International Workshop on Recent Advances in Variational/Hemivariational Inequalities and Applications

Transcription

1 International Workshop on Recent Advances in Variational/Hemivariational Inequalities and Applications June 24-26, 2017! Xi an Jiaotong University, Xi an, China

2 SCHOOL&OF&MATHEMATICS&AND&STATISTICS! International Workshop on Recent Advances in Variational/Hemivariational Inequalities and Applications! Xi an Jiaotong University, Xi an, China June 24-26, 2017 Variational and hemivariational inequalities form an important and very useful family of nonlinear problems arising in diverse application areas of physical, engineering, financial, and management sciences. The goal of this workshop is to bring mathematicians, engineers, and financiers together to exchange recent advances in the field of variational and hemivariational inequalities on theory, numerical analysis, optimization techniques, and applications, to discuss the frontiers of related subjects, and to promote collaborations among participants. The main topics of the workshop include, but are not limited to, analysis of problems leading to variational and hemivariational inequalities, properties of solutions of inequalities, numerical analysis of inequalities, optimization techniques, and applications. Organizing Committee: Honorary Chairs: Zongben Xu (Xi'an Jiaotong University) Co-Chairs: Weimin Han (Xi an Jiaotong University, University of Iowa) Fei Wang (Xi an Jiaotong University) Members: Xiaoliang Cheng (Zhejiang University) Stanislaw Migorski (Jagiellonian University) Jigen Peng (Xi an Jiaotong University) Mircea Sofonea (University of Perpignan) Wenjing Yan (Xi an Jiaotong University) Xu Zhu (Xi an Jiaotong University) Yixin Liu (Workshop Secretary, Xi an Jiaotong University) Conference Webpage:

3 International Workshop on Recent Advances in Variational/Hemivariational Inequalities and Applications June 24-26, 2017, Xi an China Workshop Program Friday, June 23, :00-20:00 Registration (Nanyang Hotel) Saturday, June 24, :40-8:15 Registration (International Conference Hall) International Conference Hall Opening Ceremony 8:15-8:30 Opening Ceremony International Conference Hall Chair: Fei Wang 8:30-9:10 Stanislaw Migorski: Evolution history-dependent variational-hemivariational inequalities with their applications 9:10-9:50 Zhenhai Liu: Nonlocal differential variational Inequalities 9:50-10:20 Group Photo, refreshment International Conference Hall Chair: Jianguo Huang 10:20-11:00 Bingsheng He: From projection and contraction methods for variational inequality to splitting methods for convex optimization 11:00-11:40 Dongyang Shi: Nonconforming FEMs for variational inequalities 11:40-12:20 Nanjing Huang: Existence and stability of solutions for inverse variational inequality problems 12:20-2:20 Nanyang Hotel Lunch International Conference Hall Chair: Deren Han

4 2:20-3:00 Xiaobing Feng: Mathematical and numerical analysis of the total variation flow based on a variational inequality approach 3:00-3:40 Xiaoliang Cheng: Numerical analysis of some inverse elasticity problems 3:40-4:20 Krzysztof Bartosz: Convergence of Rothe scheme for second order varitaional-hemivariational inequality 4:20-4:40 Refreshment International Conference Hall Chair: Xiaoming Yuan 4:40-5:10 Wei Gong: Multilevel method for control constrained elliptic optimal control problems 5:10-5:40 Jiangfeng Han: Fractional calculus with applications to viscoelastic contact mechanics 5:40-6:10 Wenqiang Xiao: Discontinuous Galerkin methods for solving a frictional contact problem with normal compliance 6:10-8:00 Nanyang Hotel Dinner Sunday, June 25, 2017 Multi-functional Hall Chair: Xiaoliang Cheng 8:00-8:40 Mircea Sofonea: Optimal Control of a Class of Variational-Hemivariational Inequalities in Reflexive Banach Spaces 8:40-9:20 Deren Han: Asymmetric proximal point algorithms with moving proximal centers 9:20-10:00 Anna Ochal: Elliptic variational-hemivariational inequalities in mathematical modeling 10:00-10:20 Refreshment Multi-functional Hall Chair: Yuhong Dai 10:20-11:00 Dan Tiba: A direct approach to variational inequalities and free boundary problems 11:00-11:40 Xiaoming Yuan: How to implement ADMM to large-scale datasets 11:40-12:20 Yanfang Zhang: Smoothing methods for the hemivariational inequalities

