An implicit Landweber method for nonlinear ill-posed operator equations

In this paper, we are interested in the solution of nonlinear inverse problems of the form F(x)=y. We propose an implicit Landweber method, which is similar to the third-order midpoint Newton method in form, and consider the convergence behavior of the implicit Landweber method. Using the discrepancy principle as a stopping criterion, we obtain a regularization method for ill-posed problems. We conclude with numerical examples confirming the theoretical results, including comparisons with the classical Landweber iteration and presented modified Landweber methods.     

A coordinate gradient descent method for <Emphasis Type="Italic">ℓ</Emphasis><Subscript>1</Subscript>-regularized convex minimization

In applications such as signal processing and statistics, many problems involve finding sparse solutions to under-determined linear systems of equations. These problems can be formulated as a structured nonsmooth optimization problems, i.e., the problem of minimizing ℓ ₁-regularized linear least squares problems. In this paper, we propose a block coordinate gradient descent method (abbreviated as CGD) to solve the more general ℓ ₁-regularized convex minimization problems, i.e., the problem of minimizing an ℓ ₁-regularized convex smooth function. We establish a Q-linear convergence rate for our method when the coordinate block is chosen by a Gauss-Southwell-type rule to ensure sufficient descent. We propose efficient implementations of the CGD method and report numerical results for solving large-scale ℓ ₁-regularized linear least squares problems arising in compressed sensing and image deconvolution as well as large-scale ℓ ₁-regularized logistic regression problems for feature selection in data classification. Comparison with several state-of-the-art algorithms specifically designed for solving large-scale ℓ ₁-regularized linear least squares or logistic regression problems suggests that an efficiently implemented CGD method may outperform these algorithms despite the fact that the CGD method is not specifically designed just to solve these special classes of problems.     

A rapid method of solving differential equations with a small parameter

We propose and justify a new method of solving differential equations with small parameter a method that fulfills the requirements of the problems of celestial mechanics. In contrast to the classical power-series method the proposed method converges rapidly. We discuss new formulations of problems and the promise of the method. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 20–27     

Parameterized complexity of even/odd subgraph problems

We study the parameterized complexity of the problems of determining whether a graph contains a k-edge subgraph (k-vertex induced subgraph) that is a Π-graph for Π-graphs being one of the following four classes of graphs: Eulerian graphs, even graphs, odd graphs, and connected odd graphs. We also consider the parameterized complexity of their parametric dual problems.For these sixteen problems, we show that eight of them are fixed parameter tractable and four are W[1]-hard. Our main techniques are the color-coding method of Alon, Yuster and Zwick, and the random separation method of Cai, Chan and Chan.     

A new logarithmic-quadratic proximal method for nonlinear complementarity problems

In this paper, we propose a new modified logarithmic-quadratic proximal (LQP) method for solving nonlinear complementarity problems (NCP). We suggest using a prediction-correction method to solve NCP. The predictor is obtained via solving the LQP system approximately under significantly relaxed accuracy criterion and the new iterate is computed by using a new step size α_k. Under suitable conditions, we prove that the new method is globally convergent. We report preliminary computational results to illustrate the efficiency of the proposed method. This new method can be considered as a significant refinement of the previously known methods for solving nonlinear complementarity problems.     

Entropic proximal decomposition methods for convex programs and variational inequalities

We consider convex optimization and variational inequality problems with a given separable structure. We propose a new decomposition method for these problems which combines the recent logarithmic-quadratic proximal theory introduced by the authors with a decomposition method given by Chen-Teboulle for convex problems with particular structure. The resulting method allows to produce for the first time provably convergent decomposition schemes based on C ^∞ Lagrangians for solving convex structured problems. Under the only assumption that the primal-dual problems have nonempty solution sets, global convergence of the primal-dual sequences produced by the algorithm is established. Received: October 6, 1999 / Accepted: February 2001?Published online September 17, 2001     

A block coordinate gradient descent method for regularized convex separable optimization and covariance selection

