This paper is devoted to the numerical solution of the non-isothermal instationary Bingham flow with temperature dependent parameters by semismooth Newton methods. We discuss the main theoretical aspects regarding this problem. Mainly, we focus on existence of solutions and a multiplier formulation which leads us to a coupled system of PDEs involving a Navier-Stokes type equation and a parabolic energy PDE. Further, we propose a Huber regularization for this coupled system of partial differential equations, and we briefly discuss the well posedness of these regularized problems. A detailed finite element discretization, based on the so called (cross-grid $\mathbb{P}_1$) - $\mathbb{Q}_0$ elements, is proposed for the space variable, involving weighted stiffness and mass matrices. After discretization in space, a second order BDF method is used as a time advancing technique, leading, in each time iteration, to a nonsmooth system of equations, which is suitable to be solved by a semismooth Newton algorithm. Therefore, we propose and discuss the main properties of a SSN algorithm, including the convergence properties. The paper finishes with two computational experiment that exhibit the main properties of the numerical approach.
We propose a semismooth Newton method for non-Newtonian models of incompressible flow where the constitutive relation between the shear stress and the symmetric velocity gradient is given implicitly; this class of constitutive relations captures for instance the models of Bingham and Herschel-Bulkley. The proposed method avoids the use of variational inequalities and is based on a particularly simple regularisation for which the (weak) convergence of the approximate stresses is known to hold. The system is analysed at the function space level and results in mesh-independent behaviour of the no…
In the paper, a variant of the \ssstar Newton method is developed for the numerical solution of generalized equations, in which the multi-valued part is a so-called SCD (subspace containing derivative) mapping. Under a rather mild regularity requirement, the method exhibits (locally) superlinear convergence behavior. From the main conceptual algorithm, two implementable variants are derived whose efficiency is tested via a generalized equation modeling a discretized static contact problem with Coulomb friction.
On the application of the SCD semismooth* Newton method to solving Stokes problem with stick-slip boundary conditions
VSB-Technical University of Ostrava（俄斯特拉发理工大学）；Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences（奥地利科学院约翰·拉德纳计算与应用数学研究所）；Institute of Information Theory and Automation, Czech Academy of Sciences（捷克科学院信息论与自动化研究所）
Efficient TV regularization of large-scale linear inverse problems via the SCD semismooth* Newton method with applications in tomography
Johann Radon Institute for Computational and Applied Mathematics（约翰·拉登计算与应用数学研究所）；Institute of Information Theory and Automation, Czech Academy of Sciences（捷克科学院信息理论与自动化研究所）；Johannes Kepler University Linz（林茨约翰·开普勒大学）；University of Eastern Finland（芬兰东部大学）
Total generalization variation (TGV) is a very powerful and important regularization for various inverse problems and computer vision tasks. In this paper, we proposed a semismooth Newton based augmented Lagrangian method to solve this problem. The augmented Lagrangian method (also called as method of multipliers) is widely used for lots of smooth or nonsmooth variational problems. However, its efficiency usually heavily depends on solving the coupled and nonlinear system together and simultaneously, which is very complicated and highly coupled for total generalization variation. With efficien…
An equilibrium of a linear elastic body subject to loading and satisfying the friction and contact conditions can be described by a variational inequality of the second kind and the respective discrete model attains the form of a generalized equation. To its numerical solution we apply the semismooth* Newton method by Gfrerer and Outrata (2019) in which, in contrast to most available Newton-type methods for inclusions, one approximates not only the single-valued but also the multi-valued part. This is performed on the basis of limiting (Morduchovich) coderivative. In our case of the Tresca fri…
A distributed semismooth Newton based augmented Lagrangian method for distributed optimization
School of Mathematics, Southwest Jiaotong University（西南交通大学数学学院）；School of Computer and Computing Science, Hangzhou City University（杭州城市大学计算机与计算科学学院）；Institute of Mathematics, Zhejiang Wanli University（浙江万里大学数学学院）
Zhejiang University City College（浙大城市学院）；Southwest Jiaotong University（西南交通大学）；The Hong Kong Polytechnic University（香港理工大学）；National University of Singapore（新加坡国立大学）
In this paper, we consider high-dimensional nonconvex square-root-loss regression problems and introduce a proximal majorization-minimization (PMM) algorithm for these problems. Our key idea for making the proposed PMM to be efficient is to develop a sparse semismooth Newton method to solve the corresponding subproblems. By using the Kurdyka-Łojasiewicz property exhibited in the underlining problems, we prove that the PMM algorithm converges to a d-stationary point. We also analyze the oracle property of the initial subproblem used in our algorithm. Extensive numerical experiments are presente…
Sichuan University（四川大学）；Southwest Jiaotong University（西南交通大学）；Zhejiang University City College（浙江大学城市学院）