5 12:20-2:20 Nanyang Hotel Lunch Multi-functional Hall Chair: Dongyang Shi 2:20-3:00 Jianguo Huang: A continuous DG time stepping method and its adaptive algorithms for second order evolution problems 3:00-3:40 Yinnian He: Super-convergence of center finite difference method 3:40-4:20 Yanren Hou: On the weak solution to steady Stokes/Darcy model with BJ interface condition 4:20-4:40 Refreshment Multi-functional Hall Chair: Wei Gong 4:40-5:10 Rongfang Gong: A modified coupled complex boundary method for an inverse chromatography problem 5:10-5:40 Daming Yuan: High order positivity-preserving DG methods for transport equation 5:40-6:10 Feifei Jing: Discontinuous Galerkin methods for a stationary Navier-Stokes problem with a nonlinear slip boundary condition of friction type 6:10-8:00 Xi an Restaurant Dinner 9:00-12:00 Free Discussion Monday, June 26, :00-2:30 Nanyang Hotel Lunch 2:30-6:30 Free Discussion 6:30-8:00 Nanyang Hotel Dinner

6 Abstracts of the presentations Evolution History-Dependent Variational-Hemivariational Inequalities with their Applications Stanislaw Migorski Jagiellonian University in Krakow, Poland We provide new results on existence and uniqueness for a first order variational-hemivariational inequality with history-dependent operators. The history-dependent operators appear in both a locally Lipschitz nonconvex superpotential and in a convex potential. The results are applied to a dynamic frictional viscoelastic contact problem with multivalued nonmonotone subdifferential boundary conditions. This is a joint contribution with Weimin Han and Mircea Sofonea. [1] W. Han, S. Migorski, M. Sofonea, Eds., Advances in Variational and Hemivariational Inequalities, Advances in Mechanics and Mathematics 33, Springer, New York, [2] S. Migorski, A. Ochal, M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics 26, Springer, New York, [3] M. Sofonea, S. Migorski, Variational-Hemivariational Inequalities with Applications, Pure and Applied Mathematics, Chapman & Hall/CRC Press, New York, 2017, in press. Nonlocal Differential Variational Inequalities Zhenhai Liu Guangxi University for Nationalities In this talk, we introduce a nonlocal differential system obtained by mixing a nonlocal partial differential equation and a variational inequalities. This kind of problems may be regarded as a special feedback control problem. Firstly, we talk about the research motivation and some examples. Then,based on the theory of nonlocal Laplacian operators, Filippov implicit function Lemma and fixed point theory for set-valued mappings, we show an existence results of solutions to the mentioned problem. Finally, we point out some interesting problems for further research.

7 From Projection and Contraction Methods for Variational Inequality to Splitting Methods for Convex Optimization Bingsheng He Southern University of Science and Technology, China Many problems arising from image processing, machine learning and other applied computation area can be formulated to a linearly constraint convex optimization (CP). For solving these problems, Alternating Directions Method of Multipliers (ADMM) is recognized as a powerful approach. ADMM belongs to the class of splitting contraction methods (SCM). In practice, many existing SCMs are close related to the projection and contraction (PC) methods for monotone variational inequalities. In the past 10 years, PC methods are successfully applied in the areas of robot control and geomechanics. This talk will give a further overview for developing effective ADMM-like SCMs for convex optimization guided by the PC methods for monotone variational inequalities. Nonconforming FEMs for Variational Inequalities Dongyang Shi Zhengzhou University In this talk, some new developments on the convergence and super-convergence analysis with nonconforming FEMs for the variational inequalities are presented, which include the Signorini problem, second order variational inequality problem with displacement obstacle, fourth order variational inequality with two-sided displacement obstacle and the contact problem of plate. At the same time, some numerical results are provided to confirm the theoretical analysis. Existence and Stability of Solutions for Inverse Variational Inequality Problems Nan-jing Huang with co-authors Yu Han, Jue Lu and Yi-bin Xiao Sichuan University In this talk, we will present two new existence theorems of solutions for inverse variational and quasi-variational inequality problems by using the Fan-KKM theorem and the Kakutani-Fan- Glicksberg fixed point theorem, respectively. Moreover, we will show the upper semicontinuity and the lower semicontinuity of solution mapping and approximate solution mapping to parametric inverse variational inequality problem are also discussed under some suitable conditions. Finally, we will give an application to the road pricing problem.