We consider a class of unconstrained nonsmooth convex optimization problems, in which the objective function is the sum of a convex smooth function on an open subset of matrices and a separable convex function on a set of matrices. This problem includes the covariance selection problem that can be expressed as an ℓ ₁-penalized maximum likelihood estimation problem. In this paper, we propose a block coordinate gradient descent method (abbreviated as BCGD) for solving this class of nonsmooth separable problems with the coordinate block chosen by a Gauss-Seidel rule. The method is simple, highly parallelizable, and suited for large-scale problems. We establish global convergence and, under a local Lipschizian error bound assumption, linear rate of convergence for this method. For the covariance selection problem, the method can terminate in O(n³/e){O(n^3/\epsilon)} iterations with an e{\epsilon}-optimal solution. We compare the performance of the BCGD method with the first-order methods proposed by Lu (SIAM J Optim 19:1807–1827, 2009; SIAM J Matrix Anal Appl 31:2000–2016, 2010) for solving the covariance selection problem on randomly generated instances. Our numerical experience suggests that the BCGD method can be efficient for large-scale covariance selection problems with constraints.     

Solution of boundary and eigenvalue problems for second‐order elliptic operators in the plane using pseudoanalytic formal powers

We propose a method for solving boundary value and eigenvalue problems for the elliptic operator D = div p grad + qin the plane using pseudoanalytic function theory and in particular pseudoanalytic formal powers. Under certain conditions on the coefficients p and q with the aid of pseudoanalytic function theory a complete system of null solutions of the operator can be constructed following a simple algorithm consisting in recursive integration. This system of solutions is used for solving boundary value and spectral problems for the operator D in bounded simply connected domains. We study theoretical and numerical aspects of the method. Copyright © 2010 John Wiley & Sons, Ltd.     

The axisymmetric problem of thermoelasticity of a multilayer thermosensitive tube

We propose a method of solving stationary heat-conduction problems of contacting bodies with coefficient of thermal conductivity that are linear functions of the temperature and the corresponding problems of thermoelasticity based on the method of perturbations. We give a numerical analysis of the thermal stresses in a two-layer tube. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 97–103.     

Solution of one-dimensional problems of elasticity and thermoelasticity in stresses for a cylinder

We propose a method of solving problems of elasticity and thermoelasticity in stresses, making it possible to give a simpler construction of the solutions of such one- dimensional problems for a cylinder, compared with the known method of solving in the displacements. We obtain a relation between the components of the stress tensor and the integral conditions of equilibrium and continuity that has important applications in the solution of inverse problems of thermoelasticity.Translated fromMatematichni Metodi ta Fiziko-Mechanichni Polya, Vol. 40, No. 3, 1997, pp. 103–107.     

Global optimization by continuous grasp

We introduce a novel global optimization method called Continuous GRASP (C-GRASP) which extends Feo and Resende’s greedy randomized adaptive search procedure (GRASP) from the domain of discrete optimization to that of continuous global optimization. This stochastic local search method is simple to implement, is widely applicable, and does not make use of derivative information, thus making it a well-suited approach for solving global optimization problems. We illustrate the effectiveness of the procedure on a set of standard test problems as well as two hard global optimization problems.     

and Norms Error Estimates in Finite Element Method for Linear Parabolic Interface Problems

We derive new a priori error estimates for linear parabolic equations with discontinuous coefficients. Due to low global regularity of the solutions the error analysis of the standard finite element method for parabolic problems is difficult to adopt for parabolic interface problems. A finite element procedure is, therefore, proposed and analyzed in this paper. We are able to show that the standard energy technique of finite element method for non-interface parabolic problems can be extended to parabolic interface problems if we allow interface triangles to be curved triangles. Optimal pointwise-in-time error estimates in the L ²(Ω) and H ¹(Ω) norms are shown to hold for the semidiscrete scheme. A fully discrete scheme based on backward Euler method is analyzed and pointwise-in-time error estimates are derived. The interfaces are assumed to be arbitrary shape but smooth for our purpose.     

A recurrence method for a special class of continuous time linear programming problems

This article studies a numerical solution method for a special class of continuous time linear programming problems denoted by (SP). We will present an efficient method for finding numerical solutions of (SP). The presented method is a discrete approximation algorithm, however, the main work of computing a numerical solution in our method is only to solve finite linear programming problems by using recurrence relations. By our constructive manner, we provide a computational procedure which would yield an error bound introduced by the numerical approximation. We also demonstrate that the searched approximate solutions weakly converge to an optimal solution. Some numerical examples are given to illustrate the provided procedure.     

