1)-Dimensional Continuum Model for Stepped Epitaxial Surfaces with Elastic Effects","authors":"Gang-Han Fan, T. Luo, Y. Xiang","doi":"10.4208/csiam-am.so-2022-0024","DOIUrl":"https://doi.org/10.4208/csiam-am.so-2022-0024","url":null,"abstract":"We study the 2+1 dimensional continuum model for the evolution of stepped epitaxial surface under long-range elastic interaction proposed by Xu and Xiang (SIAM J. Appl. Math. 69, 1393-1414, 2009). The long-range interaction term and the two length scales in this model makes PDE analysis challenging. Moreover, unlike in the 1+1 dimensional case, there is a nonconvexity contribution in the total energy in the 2+1 dimensional case, and it is not easy to prove that the solution is always in the well-posed regime during the evolution. In this paper, we propose a modified 2+1 dimensional continuum model based on the underlying physics. This modification fixes the problem of possible illposedness due to the nonconvexity of the energy functional. We prove the existence and uniqueness of both the static and dynamic solutions and derive a minimum energy scaling law for them. We show that the minimum energy surface profile is mainly attained by surfaces with step meandering instability. This is essentially different from the energy scaling law for the 1+1 dimensional epitaxial surfaces under elastic effects attained by step bunching surface profiles. We also discuss the transition from the step bunching instability to the step meandering instability in 2+1 dimensions.","PeriodicalId":29749,"journal":{"name":"CSIAM Transactions on Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46718709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-30DOI: 10.4208/csiam-am.SO-2020-0008
Lei Wang, Bin Gao, Xin Liu
We propose a class of multipliers correction methods to minimize a differentiable function over the Stiefel manifold. The proposed methods combine a function value reduction step with a proximal correction step. The former one searches along an arbitrary descent direction in the Euclidean space instead of a vector in the tangent space of the Stiefel manifold. Meanwhile, the latter one minimizes a first-order proximal approximation of the objective function in the range space of the current iterate to make Lagrangian multipliers associated with orthogonality constraints symmetric at any accumulation point. The global convergence has been established for the proposed methods. Preliminary numerical experiments demonstrate that the new methods significantly outperform other state-of-the-art first-order approaches in solving various kinds of testing problems.
{"title":"Multipliers Correction Methods for Optimization Problems over the Stiefel Manifold","authors":"Lei Wang, Bin Gao, Xin Liu","doi":"10.4208/csiam-am.SO-2020-0008","DOIUrl":"https://doi.org/10.4208/csiam-am.SO-2020-0008","url":null,"abstract":"We propose a class of multipliers correction methods to minimize a differentiable function over the Stiefel manifold. The proposed methods combine a function value reduction step with a proximal correction step. The former one searches along an arbitrary descent direction in the Euclidean space instead of a vector in the tangent space of the Stiefel manifold. Meanwhile, the latter one minimizes a first-order proximal approximation of the objective function in the range space of the current iterate to make Lagrangian multipliers associated with orthogonality constraints symmetric at any accumulation point. The global convergence has been established for the proposed methods. Preliminary numerical experiments demonstrate that the new methods significantly outperform other state-of-the-art first-order approaches in solving various kinds of testing problems.","PeriodicalId":29749,"journal":{"name":"CSIAM Transactions on Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2020-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47560082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-06DOI: 10.4208/csiam-am.2020-0043
Yuhong Dai, Liwei Zhang
Study about theory and algorithms for constrained optimization usually assumes that the feasible region of the optimization problem is nonempty. However, there are many important practical optimization problems whose feasible regions are not known to be nonempty or not, and optimizers of the objective function with the least constraint violation prefer to be found. A natural way for dealing with these problems is to extend the constrained optimization problem as the one optimizing the objective function over the set of points with the least constraint violation. Firstly, the minimization problem with least constraint violation is proved to be an Lipschitz equality constrained optimization problem when the original problem is a convex optimization problem with possible inconsistent conic constraints, and it can be reformulated as an MPEC problem. Secondly, for nonlinear programming problems with possible inconsistent constraints, various types of stationary points are presented for the MPCC problem which is equivalent to the minimization problem with least constraint violation, and an elegant necessary optimality condition, named as L-stationary condition, is established from the classical optimality theory of Lipschitz continuous optimization. Finally, the smoothing Fischer-Burmeister function method for nonlinear programming case is constructed for solving the problem minimizing the objective function with the least constraint violation. It is demonstrated that, when the positive smoothing parameter approaches to zero, any point in the outer limit of the KKT-point mapping is an L-stationary point of the equivalent MPCC problem.