8 Mathematical and Numerical Analysis of the Total Variation Flow Based on a Variational Inequality Approach Xiaobing Feng The University of Tennessee, U.S.A. The total variation (TV) flow is a gradient flow for the TV energy and naturally arises from the gradient descent method for minimizing the TV energy. In this talk, I shall first present a variational inequality formulation for the TV flow and its mathematical analysis including the well-posedness and interior regularity. I shall then present the finite element numerical analysis of the variational inequality problem and show some numerical experiments on test problems from image processing. If time permits, I shall also discuss a related stochastic TV flow and its numerical approximations. The materials of this talk are based on a joint of work with Andreas Prohl of University of Tubingen, Germany. Numerical Analysis of Some Inverse Elasticity Problems Xiaoliang Cheng Zhejiang University In this talk, we will discuss some numerical analysis for the inverse elasticity problems. We try to find a stable surface traction from the given displacement field and its estimation. We use the variational inequalities, Tikhonov regularization and finite element method to solve it. We propose some algorithm and derive some error estimates, and present some numerical results. Convergence of Rothe Scheme for Second Order Varitaional-Hemivariational Inequality Krzysztof Bartosz Jagiellonian University in Krakow, Poland We deal with a second order evolution variational inequality involving a multivalued term generated by a Clarke subdifferential of a locally Lipschitz potential. For this problem we construct a time-semidiscrete approximation, known as the Rothe scheme. We study a sequence of solutions of the semidiscrete approximate problems and provide its weak convergence to a limit

9 element that is a solution of the original problem. Next, we show that the solution is unique and the convergence is strong. Finally, we consider a dynamic visco-elastic problem of contact mechanics. We assume that the contact process is governed by a normal damped response condition with a unilateral constraint and the body is non-clamped. The mechanical problem in its weak formulation reduces to a variational-hemivariational inequality that can be solved by finding a solution of a corresponding abstract problem related to one studied in the first part of the presentation. Hence, we apply obtained existence result to provide the weak solvability of contact problem. Multilevel Method for Control Constrained Elliptic Optimal Control Problems Wei Gong, Hehu Xie, Ningning Yan Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences In this talk we introduce a multilevel method to solve control constrained elliptic optimal control problems with the finite element method. In this scheme, solving an optimization problem on the finest finite element space is transformed into a series of solutions of linear boundary value problems by the multigrid method on multilevel meshes and a series of solutions of optimization problems on the coarsest finite element space. Our proposed scheme, instead of solving a large scale optimization problem in the finest finite element space, solves only a series of linear boundary value problems and the optimization problems in a very low dimensional finite element space, and thus can improve the overall efficiency of the solution of optimal control problems governed by PDEs. An adaptive version of the algorithm is also investigated which shows optimal convergence rate. Fractional Calculus with Applications to Viscoelastic Contact Mechanics Jiangfeng Han Guangxi University of Finance and Economics In the presentation, we consider a quasistatic frictionless contact problem for a viscoelastic body in which the constitutive equation is modeled with the fractional Kelvin-Voigt law and the contact condition is described by the Clarke subdifferential of a nonconvex and nonsmooth functional. The variational formulation of this problem is provided in the form of a fractional hemivariational inequality. In order to solve this inequality, we apply the Rothe method and prove that the associated abstract Volterra inclusion has at least one solution.