On Backward p(x)-Parabolic Equations for Image Enhancement

In this study, we investigate the backward p(x)-parabolic equation as a new methodology to enhance images. We propose a novel iterative regularization procedure for the backward p(x)-parabolic equation based on the nonlinear Landweber method for inverse problems. The proposed scheme can also be extended to the family of iterative regularization methods involving the nonlinear Landweber method. We also investigate the connection between the variable exponent p(x) in the proposed energy functional and the diffusivity function in the corresponding Euler-Lagrange equation. It is well known that the forward problems converges to a constant solution destroying the image. The purpose of the approach of the backward problems is twofold. First, solving the backward problem by a sequence of forward problems, we obtain a smooth image which is denoised. Second, by choosing the initial data properly, we try to reduce the blurriness of the image. The numerical results for denoising appear to give improvement over standard methods as shown by preliminary results.     

On the complexity of some subgraph problems

We study the complexity of the problem of deciding the existence of a spanning subgraph of a given graph, and of that of finding a maximum (weight) such subgraph. We establish some general relations between these problems, and we use these relations to obtain new NP-completeness results for maximum (weight) spanning subgraph problems from analogous results for existence problems and from results in extremal graph theory. On the positive side, we provide a decomposition method for the maximum (weight) spanning chordal subgraph problem that can be used, e.g., to obtain a linear (or O(nlogn)) time algorithm for such problems in graphs with vertex degree bounded by 3.     

The current state of the problem of interaction of acoustic beams with obstacles in a deformable medium

We analyze the state of the problem of the formation of radiated and scattered acoustic beams in application to the development of a methodology for studying the information aspects of hydroacoustics and nondestructive control. We discuss the problems of the selective generation of characteristic vibrations of elastic objects in a deformable medium using sharply directed acoustic impulses. We study the problem of posing and methods of solving a certain class of inverse problems of scattering theory.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 56–64.In conclusion we note the papers [7, 74, 100] connected with the traditional method of solving inverse problems-the selection method.     

Solution of a mixed boundary-value problem of thermodiffusion for layered bodies of canonical shape

We propose a method of solving coupled thermodiffusion problems for layered bodies of canonical shape. The method is based on separating the coupled system of equations and boundary conditions into independent boundary-value problems. The solution contains arbitrary functions of time determined from a system of second-order integral equations of convolution type obtained as a result of satisfying the boundary conditions.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 87–91.     

Error estimates for finite element approximations of nonlinear monotone elliptic problems with application to numerical homogenization

We consider a finite element method (FEM) with arbitrary polynomial degree for nonlinear monotone elliptic problems. Using a linear elliptic projection, we first give a new short proof of the optimal convergence rate of the FEM in the L² norm. We then derive optimal a priori error estimates in the H¹ and L² norm for a FEM with variational crimes due to numerical integration. As an application, we derive a priori error estimates for a numerical homogenization method applied to nonlinear monotone elliptic problems. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 955–969, 2016     

A continuation method for (strongly) monotone variational inequalities

We consider the variational inequality problem, denoted by VIP(X, F), whereF is a strongly monotone function and the convex setX is described by some inequality (and possibly equality) constraints. This problem is solved by a continuation (or interior-point) method, which solves a sequence of certain perturbed variational inequality problems. These perturbed problems depend on a parameter > 0. It is shown that the perturbed problems have a unique solution for all values of > 0, and that any sequence generated by the continuation method converges to the unique solution of VIP(X,F) under a well-known linear independence constraint qualification (LICQ). We also discuss the extension of the continuation method to monotone variational inequalities and present some numerical results obtained with a suitable implementation of this method. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

Multi-Duality in Minimal Surface—Type Problems

The multi-duality of the nonlinear variational problem inf J(u, Λu) is studied for minimal surfaces-type problems. By using the method developed by Gao and Strang [1], the Fenchel-Rockafellar's duality theory is generalized to the problems with affine operator Λ. Two dual variational principles are established for nonparametric surfaces with constant mean curvature. We show that for the same primal problem, there may exist different dual problems. The primal problem may or may not possess a solution, whereas each dual problem possesses a unique solution. An evolutionary method for solving the nonlinear optimal-shape design problem is presented with numerical results.