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This paper presents a comprehensive study on Herschel–Bulkley flow, where the flow parameters are dependent on the density. The Herschel–Bulkley model is a generalized power-law model used to simulate viscoplastic fluids defined by a plasticity threshold. We consider the case where the plasticity threshold and the viscosity depend on the shear rate and fluid density. To analyze this model, we use a Huber regularization of the stress and propose an H(div)-conforming and discontinuous Galerkin (DG) numerical approximation for the coupled equations governing the flow. We discuss the stability and existence of discrete solutions and propose a semismooth Newton linearization for the numerical solution of the discretized system. Our numerical scheme is validated through several experiments that explore the behavior of Herschel–Bulkley flow under different conditions. The results demonstrate the robustness of our numerical method.

This paper is devoted to the study of Bingham flow with variable density. We propose a local bi-viscosity regularization of the stress tensor based on a Huber smoothing step. Next, our computational approach is based on a second-order, divergence-conforming discretization of the Huber regularized Bingham constitutive equations, coupled with a discontinuous Galerkin scheme for the mass density. We take advantage of the properties of divergence-conforming and discontinuous Galerkin formulations to effectively incorporate upwind discretizations, thereby ensuring the stability of the formulation. The stability of the continuous problem and the fully discrete scheme are analyzed. Further, a semismooth Newton method is proposed for solving the obtained fully discretized system of equations at each time step. Finally, several numerical examples that illustrate the main features of the problem and the properties of the numerical scheme are presented. [ABSTRACT FROM AUTHOR]

In this paper, we propose to solve Pareto eigenvalue complementarity problems by using interior-point methods. Precisely, we focus the study on an adaptation of the Mehrotra Predictor Corrector Method (MPCM) and a Non-Parametric Interior Point Method (NPIPM). We compare these two methods with two alternative methods, namely the Lattice Projection Method (LPM) and the Soft Max Method (SM). On a set of data generated from the Matrix Market, the performance profiles highlight the efficiency of MPCM and NPIPM for solving eigenvalue complementarity problems. We also consider an application to a concrete and large size situation corresponding to a geomechanical fracture problem. Finally, we discuss the extension of MPCM and NPIPM methods to solve quadratic pencil eigenvalue problems under conic constraints. [ABSTRACT FROM AUTHOR]

This paper is concerned with an approximation scheme for rate-independent systems governed by a non-smooth dissipation and a possibly non-convex energy functional. The scheme is based on the local minimization scheme introduced in Efendiev and Mielke [J. Convex Anal. 13 (2006) 151–167], but relies on local stationarity of the underlying minimization problem. Under the assumption of Mosco-convergence for the dissipation functional, we show that accumulation points exist and are so-called parametrized BV-solutions of the rate-independent system. In particular, this guarantees the existence of parametrized BV-solutions for a rather general setting. Afterwards, we apply the scheme to a model for the evolution of damage. [ABSTRACT FROM AUTHOR]

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 implemented. Local superlinear convergence of the method is also proved. Finally, we perform several numerical experiments in two and three dimensions to investigate the behavior and efficiency of the method. [ABSTRACT FROM AUTHOR]

We propose a semismooth Newton-type method for nonsmooth optimal control problems. Its particular feature is the combination of a quasi-Newton method with a semismooth Newton method. This reduces the computational costs in comparison to semismooth Newton methods while maintaining local superlinear convergence. The method applies to Hilbert space problems whose objective is the sum of a smooth function, a regularization term, and a nonsmooth convex function. In the theoretical part of this work we establish the local superlinear convergence of the method in an infinite-dimensional setting and discuss its application to sparse optimal control of the heat equation subject to box constraints. We verify that the assumptions for local superlinear convergence are satisfied in this application and we prove that convergence can take place in stronger norms than that of the Hilbert space if initial error and problem data permit. In the numerical part we provide a thorough study of the hybrid approach on two optimal control problems, including an engineering problem from magnetic resonance imaging that involves bilinear control of the Bloch equations. We use this problem to demonstrate that the new method is capable of solving nonconvex, nonsmooth large-scale real-world problems. Among others, the study addresses mesh independence, globalization techniques, and limited-memory methods. We observe throughout that algorithms based on the hybrid methodology are several times faster in runtime than their semismooth Newton counterparts. [ABSTRACT FROM AUTHOR]