10 Discontinuous Galerkin Methods for Solving a Frictional Contact Problem with Normal Compliance Wenqiang Xiao, Fei Wang, Weimin Han Xi an Jiaotong University We study several discontinuous Galerkin (DG) methods for solving a frictional contact problem with normal compliance, which can be modeled by a quasi-variational inequality. A unified numerical analysis of these DG methods is established, and they achieve optimal convergence order with linear elements. Some numerical examples are given and the numerical convergence orders match well with the theoretical prediction. Optimal Control of a Class of Variational-Hemivariational Inequalities in Reflexive Banach Spaces Mircea Sofonea University of Perpignan Via Domitia, France This work represents a continuation of [1], [2] and deals with a class of elliptic variationalhemivariational inequalities in reflexive Banach spaces. An inequality in the class is governed by a nonlinear operator, a convex set of constraints and two nondifferentiable functionals, among which at least one is convex. The existence of unique solution for such inequality was proved in [1]. Its numerical analysis, including convergence results and error estimates was carried out in [2]. In the current work we complete this study with new results, including a convergence result of the solution with respect to the set of constraints. Then we formulate two optimal control problems for which we prove the existence of optimal pairs, together with some convergence results. Finally, we exemplify our results in the study of a mathematical model which describes the equilibrium of an elastic rod in unilateral contact with a rigid-elastic foundation, under the action of a given body force. [1] S. Migorski, A. Ochal and M. Sofonea, A Class of Variational-Hemivariational Inequalities in Reflexive Banach Spaces, Journal of Elasticity 127 (2017), [2] W. Han, M. Sofonea and D. Danan, Numerical Analysis of Stationary Variational- Hemivariational Inequalities, Numerische Mathematik (2017), to appear. Asymmetric Proximal Point Algorithms with Moving Proximal Centers Deren Han, Xiaoming Yuan

11 School of Mathematical Sciences, Nanjing Normal University We discuss the classical proximal point algorithm (PPA) with a metric proximal parameter in the variational inequality context. The metric proximal parameter is usually required to be positive definite and symmetric in the PPA literature, because it plays the role of the measurement matrix of a norm in the convergence proof. Our main goal is to show that the metric proximal parameter can be asymmetric if the proximal center is shifted appropriately. The resulting asymmetric PPA with moving proximal centers maintains the same implementation difficulty and convergence properties as the original PPA; while the asymmetry of the metric proximal parameter allows us to design highly customized algorithms that can effectively take advantage of the structures of the model under consideration. In particular, some efficient structure-exploiting splitting algorithms can be easily developed for some special cases of the variational inequality. We illustrate these algorithmic benefits by a saddle point problem and a convex minimization model with a generic separable objective function, both of which have wide applications in various fields. We present both the exact and inexact versions of the asymmetric PPA with moving proximal centers; and analyze their convergence including the estimate of their worst-case convergence rates measured by the iteration complexity under mild assumptions and their asymptotically linear convergence rates under stronger assumptions. Elliptic Variational-Hemivariational Inequalities in Mathematical Modeling Anna Ochal Jagiellonian University in Krakow, Poland We study a class of elliptic variational-hemivariational inequalities in reflexive Banach spaces. An inequality in the class involves a nonlinear operator, a convex set of constraints and two nondifferentiable functionals, among which at least one is convex. The motivation to study the problem comes from Contact Mechanics and the fact that the inequality contains, as particular cases, various problems considered in the literature. We deliver a result on existence and uniqueness of a solution to the inequality. The proof is based on arguments of surjectivity for pseudomonotone operators and the Banach fixed point theorem. Next, we introduce a class of penalized problems, prove their unique solvability and the convergence of the corresponding solutions to the solution of the original problem, as the penalization coefficient converges to zero. We consider a mathematical model which describes the equilibrium of an elastic body in unilateral contact with a foundation. The model leads to a variational-hemivariational inequality for the displacement field, that we analyse by using our abstract results. This is a joint contribution with Stanislaw Migorski and Mircea Sofonea.

