Norm Solution Research Articles

Choice of the parameters in a primal-dual algorithm for Bregman iterated variational regularization

Focus of this work is solving a non-smooth constraint minimization problem by a primal-dual splitting algorithm involving proximity operators. The problem is penalized by the Bregman divergence associated with the non-smooth total variation (TV) functional. We analyze two aspects: Firstly, the convergence of the regularized solution of the minimization problem to the minimum norm solution. Second, the convergence of the iteratively regularized minimizer to the minimum norm solution by a primal-dual algorithm. For both aspects, we use the assumption of a variational source condition (VSC). This work emphasizes the impact of the choice of the parameters in stabilization of a primal-dual algorithm. Rates of convergence are obtained in terms of some concave, positive definite index function. The algorithm is applied to a simple two-dimensional image processing problem. Sufficient error analysis profiles are provided based on the size of the forward operator and the noise level in the measurement.

Singular Value Thresholding Algorithm for Wireless Sensor Network Localization

Wireless Sensor Networks (WSN) are of great current interest in the proliferation of technologies. Since the location of the sensors is one of the most interesting issues in WSN, the process of node localization is crucial for any WSN-based applications. Subsequently, WSN’s node estimation deals with a low-rank matrix which gives rise to the application of the Nuclear Norm Minimization (NNM) method. This paper will focus on the localization of 2-dimensional WSN with objects (obstacles). Recent studies introduce Nuclear Norm Minimization (NNM) for node estimation instead of formulating the rank minimization problem. Common way to tackle this problem is by implementing the Semidefinite Programming (SDP). However, SDP can only handle matrices with a size of less than 100 × 100. Therefore, we introduce the method of Singular Value Thresholding (SVT) which is an iterative algorithm to solve the NNM problem that produces a sequence of matrices { X k , Y k } and executes a soft-thresholding operation on the singular value of the matrix Y k . This algorithm is a user-friendly algorithm which produces a low computational cost with low storage capacity required to give the lowest-rank minimum nuclear norm solution.

Reordering sparse matrices into block-diagonal column-overlapped form

Many scientific and engineering applications necessitate computing the minimum norm solution of a sparse underdetermined linear system of equations. The minimum 2-norm solution of such systems can be obtained by a recent parallel algorithm, whose numerical effectiveness and parallel scalability are validated in both shared- and distributed-memory architectures. This parallel algorithm assumes the coefficient matrix in a block-diagonal column-overlapped (BDCO) form, which is a variant of the block-diagonal form where the successive diagonal blocks may overlap along their columns. The total overlap size of the BDCO form is an important metric in the performance of the subject parallel algorithm since it determines the size of the reduced system, solution of which is a bottleneck operation in the parallel algorithm. In this work, we propose a hypergraph partitioning model for reordering sparse matrices into BDCO form with the objective of minimizing the total overlap size and the constraint of maintaining balance on the number of nonzeros of the diagonal blocks. Our model makes use of existing partitioning tools that support fixed vertices in the recursive bipartitioning paradigm. Experimental results validate the use of our model as it achieves small overlap size and balanced diagonal blocks.

Finite element simulation of intimal thickening in 2D multi-layered arterial cross sections by morphoelasticity

Morphoelasticity is a framework commonly employed to describe tissue growth. Its central premise is the decomposition of the deformation gradient into the product of a growth tensor and an elastic tensor. In this paper we present a 2D finite element method to solve compressible morphoelasticity problems in arterial cross sections. The arterial wall is composed of three layers: the intima, media and adventitia. The intima is allowed to grow isotropically while the areas of the media and adventitia are approximately conserved. All three layers are modeled as Holzapfel–Gasser–Ogden type anisotropic hyperelastic materials with different mechanical parameters.This paper consists of three main contributions. First, a new energy functional that underpins the finite element method for morphoelasticity is presented. The bulk energy is associated with elastically deforming grown elements, rather than elements in the reference configuration which is the usual practice in classical hyperelasticity. A surface energy describes live pressure loading through a lumen blood pressure. We derive the equivalent weak form and show that stationary points correspond to the usual boundary value problem for mechanical equilibrium.Second, we present the details of a displacement-based morphoelastic finite element method. The primary variable for our method is the deformation, which we separate into two groups (horizontal and vertical deformations). We arrive at a system of nonlinear algebraic equations that are organized into self-contained blocks which can be assembled independently. We present a method to compute the Jacobian of this system which is fed into a Newton algorithm. The direction of collagen fibers in each layer is computed off-line by interpolating tangent vectors from nearby boundaries and a minimum norm solution for the grown and deformed artery is found by the QR factorization. This single-field approach produces reasonable numerical results, although the locking phenomenon is clearly present. For axisymmetric meshes, our position and stress fields are validated by the solution obtained from solving a system of ordinary differential equations.Finally, we use the code to simulate intimal thickening in pseudo-realistic arterial cross sections. We find that (i) Oscillations in the lumen-intima interface are unstable and amplify with growth (ii) Glagov remodeling occurs with lumen area initially increasing and then decreasing with growth and (iii) Maximum stresses migrate from the lumen-intima interface to the media-intima interface so that the media eventually bears the maximum stress.