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 nonlinear iterations. [ABSTRACT FROM AUTHOR]

Currently, nearly all positivity preserving discontinuous Galerkin (DG) discretizations of partial differential equations are coupled with explicit time integration methods. Unfortunately, for many problems this can result in severe time-step restrictions. The techniques used to develop explicit positivity preserving DG discretizations cannot, however, easily be combined with implicit time integration methods. In this paper, we therefore present a new approach. Using Lagrange multipliers, the conditions imposed by the positivity preserving limiters are directly coupled to a DG discretization combined with a diagonally implicit Runge--Kutta time integration method. The positivity preserving DG discretization is then reformulated as a Karush--Kuhn--Tucker (KKT) problem, which is frequently encountered in constrained optimization. Since the limiter is only active in areas where positivity must be enforced, it does not affect the higher order DG discretization elsewhere. The resulting nonsmooth nonlinear algebraic equations have, however, a different structure compared to most constrained optimization problems. We therefore develop an efficient active set semismooth Newton method that is suitable for the KKT formulation of time-implicit positivity preserving DG discretizations. Convergence of this semismooth Newton method is proven using a specially designed quasi-directional derivative of the time-implicit positivity preserving DG discretization. The time-implicit positivity preserving DG discretization is demonstrated for several nonlinear scalar conservation laws, which include the advection, Burgers, Allen--Cahn, Barenblatt, and Buckley--Leverett equations. [ABSTRACT FROM AUTHOR]

Abstract The theoretical part of the paper analyzes discretized Stokes systems with local Coulomb's slip boundary conditions. Solutions to discrete models are defined by means of fixed-points of an appropriate mapping. We prove the existence of a fixed-point, establish conditions guaranteeing its uniqueness and examine how they depend on the discretization parameter h and the slip coefficient κ. The second part of the paper is devoted to computational aspects. Numerical experiments are presented. [ABSTRACT FROM AUTHOR]

This work is concerned with optimal control problems where the objective functional consists of a tracking-type functional and an additional “multibang” regularization functional that promotes optimal control taking values from a given discrete set pointwise almost everywhere. Under a regularity condition on the set where these discrete values are attained, error estimates for the Moreau-Yosida approximation (which allows its solution by a semismooth Newton method) and the discretization of the problem are derived. Numerical results support the theoretical findings. [ABSTRACT FROM AUTHOR]

In applying the level-set method developed in [E. Van den Berg and M. P. Friedlander, SIAM J. Sci. Comput., 31 (2008), pp. 890{912] and [E. Van den Berg and M. P. Friedlander, SIAM J. Optim., 21 (2011), pp. 1201{1229] to solve the fused lasso problems, one needs to solve a sequence of regularized least squares subproblems. In order to make the level-set method practical, we develop a highly efficient inexact semismooth Newton based augmented Lagrangian method for solving these subproblems. The efficiency of our approach is based on several ingredients that constitute the main contributions of this paper. First, an explicit formula for constructing the generalized Jacobian of the proximal mapping of the fused lasso regularizer is derived. Second, the special structure of the generalized Jacobian is carefully extracted and analyzed for the efficient implementation of the semismooth Newton method. Finally, numerical results, including the comparison between our approach and several state-of-the-art solvers, on real data sets are presented to demonstrate the high efficiency and robustness of our proposed algorithm in solving challenging large-scale fused lasso problems. [ABSTRACT FROM AUTHOR]