12 A Direct Approach to Variational Inequalities and Free Boundary Problems Dan Tiba Institute of Mathematics, Romanian Academy, Bucharest We discuss a fixed domain approximation method for elliptic and parabolic variational inequalities, fourth order variational inequalities or solid-fluid interaction problems. The approach uses just linear equations in each iteration and its numerical behavior is very efficient. We also discuss theoretical properties of the method. It is to be underlined that one basic idea in this context is to use techniques from shape optimization theory due to the presence of geometric unknowns (free boundaries, coincidence sets) and it has certain implementation advantages. The presentation is based on very recent papers of the author and his collaborators. How to Implement ADMM to Large-scale Datasets Yuan Xiaoming Hong Kong Baptist University The alternating direction method of multipliers (ADMM) is being popularly used for a wide range of applications including many in data science and engineering. To tackle very large-scale datasets of some representative statistical learning problems such as the LASSO and distributed LASSO, it is neither possible nor necessary to solve the ADMM subproblems exactly or up to a high precision --- inexact solutions in low-accuracy are cheaper while indeed better towards the convergence. We try to study this implementation issue mathematically. We are particularly interested in the case where the subproblems are very large-scale systems of linear equations and they are solved iteratively by benchmark numerical linear algebra solvers such as CG or SOR. We show how to rigorously ensure the convergence for this case. More specifically, we estimate precisely how many iterations of these numerical linear algebra solvers are needed to ensure the convergence. We thus make the implementation of ADMM embedded with benchmark numerical linear algebra solvers for large-scale cases of some statistical learning and engineering problems fully automatic, with proved convergence. Smoothing Methods for the Hemivariational Inequalities Yanfang Zhang Minzu University of China

13 In this paper, we employ the smoothing methods to solve the hemivariational inequalities. The hemivariational inequalities arise from the nonsmooth mechanics of solid, especially from the nonmonotone contact problems. The smoothing quadratic regularization method is considered, and it is not expensive to calculate at each iteration since we can get the closed form of the problems. The worst-case complexity is given for the method. The numerical experiments are realized to show the effectiveness of the smoothing method. A Continuous DG Time Stepping Method and Its Adaptive Algorithms for Second Order Evolution Problems Jianguo Huang Shanghai Jiao Tong University Second order dynamical systems frequently occur in the areas of civil engineering, material science, structural analysis and electrical engineering. The finite difference method is a typical approach for discretization in time. However, such treatment is not convenient to produce adaptive time stepping methods. In this talk, I will first introduce the continuous P 2 discontinuous Galerkin (DG) method for solving a second order evolution problem. Then, I will propose an adaptive time stepping method based on a posteriori error estimator for the previous method. To further increase the computational efficiency, I am going to devise a hybrid time stepping method, which is the combination of an adaptive finite element method and the spectral Picard method. This method has the advantage of small amount of computational cost and high accuracy of numerical solution, and is particularly suitable for capturing the solution with rapid change in time. Several numerical experiments are provided to show the computational performance and efficiency of the proposed methods. This is a joint work with Junjiang Lai, Huashan Sheng and Tao Tang. Super-convergence of Center Finite Difference Method Xinlong Feng, Yinnian He Xinjiang University and Xi'an Jiaotong University In this paper, the center finite difference (CFD) method for the elliptic equation is deduced by the P1-element and the first-order discrete partial differential equation over the dual element K* in the 1D or 2D domain. Next, the coefficient matrix of the CFD method is explicitly reduced and the H 1 -stability and convergence of the CFD solution u h is provided. Furthermore, the H 1 -superconvergence of uh to I h u is obtained under the case of the almost-uniform mesh. Based on the H1- super-convergence of u h to I h u, the optimal L 2 -error estimate of the numerical solution u h and the H 1 -super-convergence error estimate of the interpolation solution are derived respectively. Finally, some numerical tests are made to show the analytical results of the CFD method.