{"title":"Optimization with Least Constraint Violation","authors":"Yuhong Dai, Liwei Zhang","doi":"10.4208/csiam-am.2020-0043","DOIUrl":"https://doi.org/10.4208/csiam-am.2020-0043","url":null,"abstract":"Study about theory and algorithms for constrained optimization usually assumes that the feasible region of the optimization problem is nonempty. However, there are many important practical optimization problems whose feasible regions are not known to be nonempty or not, and optimizers of the objective function with the least constraint violation prefer to be found. A natural way for dealing with these problems is to extend the constrained optimization problem as the one optimizing the objective function over the set of points with the least constraint violation. Firstly, the minimization problem with least constraint violation is proved to be an Lipschitz equality constrained optimization problem when the original problem is a convex optimization problem with possible inconsistent conic constraints, and it can be reformulated as an MPEC problem. Secondly, for nonlinear programming problems with possible inconsistent constraints, various types of stationary points are presented for the MPCC problem which is equivalent to the minimization problem with least constraint violation, and an elegant necessary optimality condition, named as L-stationary condition, is established from the classical optimality theory of Lipschitz continuous optimization. Finally, the smoothing Fischer-Burmeister function method for nonlinear programming case is constructed for solving the problem minimizing the objective function with the least constraint violation. It is demonstrated that, when the positive smoothing parameter approaches to zero, any point in the outer limit of the KKT-point mapping is an L-stationary point of the equivalent MPCC problem.","PeriodicalId":29749,"journal":{"name":"CSIAM Transactions on Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2020-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48541755","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-17DOI: 10.4208/csiam-am.SO-2021-0034
Jie Xu
We discuss the expansion of interaction kernels between anisotropic rigid molecules. The expansion decouples the correlated orientational variables so that it can be utilized to derive macroscopic models. Symmetries of two types are considered. First, we examine the symmetry of the interacting cluster, including the translation and rotation of the whole cluster, and label permutation within the cluster. The expansion is expressed by symmetric traceless tensors, and the linearly independent terms are identified. Then, we study the molecular symmetry characterized by a point group in $O(3)$. The proper rotations determine what symmetric traceless tensors can appear. The improper rotations decompose these tensors into two subspaces and determine how the tensors in the two subspaces are coupled. For each point group, we identify the two subspaces, so that the expansion consistent with the point group is established.
{"title":"Symmetry-Consistent Expansion of Interaction Kernels Between Rigid Molecules","authors":"Jie Xu","doi":"10.4208/csiam-am.SO-2021-0034","DOIUrl":"https://doi.org/10.4208/csiam-am.SO-2021-0034","url":null,"abstract":"We discuss the expansion of interaction kernels between anisotropic rigid molecules. The expansion decouples the correlated orientational variables so that it can be utilized to derive macroscopic models. Symmetries of two types are considered. First, we examine the symmetry of the interacting cluster, including the translation and rotation of the whole cluster, and label permutation within the cluster. The expansion is expressed by symmetric traceless tensors, and the linearly independent terms are identified. Then, we study the molecular symmetry characterized by a point group in $O(3)$. The proper rotations determine what symmetric traceless tensors can appear. The improper rotations decompose these tensors into two subspaces and determine how the tensors in the two subspaces are coupled. For each point group, we identify the two subspaces, so that the expansion consistent with the point group is established.","PeriodicalId":29749,"journal":{"name":"CSIAM Transactions on Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2020-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43432088","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-01DOI: 10.4208/csiam-am.2020-0010
Yiming Gao
. A good statistical model of speckle formation is useful to design a good speckle reduction model for clinical ultrasound images. We propose a new general distribution to describe the distribution of speckle in clinical ultrasound images accord-ing to a log-compression algorithm of clinical ultrasound imaging. A new variational model is designed to remove the speckle noise with the proposed general distribution. The efficiency of the proposed model is confirmed by experiments on synthetic images and real ultrasound images. Compared with previous variational methods which as-sign a designated distribution, the proposed method is adaptive to remove different kinds of speckle noise by estimating parameters to find suitable distribution. The experiments show that the proposed method can adaptively remove different types of speckle noise.
{"title":"An Adaptive Data-Fitting Model for Speckle Reduction of Log-Compressed Ultrasound Images","authors":"Yiming Gao","doi":"10.4208/csiam-am.2020-0010","DOIUrl":"https://doi.org/10.4208/csiam-am.2020-0010","url":null,"abstract":". A good statistical model of speckle formation is useful to design a good speckle reduction model for clinical ultrasound images. We propose a new general distribution to describe the distribution of speckle in clinical ultrasound images accord-ing to a log-compression algorithm of clinical ultrasound imaging. A new variational model is designed to remove the speckle noise with the proposed general distribution. The efficiency of the proposed model is confirmed by experiments on synthetic images and real ultrasound images. Compared with previous variational methods which as-sign a designated distribution, the proposed method is adaptive to remove different kinds of speckle noise by estimating parameters to find suitable distribution. The experiments show that the proposed method can adaptively remove different types of speckle noise.","PeriodicalId":29749,"journal":{"name":"CSIAM Transactions on Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45191729","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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