We discuss several optimization procedures to solve finite element approximations of linear-quadratic Dirichlet optimal control problems governed by an elliptic partial differential equation posed on a 2D or 3D Lipschitz domain. The control is discretized explicitly using continuous piecewise linear approximations. Unconstrained, control-constrained, state-constrained and control-and-state constrained problems are analysed. A preconditioned conjugate method for a reduced problem in the control variable is proposed to solve the unconstrained problem, whereas semismooth Newton methods are discussed for the solution of constrained problems. State constraints are treated via a Moreau-Yosida penalization. Convergence is studied for both the continuous problems and the finite dimensional approximations. In the finite dimensional case, we are able to show convergence of the optimization procedures even in the absence of Tikhonov regularization parameter. Computational aspects are also treated and several numerical examples are included to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

It is well known that optimal control problems with L1-control costs produce sparse solutions, i.e., the optimal control is zero on whole intervals. In this paper, we study a general class of convex linear-quadratic optimal control problems with a sparsity functional that promotes a so-called group sparsity structure of the optimal controls. In this case, the components of the control function take the value of zero on parts of the time interval, simultaneously. These problems are both theoretically interesting and practically relevant. After obtaining results about the structure of the optimal controls, we derive stability estimates for the solution of the problem w.r.t. perturbations and L2-regularization. These results are consequently applied to prove convergence of the Euler discretization. Finally, the usefulness of our approach is demonstrated by solving an illustrative example using a semismooth Newton method. [ABSTRACT FROM AUTHOR]

We develop a fast and robust algorithm for solving large-scale convex composite optimization models with an emphasis on the ℓ1-regularized least squares regression (lasso) problems. Despite the fact that there exist a large number of solvers in the literature for the lasso problems, we found that no solver can efficiently handle difficult large-scale regression problems with real data. By leveraging on available error bound results to realize the asymptotic superlinear convergence property of the augmented Lagrangian algorithm, and by exploiting the second order sparsity of the problem through the semismooth Newton method, we are able to propose an algorithm, called Ssnal, to efficiently solve the aforementioned difficult problems. Under very mild conditions, which hold automatically for lasso problems, both the primal and the dual iteration sequences generated by Ssnal possess a fast linear convergence rate, which can even be superlinear asymptotically. Numerical comparisons between our approach and a number of state-of-the-art solvers, on real data sets, are presented to demonstrate the high efficiency and robustness of our proposed algorithm in solving difficult large-scale lasso problems. [ABSTRACT FROM AUTHOR]

PDE-constrained optimization aims at finding optimal setups for partial differential equations so that relevant quantities are minimized. Including nonsmooth L 1 sparsity promoting terms in the formulation of such problems results in more practically relevant computed controls but adds more challenges to the numerical solution of these problems. The needed L 1 -terms as well as additional inclusion of box control constraints require the use of semismooth Newton methods. We propose robust preconditioners for different formulations of the Newton equation. With the inclusion of a line-search strategy and an inexact approach for the solution of the linear systems, the resulting semismooth Newton’s method is reliable for practical problems. Our results are underpinned by a theoretical analysis of the preconditioned matrix. Numerical experiments illustrate the robustness of the proposed scheme. [ABSTRACT FROM AUTHOR]

We propose an algorithm, semismooth Newton coordinate descent (SNCD), for the elastic-net penalized Huber loss regression and quantile regression in high dimensional settings. Unlike existing coordinate descent type algorithms, the SNCD updates a regression coefficient and its corresponding subgradient simultaneously in each iteration. It combines the strengths of the coordinate descent and the semismooth Newton algorithm, and effectively solves the computational challenges posed by dimensionality and nonsmoothness. We establish the convergence properties of the algorithm. In addition, we present an adaptive version of the “strong rule” for screening predictors to gain extra efficiency. Through numerical experiments, we demonstrate that the proposed algorithm is very efficient and scalable to ultrahigh dimensions. We illustrate the application via a real data example. Supplementary materials for this article are available online. [ABSTRACT FROM PUBLISHER]