14 On the Weak Solution to Steady Stokes/Darcy Model with BJ Interface Condition Yanren Hou Xi an Jiaotong University In this talk, we consider the well-posedness of the steady-state coupled Stokes/Darcy model with Beavers-Joseph interface condition. By constructing an expanded system, we show that this coupled problem is well-posed for any physical parameters, while some smallness requirement of an experimental determined physical parameter is needed for obtaining the well-posedness of this coupled system in former literatures. A Modified Coupled Complex Boundary Method for an Inverse Chromatography Problem Rongfang Gong Nanjing University of Aeronautics and Astronautics Adsorption isotherms are the most important parameters in rigorous models of chromatographic processes. In this talk, in order to recover adsorption isotherms, we consider a coupled complex boundary method (CCBM), which was previously proposed for solving an inverse source problem, [Cheng, et al. A novel coupled complex boundary method for inverse source problems, Inverse Problems, 2014, 30, ]. With CCBM, the original boundary fitting problem is transferred to a domain fitting problem. Thus, this method has advantages regarding robustness and computation in reconstruction. In contrast to the traditional CCBM, for the sake of the reduction of computational complexity and computational cost, the recovered adsorption isotherm only corresponds to the real part of the solution of a forward complex initial boundary value problem. Furthermore, we take into account the position of the profiles and apply the momentum criterion to improve the optimization progress. Using Tikhonov regularization, the well-posedness, convergence properties and regularization parameter selection methods are studied. Based on an adjoint technique, we derive the exact Jacobian of the objective function and give an algorithm to reconstruct the adsorption isotherm. Finally, numerical simulations are given to show the feasibility and efficiency of the proposed regularization method. This is a joint work with Xiaoliang Cheng and Guangliang Lin of Zhejiang University, China, Marten.Gulliksson and Ye Zhang of Orebro University, Sweden.

15 High Order Positivity-preserving DG Methods for Transport Equation Daming Yuan 1, Juan Cheng 2, Chi-Wang Shu 3 1.College of Mathematics and Information Science, Jiangxi Normal University 2.Institute of Applied Physics and Computational Mathematics, Beijing, China 3.Division of Applied Mathematics, Brown University, USA The positivity-preserving property is an important and challenging issue for the numerical solution of radiative transfer equations. In the past few decades, di- fferent numerical techniques have been proposed to guarantee positivity of the radiative intensity in several schemes; however, it is difficult to maintain both high order accuracy and positivity. The discontinuous Galerkin (DG) finite element method is a high order numerical method which is widely used to solve the neutron/photon transfer equations, due to its distinguished advantages such as high order accuracy, geometric flexibility, suitability for h- and p-adaptivity, parallel efficiency, and a good theoretical foundation for stability and error estimates. In this paper, we construct arbitrarily high order accurate DG schemes which preserve positivity of the radiative intensity in the simulation of both steady and unsteady radiative transfer equations in one- and two-dimensional geometry by using a combined technique of the scaling positivity-preserving limiter in [X. Zhang and C.-W. Shu, J. Comput. Phys., 229 (2010), pp ] and a new rotational positivity-preserving limiter. This combined limiter is simple to implement and we prove the properties of positivity-preserving and high order accuracy rigorously. One- and two-dimensional numerical results are provided to verify the good properties of the positivity-preserving DG schemes. Discontinuous Galerkin methods for the incompressible flow with a nonlinear slip boundary condition of friction type Feifei Jing, Weimin Han, Wenjing Yan, Fei Wang Xi an Jiaotong University In this work, interior penalty discontinuous Galerkin methods are introduced and analyzed to solve a variational inequality from the stationary Stokes and Navier-Stokes equations with a nonlinear slip boundary condition of friction type. Stability, existence and uniqueness of the numerical solutions are shown for the discontinuous Galerkin methods. Error estimates are derived for the velocity in a broken H 1 -norm and for the pressure in a L 2 -norm, with the optimal convergence order when linear elements for the velocity and piecewise constants for the pressure are used. Numerical results are reported to demonstrate the theoretically predicted convergence orders, and the results also show the capability of the discontinuous Galerkin methods to capture the discontinuity.