Implicit Fixed-point Equation Research Articles

Computationally Efficient IMplicit Training Strategy for UNrolled NEtworks (IMUNNE): A preliminary analysis using accelerated real-time cardiac cine MRI.

Highly-undersampled, dynamic MRI reconstruction, particularly in multi-coil scenarios, is a challenging inverse problem. Unrolled networks achieve state-of-the-art performance in MRI reconstruction but suffer from long training times and extensive GPU memory cost. In this work, we propose a novel training strategy for IMplicit UNrolled NEtworks (IMUNNE) for highly-undersampled, multi-coil dynamic MRI reconstruction. It formulates the MRI reconstruction problem as an implicit fixed-point equation and leverages gradient approximation for backpropagation, enabling training of deep architectures with fixed memory cost. This study represents the first application of implicit network theory in the context of real-time cine MRI. The proposed method is evaluated using a prospectively undersampled, real-time cine dataset using radial k-space sampling, comprising balanced steady-state free precession (b-SSFP) readouts. Experiments include a hyperparameter search, head-to-head comparisons with a complex U-Net (CU-Net) and an alternating unrolled network (Alt-UN), and an analysis of robustness under noise perturbations; peak signal-to-noise ratio, structural similarity index, normalized root mean-square error, spatio-temporal entropic difference, and a blur metric were used. IMUNNE produced significantly and slightly better image quality compared to CU-Net and Alt-UN, respectively. Compared with Alt-UN, IMUNNE significantly reduced training and inference times, making it a promising approach for highly-accelerated, multi-coil real-time cine MRI reconstruction. IMUNNE strategy successfully applies unrolled networks to image reconstruction of highly-accelerated, real-time radial cine MRI. Implicit training enables rapid, high-quality, and cost-effective CMR exams by reducing training and inference times and lowering memory cost associated with advanced reconstruction methods.

Modified Picard-like Method for Solving Absolute Value Equations

We present a modified Picard-like method to solve absolute value equations by equivalently expressing the implicit fixed-point equation form of the absolute value equations as a two-by-two block nonlinear equation. This unifies some existing matrix splitting algorithms and improves the efficiency of the algorithm by introducing the parameter ω. For the choice of ω in the new method, we give a way to determine the quasi-optimal values. Numerical examples are given to show the feasibility of the proposed method. It is also shown that the new method is better than those proposed by Ke and Ma in 2017 and Dehghan and Shirilord in 2020.

A Robust Two-Step Modulus-Based Matrix Splitting Iteration Method for Mixed-Size Cell Circuit Legalization Problem

As semiconductor manufacturing moves into the nanotechnology era, mixed-size standard cell circuit designs have become mainstream, but they also pose significant challenges to placement. To address the mixed-size standard cell circuit legalization problem, in this paper, we first formulate it as a quadratic programming problem with nonnegative constraints, resulting in an equivalent linear complementarity problem (LCP). In particular, the system matrix involved in the LCP is positive semi-definite, so we perturb it to a positive-definite matrix. Then, we cast the linear complementarity problem as an equivalent implicit fixed-point equation. To improve the convergence rate of the class of modulus-based matrix splitting iteration method (MMS), we develop a robust two-step MMS to solve the fixed-point equation. Finally, we analyze the convergence property and illustrate the approximation of the solution between the perturbed LCP and the original LCP. Numerical experiments for the legalization problem are provided, and the results reveal that our approach is competitive with the existing state-of-the-art methods for solving mixed-size cell circuit legalization problems.

Modulus-based matrix splitting iteration methods with new splitting scheme for horizontal implicit complementarity problems

In this paper, we propose the horizontal implicit complementarity problems, which cover many complementarity problems that have wide applications. After reformulating the horizontal implicit complementarity problem as an implicit fixed-point equation, the modulus-based matrix splitting iteration methods are applied to solve the problems. Besides, this paper introduces a new kind of matrix splitting, called modulus-based Hermitian and skew-Hermitian matrix relaxation splitting. The corresponding modulus-based matrix splitting iteration method is shown superior to the nonsmooth Newton's method and the existing matrix splitting schemes by some numerical experiments. The convergence analysis is given and validates that our methods are more practical in applications.