In this paper, we present a semismooth Newton method for a kind of HJB equation. By suitably choosing the initial iterative point, the method is proved to have monotone convergence. Moreover, the semismooth Newton method has local superlinear convergence rate. Some simple numerical results are reported. [ABSTRACT FROM AUTHOR]

An algorithm for solving nearly-separable quadratic optimization problems (QPs) is presented. The approach is based on applying a semismooth Newton method to solve the implicit complementarity problem arising as the first-order stationarity conditions of such a QP. An important feature of the approach is that, as in dual decomposition methods, separability of the dual function of the QP can be exploited in the search direction computation. Global convergence of the method is promoted by enforcing decrease in component(s) of a Fischer-Burmeister formulation of the complementarity conditions, either via a merit function or through a filter mechanism. The results of numerical experiments when solving convex and nonconvex instances are provided to illustrate the efficacy of the method. [ABSTRACT FROM AUTHOR]

This work is concerned with a class of PDE-constrained optimization problems that are motivated by an application in radiotherapy treatment planning. Here the primary design objective is to minimize the volume where a functional of the state violates a prescribed level, but prescribing these levels in the form of pointwise state constraints leads to infeasible problems. We therefore propose an alternative approach based on $$L^1$$ penalization of the violation that is also applicable when state constraints are infeasible. We establish well-posedness of the corresponding optimal control problem, derive first-order optimality conditions, discuss convergence of minimizers as the penalty parameter tends to infinity, and present a semismooth Newton method for their efficient numerical solution. The performance of this method for a model problem is illustrated and contrasted with an alternative approach based on (regularized) state constraints. [ABSTRACT FROM AUTHOR]

We present a second order algorithm, based on orthantwise directions, for solving optimization problems involving the sparsity enhancing $$\ell _1$$ -norm. The main idea of our method consists in modifying the descent orthantwise directions by using second order information both of the regular term and (in weak sense) of the $$\ell _1$$ -norm. The weak second order information behind the $$\ell _1$$ -term is incorporated via a partial Huber regularization. One of the main features of our algorithm consists in a faster identification of the active set. We also prove that a reduced version of our method is equivalent to a semismooth Newton algorithm applied to the optimality condition, under a specific choice of the algorithm parameters. We present several computational experiments to show the efficiency of our approach compared to other state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

In this paper we propose Jacobian smoothing inexact Newton method for nonlinear complementarity problems (NCP) with derivative-free nonmonotone line search. This nonmonotone line search technique ensures globalization and is a combination of Grippo-Lampariello-Lucidi (GLL) and Li-Fukushima (LF) strategies, with the aim to take into account their advantages. The method is based on very well known Fischer-Burmeister reformulation of NCP and its smoothing Kanzow's approximation. The mixed Newton equation, which combines the semismooth function with the Jacobian of its smooth operator, is solved approximately in every iteration, so the method belongs to the class of Jacobian smoothing inexact Newton methods. The inexact search direction is not in general a descent direction and this is the reason why nonmonotone scheme is used for globalization. Global convergence and local superlinear convergence of method are proved. Numerical performances are also analyzed and point out that high level of nonmonotonicity of this line search rule enables robust and efficient method. [ABSTRACT FROM AUTHOR]

Most existing methods for solving two-phase flow problems in porous media do not take the physically feasible saturation fractions between 0 and 1 into account, which often destroys the numerical accuracy and physical interpretability of the simulation. To calculate the solution without the loss of this basic requirement, we introduce a variational inequality formulation of the saturation equilibrium with a box inequality constraint, and use a conservative finite element method for the spatial discretization and a backward differentiation formula with adaptive time stepping for the temporal integration. The resulting variational inequality system at each time step is solved by using a semismooth Newton algorithm. To accelerate the Newton convergence and improve the robustness, we employ a family of adaptive nonlinear elimination methods as a nonlinear preconditioner. Some numerical results are presented to demonstrate the robustness and efficiency of the proposed algorithm. A comparison is also included to show the superiority of the proposed fully implicit approach over the classical IMplicit Pressure-Explicit Saturation (IMPES) method in terms of the time step size and the total execution time measured on a parallel computer. [ABSTRACT FROM AUTHOR]

