In this paper, a novel alternating direction method of multiplier (ADMM) is proposed to solve the inverse scattering problems. The proposed method is suitable for a wide range of applications with electromagnetic detection. In order to solve the internal ill-posed problem of the integral equation and make the reconstructed images more closer to the ground truth, first, the augmented Lagrangian method is introduced to transform the complex constrained optimization problem into the extremum problem of unconstrained cost function. Therefore, two artificial regularization factors of the cost function are optimized. Then, this proposed method decomposes the unconstrained global problem in the inversion process into three linear sub-problem forms of contrast source function, contrast function, and dual variables. And the form of the updated algebra for each sub-problem is not complicated. By cross-iterating and updating contrast source function, contrast function, and dual variables, the global minimization of the cost function can be accurately found. Finally, the proposed method is compared with the existing well-known iterative method for solving the inverse scattering problem. Both the numerical and experimental tests verify the validity and accuracy of the proposed ADMM.

The Douglas–Rachford method is a splitting method frequently employed for finding zeros of sums of maximally monotone operators. When the operators in question are normal cone operators, the iterated process may be used to solve feasibility problems of the following form: Find $x\in \bigcap _{k=1}^{N}S_{k}$. The success of the method in the context of closed, convex, nonempty sets $S_{1},\ldots ,S_{N}$ is well known and understood from a theoretical standpoint. However, its performance in the nonconvex context is less well understood, yet it is surprisingly impressive. This was particularly compelling to Jonathan M. Borwein who, intrigued by Elser, Rankenburg and Thibault’s success in applying the method to solving sudoku puzzles, began an investigation of his own. We survey the current body of literature on the subject, and we summarize its history. We especially commemorate Professor Borwein’s celebrated contributions to the area.

For problems governed by a non-normal operator, the leading eigenvalue of the operator is of limited interest and a more relevant measure of the stability is obtained by considering the harmonic forcing causing the largest system response. Various methods for determining this so-called optimal forcing exist, but they all suffer from great computational expense and are hence not practical for large-scale problems. In the present paper a new method is presented, which is applicable to problems of arbitrary size. The method does not rely on timestepping, but on the solution of linear systems, in which the inverse Laplacian acts as a preconditioner. By formulating the search for the optimal forcing as an eigenvalue problem based on the resolvent operator, repeated system solves amount to power iterations, in which the dominant eigenvalue is seen to correspond to the energy amplification in a system for a given frequency, and the eigenfunction to the corresponding forcing function. Implementation of the method requires only minor modifications of an existing timestepping code, and is applicable to any partial differential equation containing the Laplacian, such as the Navier-Stokes equations. We discuss the method, first, in the context of the linear Ginzburg-Landau equation and then, the two-dimensional lid-driven cavity flow governed by the Navier-Stokes equations. Most importantly, we demonstrate that for the lid-driven cavity, the optimal forcing can be computed using a factor of up to 500 times fewer operator evaluations than the standard method based on exponential timestepping.

For solving linear complementarity problems LCP more attention has recently been paid on a class of iterative methods called the matrix-splitting. But up to now, no paper has discussed the effect of preconditioning technique for matrix-splitting methods in LCP. So, this paper is planning to fill in this gap and we use a class of preconditioners with generalized Accelerated Overrelaxation (GAOR) methods and analyze the convergence of these methods for LCP. Furthermore, Comparison between our methods and other non-preconditioned methods for the studied problem shows a remarkable agreement and reveals that our models are superior in point of view of convergence rate and computing efficiency. Besides, by choosing the appropriate parameters of these methods, we derive same results as the other iterative methods such as AOR, JOR, SOR etc. Finally the method is tested by some numerical experiments.

We present a simple yet effective and applicable scheme, based on quadrature, for constructing optimal iterative methods. According to the, still unproved, Kung-Traub conjecture an optimal iterative method based on n + 1 evaluations could achieve a maximum convergence order of 2n. Through quadrature, we develop optimal iterative methods of orders four and eight. The scheme can further be applied to develop iterative methods of even higher orders. Computational results demonstrate that the developed methods are efficient as compared with many well known methods.

In this paper we develop a Kantorovich-like theory for Chebyshev’s method, a well-known iterative method for solving nonlinear equations in Banach spaces. We improve the results obtained previously by considering Chebyshev’s method as an element of a family of iterative processes.

With the aid of index functions, we re-derive the ML(n)BiCGStab algorithm in [Yeung and Chan, SIAM J. Sci. Comput., 21 (1999), pp. 1263-1290] systematically. There are n ways to define the ML(n)BiCGStab residual vector. Each definition leads to a different ML(n)BiCGStab algorithm. We demonstrate this by presenting a second algorithm which requires less storage. In theory, this second algorithm serves as a bridge that connects the Lanczos-based BiCGStab and the Arnoldi-based FOM while ML(n)BiCG is a bridge connecting BiCG and FOM. We also analyze the breakdown situation from the probabilistic point of view and summarize some useful properties of ML(n)BiCGStab. Implementation issues are also addressed.

Inexact Uzawa algorithms for solving nonlinear saddle-point problems are proposed. A simple sufficient condition for the convergence of the inexact Uzawa algorithms is obtained. Numerical experiments show that the inexact Uzawa algorithms are convergent.

We present an iterative method based on an infinite dimensional adaptation of thesuccessive overrelaxation (SOR) algorithm for solving the 2-D neutron transport equation.In a wide range of application, the neutron transport operator admits a Self-Adjoint andm-Accretive Splitting (SAS). This splitting leads to an ADI-like iterative method whichconverges unconditionally and is equivalent to a fixed point problem where the operator isa 2 by 2 matrix of operators. An infinite dimensional adaptation of a SOR algorithm isthen applied to solve the matrix operator equation. Theoretical and numerical results ofconvergence are given

We develop several iterative methods for computing generalized inverses using both first and second order optimization methods in C*-algebras. Known steepest descent iterative methods are generalized in C*-algebras. We introduce second order methods based on the minimization of the norms ‖Ax − b‖2 and ‖x‖2 by means of the known second order unconstrained minimization methods. We give several examples which illustrate our theory.