Modulus-based inexact non-alternating preconditioned splitting method for linear complementarity problems

For the large sparse linear complementarity problem, by reformulating it as implicit fixed-point equations, we establish a modulus-based iteration method with the aid of an inexact non-alternating preconditioned matrix splitting iteration method used as an inner iteration to solve the module equations rapidly in each iteration step. The convergence properties of this method are carefully demonstrated under certain conditions. Numerical results, including the numbers of iteration steps and the computing times, validate that our method is superior to the other three iteration methods.

A modified LM algorithm for tensor complementarity problems over the circular cone

The tensor complementarity problem over circular cone (CCTCP for short) is studied, which is a specially structured nonlinear complementarity problem. Useful properties of the circular cone help to reformulate equivalently CCTCP as an implicit fixed-point equation. Based on the smoothing functions, we reformulate the obtained fixed-point equation as a family of parameterized smoothing equations. Moreover, we propose a modified Levenberg–Marquardt (LM) algorithm to solve the problem iteratively and show that the sequence generated by the new algorithm converges to a solution quadratically under suitable conditions. Preliminary numerical results demonstrate that the proposed algorithm is effective.

Modified SOR-Like Method for Absolute Value Equations

In this paper, based on the work of Ke and Ma, a modified SOR-like method is presented to solve the absolute value equations (AVE), which is gained by equivalently expressing the implicit fixed-point equation form of the AVE as a two-by-two block nonlinear equation. Under certain conditions, the convergence conditions for the modified SOR-like method are presented. The computational efficiency of the modified SOR-like method is better than that of the SOR-like method by some numerical experiments.

Generalized SOR-Like Iteration Method for Linear Complementarity Problem

In this paper, we present a generalized SOR-like iteration method to solve the non-Hermitian positive definite linear complementarity problem (LCP), which is obtained by reformulating equivalently the implicit fixed-point equation of the LCP as a two-by-two block nonlinear equation. The convergence properties of the generalized SOR-like iteration method are discussed under certain conditions. Numerical experiments show that the generalized SOR-like method is efficient, compared with the SOR-like method and the modulus-based SOR method.

Modulus-based matrix splitting methods for a class of horizontal nonlinear complementarity problems

In this paper, we generalize modulus-based matrix splitting methods to a class of horizontal nonlinear complementarity problems (HNCPs). First, we write the HNCP as an implicit fixed-point equation and we introduce the proposed solution procedures. We then prove the convergence of the methods under some assumptions. We also comment on how the proposed methods and convergence theorems generalize existing results on (standard) linear and nonlinear complementarity problems and on horizontal linear complementarity problems. Finally, numerical experiments are solved to demonstrate the efficiency of the procedures. In this context, the effects of the splitting, of the dimension of the matrices, and of the nonlinear term of the problem are analyzed.

The Matrix Splitting Iteration Method for Nonlinear Complementarity Problems Associated with Second-Order Cone

For a class of second-order cone nonlinear complementarity problems, abbreviated as SOCNCPs, we establish the modulus-based matrix splitting relaxation iteration methods, which are obtained by reformulating equivalently SOCNCP as an implicit fixed-point equation based on Jordan algebra associated with the second-order cone. The global convergence theorems are given under suitable choices of the involved splitting matrix and parameter matrices. When the splitting matrix is symmetric positive definite, the strategy choice of the parameters is discussed. Numerical experiments have shown that the modulus-based iteration methods are effective for solving SOCNCPs.

The modulus-based matrix double splitting iteration method for linear complementarity problems

For large sparse linear complementarity problems, through reformulating them as implicit fixed-point equations, we propose a modulus-based matrix double splitting (MB-DS) iteration method by splitting the system matrices twice. Besides, the convergence of this method is proved when the system matrix is a P-matrix and an H+-matrix. In some special cases, we present the convergence regions and the optimal values for the parameter ω. In order to show the efficiency of the MB-DS iteration method and the effectiveness of the optimal parameter, some corresponding numerical experiments are performed.

The relaxation modulus-based matrix splitting iteration methods for circular cone nonlinear complementarity problems

In this paper, we study a class of nonlinear complementarity problems associated with circular cone (CCNCP for short), which is a type of non-symmetric cone complementarity problems. Useful properties of the circular cone are investigated, which help to reformulate equivalently CCNCP as an implicit fixed-point equation. Based on the implicit fixed-point equation and splittings of the system matrix, we establish a class of relaxation modulus-based matrix splitting iteration methods for solving such a complementarity problem. The convergence of the proposed modulus-based matrix splitting iteration methods has been analyzed and the strategy choice of the parameters are discussed when the splitting matrix is symmetric positive definite. Numerical experiments have shown that the modulus-based iteration methods are effective for solving CCNCP.