In this paper a special semi-smooth equation associated to the second order cone is studied. It is shown that, under mild assumptions, the semi-smooth Newton method applied to this equation is well-defined and the generated sequence is globally and Q-linearly convergent to a solution. As an application, the obtained results are used to study the linear second order cone complementarity problem, with special emphasis on the particular case of positive definite matrices. Moreover, some computational experiments designed to investigate the practical viability of the method are presented. [ABSTRACT FROM AUTHOR]

Low-rank tensor methods provide efficient representations and computations for high-dimensional problems and are able to break the curse of dimensionality when dealing with systems involving multiple parameters. Motivated by optimal control problems with PDEs under uncertainty, we present algorithms for constrained nonlinear optimization problems that use low-rank tensors. They are applied to optimal control of PDEs with uncertain parameters and to parametrized variational inequalities of obstacle type. These methods are tailored to the usage of low-rank tensor arithmetics and allow us to solve huge scale optimization problems. In particular, we consider a semismooth Newton method for an optimal control problem with pointwise control constraints and an interior point algorithm for an obstacle problem, both with uncertainties in the coefficients. [ABSTRACT FROM AUTHOR]

The signed distance function d to an embedded (hyper-)surface r is required in the analysis and implementation of some higher-order methods for the numerical treatment of partial differential equations on surfaces. Two algorithms for the approximation of d are presented in this paper which require only a finite element approximation of a (smooth) level set function of Γ. One method is based on a semismooth Newton method; the other method is a nested fixed point iteration. Both are generalizations of known methods. We provide full (local) convergence analyses. Moreover, the methods are compared in two numerical experiments. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of nonlinear and nonsmooth operator equations in Hilbert spaces. A semismooth Newton method is used for search direction generation. The operator equation is solved by a globalized semismooth Newton method that is equipped with an Armijo linesearch using a semismooth merit function. We prove that an accumulation point of the globalized algorithm is a solution and transition to fast local convergence under a directional Hadamard-like continuity assumption on the Newton matrix. In particular, no auxiliary descent directions or smoothing steps are required. Finally, we apply this method to a control-constrained and also to a regularized state-constrained optimal control problem subject to partial differential equations. [ABSTRACT FROM AUTHOR]

A class of abstract nonlinear evolution quasi-variational inequality (QVI) problems in function space is considered. The framework developed includes constraint sets of obstacle and gradient type. The paper addresses the existence, uniqueness and approximation of solutions when the constraint set mapping is of a special form. Uniqueness is addressed through contractive behavior of a nonlinear mapping whose fixed points are solutions to the QVI. An axiomatic semi-discrete approximation scheme is developed, which is proven to be convergent and is numerically implemented. The paper ends by a report on numerical tests for several nonlinear constraints of gradient-type. [ABSTRACT FROM AUTHOR]

We propose primal-dual active-set (PDAS) methods for solving large-scale instances of an important class of convex quadratic optimization problems (QPs). The iterates of the algorithms are partitions of the index set of variables, where corresponding to each partition there exist unique primal-dual variables that can be obtained by solving a (reduced) linear system. Algorithms of this type have recently received attention when solving certain QPs and linear complementarity problems since, with rapid changes in the active set estimate, they often converge in few iterations. Indeed, as discussed in this paper, convergence in a finite number of iterations is guaranteed when a basic PDAS method is employed to solve certain QPs for which a reduced Hessian of the objective function is (a perturbation of) an M-matrix. We propose three PDAS algorithms. The novelty of the algorithms is that they allow inexactness in the (reduced) linear system solves at all partitions except optimal ones. Such a feature is particularly important in large-scale settings when one employs iterative Krylov subspace methods to solve these systems. Our first algorithm is convergent when solving problems for which properties of the Hessian can be exploited to derive explicit bounds to be enforced on the (reduced) linear system residuals, whereas our second and third algorithms employ dynamic parameters to obviate the need of such explicit bounds. We prove that when applied to solve an important class of convex QPs, our algorithms converge from any initial partition. We also illustrate their practical behavior by providing the results of numerical experiments on a set of discretized optimal control problems, some of which are explicitly formulated to exhibit degeneracy. [ABSTRACT FROM AUTHOR]