Support vector machines (SVMs) are successful modeling and prediction tools with a variety of applications. Previous work has demonstrated the superiority of the SVMs in dealing with the high dimensional, low sample size problems. However, the numerical difficulties of the SVMs will become severe with the increase of the sample size. Although there exist many solvers for the SVMs, only few of them are designed by exploiting the special structures of the SVMs. In this paper, we propose a highly efficient sparse semismooth Newton based augmented Lagrangian method for solving a large-scale convex…
Southwest Jiaotong University（西南交通大学）；Zhejiang University City College（浙江大学城市学院）
Square-root Lasso problems are proven robust regression problems. Furthermore, square-root regression problems with structured sparsity also plays an important role in statistics and machine learning. In this paper, we focus on the numerical computation of large-scale linearly constrained sparse group square-root Lasso problems. In order to overcome the difficulty that there are two nonsmooth terms in the objective function, we propose a dual semismooth Newton (SSN) based augmented Lagrangian method (ALM) for it. That is, we apply the ALM to the dual problem with the subproblem solved by the S…
A TFETI Domain Decomposition Solver for Elastoplastic Problems
VŠB-Technical University of Ostrava（俄斯特拉发理工大学）；Institute of Geonics AS CR, v.v.i.（捷克科学院地质力学研究所）；The Institute of Information Theory and Automation（信息与自动化研究所）
提出基于 TFETI 区域分解方法的并行算法，结合半光滑牛顿法和隐式欧拉时间离散，高效求解具有线性各向同性硬化 von Mises 准则的弹塑性问题。
We introduce a domain decomposition-based nonlinear preconditioned iteration for solving nonlinear, nonsmooth elliptic optimal control problems, with a nonlinear reaction term, $L^1$ regularization and box constraints on the control function. The method is obtained by applying semismooth Newton to the fixed-point equation of the parallel optimized Schwarz iteration. As a proof of concept, numerical experiments are performed on two subdomains, as well as on a multi-subdomain test case. The results show that it is possible to obtain substantial improvements in robustness and efficiency with the …
In this paper, we propose a dual-mixed formulation for stationary viscoplastic flows with yield, such as the Bingham or the Herschel-Bulkley flow. The approach is based on a Huber regularization of the viscosity term and a two-fold saddle point nonlinear operator equation for the resulting weak formulation. We provide the uniqueness of solutions for the continuous formulation and propose a discrete scheme based on Arnold-Falk-Winther finite elements. The discretization scheme yields a system of slantly differentiable nonlinear equations, for which a semismooth Newton algorithm is proposed and …
Preconditioning PDE-constrained optimization with $\rm L^1$-sparsity and control constraints
Università degli Studi di Firenze（佛罗伦萨大学）；Università di Bologna（博洛尼亚大学）；Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）
This paper develops the proximal method of multipliers for a class of nonsmooth convex optimization. The method generates a sequence of minimization problems (subproblems). We show that the sequence of approximations to the solutions of the subproblems converges to a saddle point of the Lagrangian even if the original optimization problem may possess multiple solutions. The augmented Lagrangian due to Fortin appears in the subproblem. The remarkable property of the augmented Lagrangian over the standard Lagrangian is that it is always differentiable, and it is often semismoothly differentiable…
A Multigrid Optimization Algorithm for the Numerical Solution of Quasilinear Variational Inequalities Involving the $p$-Laplacian
Escuela Politécnica Nacional（厄瓜多尔国立理工学院）；Research Center on Mathematical Modeling (MODEMAT)（数学建模研究中心）；Departamento de Matemática - Escuela Politécnica Nacional（厄瓜多尔国立理工学院数学系）
In this work, a class of parabolic economic optimal control problems is considered. These problems are characterized by pointwise state constraints regularized by a parameter, which transforms the pure state constraints in mixed control-state ones. However, the convergence of classical (semismooth) Newton methods deteriorates for decreasing values of the regularization parameter. To tackle this problem, a nonlinear preconditioner is introduced. This is based on an overlapping optimized waveform-relaxation method characterized by Robin transmission conditions. Numerical experiments show that ap…
We propose a least squares Galerkin based gradient recovery to approximate Dirichlet problems for strong solutions of linear elliptic problems in nondivergence form and corresponding apriori and aposteriori error bounds. This approach is used to tackle fully nonlinear elliptic problems, e.g., Monge-Ampère, Hamilton-Jacobi-Bellman, using the smooth (vanilla) and the semismooth Newton linearization. We discuss numerical results, including adaptive methods based on the aposteriori error indicators.
Zhejiang University City College（浙大城市学院）；Southwest Jiaotong University（西南交通大学）；Zhejiang University（浙江大学）
In this paper, we introduce a proximal-proximal majorization-minimization (PPMM) algorithm for nonconvex tuning-free robust regression problems. The basic idea is to apply the proximal majorization-minimization algorithm to solve the nonconvex problem with the inner subproblems solved by a sparse semismooth Newton (SSN) method based proximal point algorithm (PPA). We must emphasize that the main difficulty in the design of the algorithm lies in how to overcome the singular difficulty of the inner subproblem. Furthermore, we also prove that the PPMM algorithm converges to a d-stationary point. …