AN ITERATIVE ALGORITHM FOR SOLVING A CLASS OF GENERALIZED COUPLED SYLVESTER-TRANSPOSE MATRIX EQUATIONS OVER BISYMMETRIC OR SKEW-ANTI-SYMMETRIC MATRICES

This paper presents an iterative algorithm to solve a class of generalized coupled Sylvester-transpose matrix equations over bisymmetric or skew-anti-symmetric matrices. When the matrix equations are consistent, the bisymmetric or skew-anti-symmetric solutions can be obtained within finite iteration steps in the absence of round-off errors for any initial bisymmetric or skew-anti-symmetric matrices by the proposed iterative algorithm. In addition, we can obtain the least norm solution by choosing the special initial matrices. Finally, numerical examples are given to demonstrate the iterative algorithm is quite efficient. The merit of our method is that it is easy to implement.

A Split-Future MPC Algorithm for Lithium-Ion Battery Cell-Level Fast-Charge Control

Constrained predictive control has emerged as a viable candidate for generating optimal real-time charging strategies for lithium ion batteries. Able to conform to hard constraints on problem variables, model predictive control (MPC) formulates an optimal 2-norm solution at each time step and can thus assure safe and reliable fast-charge operation. Standard MPC implementations make certain simplifying assumptions regarding future control actions beyond a specified control horizon. This paper demonstrates the potential gains that can be realized for the battery charge problem by relaxing these assumptions.

Modified CGLS iterative algorithm for solving the generalized Sylvester-conjugate matrix equation

By introducing the real inner product, this paper offers an modified conjugate gradient least squares iterative algorithm (MCGLS)for solving the generalized Sylvester-conjugate matrix equation. The properties of this algorithm are discussed and the finite convergence of this algorithm is proven. This new iterative method can obtain the symmetric least squares Frobenius norm solution within finite iteration steps in the absence of roundoff errors. Finally, two numerical examples are offered to illustrate the effectiveness of the proposed algorithm.

The generalized conjugate direction method for solving quadratic inverse eigenvalue problems over generalized skew Hamiltonian matrices with a submatrix constraint

In this paper, we consider a class of constrained quadratic inverse eigenvalue Problem 1.1. Then, a generalized conjugate direction method is proposed to obtain the generalized skew Hamiltonian matrix solutions with a submatrix constraint. In addition, by choosing a special kind of initial matrices, it is shown that the unique least Frobenius norm solutions can be obtained consequently. Some numerical results are reported to demonstrate the efficiency of our algorithm.

Hybrid Algorithms for Variational Inequalities Involving a Strict Pseudocontraction

In a real Hilbert space, we investigate the Tseng’s extragradient algorithms with hybrid adaptive step-sizes for treating a Lipschitzian pseudomonotone variational inequality problem and a strict pseudocontraction fixed-point problem, which are symmetry. By imposing some appropriate weak assumptions on parameters, we obtain a norm solution of the problems, which solves a certain hierarchical variational inequality.

On a backward problem for fractional diffusion equation with Riemann‐Liouville derivative

In the present paper, we study the initial inverse problem (backward problem) for a two‐dimensional fractional differential equation with Riemann‐Liouville derivative. Our model is considered in the random noise of the given data. We show that our problem is not well‐posed in the sense of Hadamard. A truncated method is used to construct an approximate function for the solution (called the regularized solution). Furthermore, the error estimate of the regularized solution in L2 and Hτ norms is considered and illustrated by numerical example.

Finite difference schemes for a structured population model in the space of measures.

We present two finite-difference methods for approximating solutions to a structured population model in the space of non-negative Radon Measures. The first method is a first-order upwind-based scheme and the second is high-resolution method of second-order. We prove that the two schemes converge to the solution in the Bounded-Lipschitz norm. Several numerical examples demonstrating the order of convergence and behavior of the schemes around singularities are provided. In particular, these numerical results show that for smooth solutions the upwind and high-resolution methods provide a first-order and a second-order approximation, respectively. Furthermore, for singular solutions the second-order high-resolution method is superior to the first-order method.

Two-grid methods for semi-linear elliptic interface problems by immersed finite element methods

In this paper, two-grid immersed finite element (IFE) algorithms are proposed and analyzed for semi-linear interface problems with discontinuous diffusion coefficients in two dimension. Because of the advantages of finite element (FE) formulation and the simple structure of Cartesian grids, the IFE discretization is used in this paper. Two-grid schemes are formulated to linearize the FE equations. It is theoretically and numerically illustrated that the coarse space can be selected as coarse as H = O(h1/4) (or H = O(h1/8)), and the asymptotically optimal approximation can be achieved as the nonlinear schemes. As a result, we can settle a great majority of nonlinear equations as easy as linearized problems. In order to estimate the present two-grid algorithms, we derive the optimal error estimates of the IFE solution in the Lp norm. Numerical experiments are given to verify the theorems and indicate that the present two-grid algorithms can greatly improve the computing efficiency.