In this paper we present a new multigrid preconditioner for the linear systems arising in the semismooth Newton method solution of certain control-constrained, quadratic distributed optimal control problems. Using a piecewise constant discretization of the control space, each semismooth Newton iteration essentially requires inverting a principal submatrix of the matrix entering the normal equations of the associated unconstrained optimal control problem, the rows (and columns) of the submatrix representing the constraints deemed inactive at the current iteration. Previously developed multigrid preconditioners for the aforementioned submatrices were based on constructing a sequence of conforming coarser spaces and proved to be of suboptimal quality for the class of problems considered. Instead, the multigrid preconditioner introduced in this work uses nonconforming coarse spaces, and it is shown that, under reasonable geometric assumptions on the constraints that are deemed inactive, the preconditioner approximates the inverse of the desired submatrix to optimal order. The preconditioner is tested numerically on a classical elliptic-constrained optimal control problem and further on a constrained image-deblurring problem. [ABSTRACT FROM AUTHOR]

In this paper we explore a numerical solution to elastoplastic constitutive initial value problems. An improved form of the implicit return-mapping scheme for nonsmooth yield surfaces is proposed that systematically builds upon a subdifferential formulation of the flow rule. The main advantage of this approach is that the treatment of singular points - apices or edges at which the flow direction is multivalued - only involves a uniquely defined set of non-linear equations, similarly to smooth yield surfaces. This paper focuses on isotropic models containing: a) yield surfaces with one or two apices (singular points) on the hydrostatic axis, b) plastic pseudo-potentials that are independent of the Lode angle, and c) possibly nonlinear isotropic hardening. We show that for some models the improved integration scheme also enables us to a priori decide about a type of the return and to investigate the existence, uniqueness, and semismoothness of discretized constitutive operators. The semismooth Newton method is also introduced for solving the incremental boundary-value problems. The paper contains numerical examples related to slope stability with publicly available Matlab implementations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the tensor eigenvalue complementarity problem which is closely related to the optimality conditions for polynomial optimization, as well as a class of differential inclusions with nonconvex processes. By introducing an NCP-function, we reformulate the tensor eigenvalue complementarity problem as a system of nonlinear equations. We show that this function is strongly semismooth but not differentiable, in which case the classical smooth methods cannot apply. Furthermore, we propose a damped semismooth Newton method for tensor eigenvalue complementarity problem. A new procedure to evaluate an element of the generalized Jacobian is given, which turns out to be an element of the B-subdifferential under mild assumptions. As a result, the convergence of the damped semismooth Newton method is guaranteed by existing results. The numerical experiments also show that our method is efficient and promising. [ABSTRACT FROM AUTHOR]

The numerical realization of the dynamic programming principle for continuous-time optimal control leads to nonlinear Hamilton-Jacobi-Bellman equations which require the minimization of a nonlinear mapping over the set of admissible controls. This minimization is often performed by comparison over a finite number of elements of the control set. In this paper we demonstrate the importance of an accurate realization of these minimization problems and propose algorithms by which this can be achieved effectively. The considered class of equations includes nonsmooth control problems with ℓ1-penalization which lead to sparse controls. [ABSTRACT FROM AUTHOR]