The modulus-based matrix splitting iteration methods for second-order cone linear complementarity problems

For the second-order cone linear complementarity problems, abbreviated as SOCLCPs, we establish two classes of modulus-based matrix splitting iteration methods, which are obtained by reformulating equivalently the SOCLCP as an implicit fixed-point equation based on Jordan algebra associated with the second-order cone. The convergence of these modulus-based matrix splitting iteration methods has been established and the optimal iteration parameters of these methods are discussed when the splitting matrix is symmetric positive definite. Numerical experiments have shown that the modulus-based iteration methods are effective for solving the SOCLCPs.

On the convergence of modulus-based matrix splitting iteration methods for a class of nonlinear complementarity problems with H+-matrices

A class of modulus-based iteration methods for the nonlinear complementarity problem are presented by reformulating the complementarity problem to an implicit fixed-point equation. In order to efficiently implement these methods, practical iteration schemes based on matrix splitting and the choices of the parameters are presented and well studied. Moreover, we extend the convergence theory from the H -compatible splitting to H -splitting when the matrix is an H + -matrix, which results in more choices for the parameters. Numerical experiments are given to verify the theoretical results and illustrate the efficiency of the proposed methods.

Accelerated modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems

In this paper, we reformulate a class of nonlinear complementarity problems as the implicit fixed-point equations. We demonstrate accelerated modulus-based matrix splitting iteration method. We show their convergence by assuming that the system matrix is positive definite or the splitting of the system matrix are $$H_+$$ -compatible splitting and discuss the choice of the optimal parameter. Furthermore, we give two-step accelerated modulus-based matrix splitting iteration method, which may achieve higher computing efficiency. Numerical experiments are presented to show the effectiveness of the method.

Convergence results of a matrix splitting algorithm for solving weakly nonlinear complementarity problems

In this paper, we consider a class of weakly nonlinear complementarity problems (WNCP) with large sparse matrix. We present an accelerated modulus-based matrix splitting algorithm by reformulating the WNCP as implicit fixed point equations based on two splittings of the system matrixes. We show that, if the system matrix is a P-matrix, then under some mild conditions the sequence generated by the algorithm is convergent to the solution of WNCP.

Modified modulus-based matrix splitting algorithms for a class of weakly nondifferentiable nonlinear complementarity problems

By reformulating a class of weakly nonlinear complementarity problems as implicit fixed-point equations based on splitting of the system matrix, a modified modulus-based matrix splitting algorithm is presented. The convergence analysis of proposed algorithm is established for the case that the splitting of the system matrix is an H-splitting. Numerical experiments on two model problems are given to illustrate the theoretical results and examine the numerical effectiveness.

Two-sweep modulus-based matrix splitting iteration methods for linear complementarity problems

In this paper, we will extend the two-sweep iteration methods to solve the linear complementarity problems and establish a class of two-sweep modulus-based matrix splitting iteration methods for the implicit fixed-point equation of the linear complementarity problems. Some convergence properties of two-sweep modulus-based matrix splitting iteration methods are discussed when the system matrices are positive-definite matrices and H+-matrices. Numerical experiments are presented to illustrate the efficiency of the proposed methods.

A generalized Newton method for non-Hermitian positive definite linear complementarity problem

In this paper, based on the implicit fixed-point equation of the linear complementarity problem (LCP), a generalized Newton method is presented to solve the non-Hermitian positive definite linear complementarity problem. Some convergence properties of the proposed generalized Newton method are discussed. Numerical experiments are presented to illustrate the efficiency of the proposed method.

Two-step modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems

In this paper, we reformulate the nonlinear complementarity problem as an implicit fixed-point equation. We establish a modulus-based matrix splitting iteration method based on the implicit fixed-point equation and prove its convergence theorem under suitable conditions. Furthermore, we propose a two-step modulus-based matrix splitting iteration method, which may achieve higher computing efficiency. We can obtain many matrix splitting iteration methods by suitably choosing the matrix splittings and the parameters. The proposed methods can be regarded as extensions of the methods for linear complementarity problem. Numerical experiments are presented to show the effectiveness of the proposed methods.