A unified weighted minimum norm solution for the reference inverse problem in EEG

A well known problem in EEG recordings deals with the unknown potential of the reference electrode. In the last years several authors presented comparisons among the most popular solutions, the global conclusion being that the traditional Average Reference (AR) and the Reference Standardization Technique (REST) are the best approximations (Nunez, 2010; Kayser and Tenke, 2010; Liu et al., 2015; Chella et al., 2016). In this work we do not aim to further compare these techniques but to support the fact that both solutions can be derived from a general inverse problem formalism for reference estimation (Hu et al., 2019; Hu et al., 2018; Salido-Ruiz et al., 2011). Using the alternative approach of least squares, our findings are consistent with the theoretical findings in Hu et al. (2019) and Hu et al. (2018) showing that the AR is the minimum norm solution, while REST is a weighted minimum norm including some approximate propagation model. AR is thus a particular case of REST, which itself uses a particular formulation of the source estimation inverse problem. With a different derivation, we provide the additional powerful evidences to reinforce the cited findings.

Fitted finite volume method for indifference pricing in an exponential utility regime-switching model

We consider one-dimensional systems of weakly coupled degenerate semi-linear parabolic equations for buyer’s and writer’s pricing of European options under regime-switching with exponential utility function. First, we prove a comparison principle and then we establish a maximum principle for the indifference pricing differential problems. The differential problems are solved numerically by fitted finite volume method with second-order of accuracy. We prove the discrete maximum principle and convergence of the numerical solutions in maximum norm. Numerical results, illustrating the theoretical statements are presented and discussed.

Mildly Inertial Subgradient Extragradient Method for Variational Inequalities Involving an Asymptotically Nonexpansive and Finitely Many Nonexpansive Mappings

In a real Hilbert space, let the notation VIP indicate a variational inequality problem for a Lipschitzian, pseudomonotone operator, and let CFPP denote a common fixed-point problem of an asymptotically nonexpansive mapping and finitely many nonexpansive mappings. This paper introduces mildly inertial algorithms with linesearch process for finding a common solution of the VIP and the CFPP by using a subgradient approach. These fully absorb hybrid steepest-descent ideas, viscosity iteration ideas, and composite Mann-type iterative ideas. With suitable conditions on real parameters, it is shown that the sequences generated our algorithms converge to a common solution in norm, which is a unique solution of a hierarchical variational inequality (HVI).

Inertial self‐adaptive algorithm for solving split feasible problems with applications to image restoration

We introduce a new self‐adaptive algorithm for applications to image restoration problems. In order to study an image restoration, we consider the algorithm that contains inertial effects and step sizes, which is independent from the norm of the bounded linear operator. With some control conditions, the strong convergence to the minimum norm solution of the algorithm is obtained. Convergence analysis of the proposed algorithm is also discussed. Moreover, numerical results of image restoration problems illustrate that the proposed algorithm is efficient and outperforms other ones.

Least-Norm of the General Solution to Some System of Quaternion Matrix Equations and Its Determinantal Representations

We constitute some necessary and sufficient conditions for the system A1X1=C1, X1B1=C2, A2X2=C3, X2B2=C4, A3X1B3+A4X2B4=Cc, to have a solution over the quaternion skew field in this paper. A novel expression of general solution to this system is also established when it has a solution. The least norm of the solution to this system is also researched in this article. Some former consequences can be regarded as particular cases of this article. Finally, we give determinantal representations (analogs of Cramer’s rule) of the least norm solution to the system using row-column noncommutative determinants. An algorithm and numerical examples are given to elaborate our results.

Finite-Time Distributed Linear Equation Solver for Solutions With Minimum $l_1$-Norm

This paper proposes a continuous-time distributed algorithm for multiagent networks to achieve a solution with the minimum $l_1$ -norm to underdetermined linear equations. The proposed algorithm comes from a combination of the Filippov set-valued map with the projection-consensus flow. Given the underlying network is undirected and fixed, it is shown that the proposed algorithm drives all agents’ individual states to converge in finite time to a common value, which is the minimum $l_1$ -norm solution.

Condition numbers of the multidimensional total least squares problems having more than one solution

Recently, the condition numbers of the total least squares (TLS) problems having a unique solution have been studied at length in Zheng et al. (SIAM J. Matrix Anal. Appl. 38: 924–948, 2017). However, it is known that the TLS problem may have no solution, and even if an existing solution, it may not be unique. As a continuation of their work, in this paper, we investigate the condition numbers of the minimum Frobenius norm solution of the (multidimensional) TLS problem when having more than one solution. The tight and computable upper bound estimates of the normwise, mixed, and componentwise condition numbers are respectively derived. Some numerical experiments are performed to illustrate the tightness of these upper bounds.

Some regularization methods for a class of nonlinear fractional evolution equations

In this paper, we study initial value problems for nonlinear fractional elliptic equations. In general, the problem is not well-posed, and herein the Hadamard-instability occurs. Under some weak a priori assumptions on the sought solution, we propose two new regularization methods to stabilize the problem when the source term is a globally or locally Lipschitz function. Furthermore, we also investigate the error estimate and show that the approximate solution converges to the exact solution in L2 and H2 norms.