Elliptic optimal control problems with pointwise state gradient constraints are considered. A quadratic penalty approach is employed together with a semismooth Newton iteration. Three different preconditioners are proposed and the ensuing spectral properties of the preconditioned linear Newton saddle-point systems are analyzed dependent on the penalty parameter. A new bound for the smallest positive eigenvalue is proved. Since the analysis is carried out in function space it will ensure mesh independent convergence behavior of suitable Krylov subspace methods such as Minres, also in discretized settings. A path-following strategy with a preconditioned inexact Newton solver is implemented and numerical results are provided. [ABSTRACT FROM AUTHOR]

We analyze augmented Lagrangian and boundary element methods for the Signorini boundary value problem of Laplacian. The boundary variational formulation is presented by the boundary integral operators, and the Signorini boundary conditions are formulated as a fixed point problem. Semismooth Newton methods are applied for the numerical solution of the problem. We prove the convergence of the method and confirm the theory by some numerical experiments. [ABSTRACT FROM AUTHOR]

A convex penalty for promoting switching controls for partial differential equations is introduced; such controls consist of an arbitrary number of components of which at most one should be simultaneously active. Using a Moreau–Yosida approximation, a family of approximating problems is obtained that is amenable to solution by a semismooth Newton method. The efficiency of this approach and the structure of the obtained controls are demonstrated by numerical examples. [ABSTRACT FROM AUTHOR]

A Fenchel dualization scheme for the one-step time-discretized contact problem of quasi-static elasto-plasticity with combined kinematic–isotropic hardening is considered. The associated path is induced by a coupled Moreau–Yosida/Tikhonov regularization of the dual problem. The sequence of solutions to the regularized problems is shown to converge strongly to the optimal displacement–stress–strain triple of the original elasto-plastic contact problem in the space-continuous setting. This property relies on the density of the intersection of certain convex sets which is shown as well. It is also argued that the mappings associated with the resulting problems are Newton—or slantly differentiable. Consequently, each regularized subsystem can be solved mesh-independently at a local superlinear rate of convergence. For efficiency purposes, an inexact path-following approach is proposed and a numerical validation of the theoretical results is given. [ABSTRACT FROM AUTHOR]

Numerical methods for non-smooth equation-solving and optimization often require generalized derivative information in the form of elements of the Clarke Jacobian or the B-subdifferential. It is shown here that piecewise differentiable functions are lexicographically smooth in the sense of Nesterov, and that lexicographic derivatives of these functions comprise a particular subset of both the B-subdifferential and the Clarke Jacobian. Several recently developed methods for generalized derivative evaluation of composite piecewise differentiable functions are shown to produce identical results, which are also lexicographic derivatives. A vector forward mode of automatic differentiation (AD) is presented for evaluation of these derivatives, generalizing established methods and combining their computational benefits. This forward AD mode may be applied to any finite composition of known smooth functions, piecewise differentiable functions such as the absolute value function,, and, and certain non-smooth functions which are not piecewise differentiable, such as the Euclidean norm. This forward AD mode may be implemented using operator overloading, does not require storage of a computational graph, and is computationally tractable relative to the cost of a function evaluation. An implementation in Cis discussed. [ABSTRACT FROM AUTHOR]

The solution of vector-valued Cahn-Hilliard systems is of interest in many applications. We discuss strategies for the handling of smooth and nonsmooth potentials as well as for different types of constant mobilities. Concerning the nonsmooth systems, the necessary bound constraints are incorporated via the Moreau-Yosida regularization technique. We develop effective preconditioners for the efficient solution of the linear systems in saddle point form. Numerical results illustrate the efficiency of our approach. In particular, we numerically show mesh and phase independence of the developed preconditioner in the smooth case. The results in the nonsmooth case are also satisfying, and the preconditioned version always outperforms the unpreconditioned one. [ABSTRACT FROM AUTHOR]

Seismic tomography is a technique to determine the material properties of the Earth's subsurface based on the observation of seismograms. This can be stated as a PDE-constrained optimization problem governed by the elastic wave equation. We present a semismooth Newton-PCG method with a trust-region globalization for full-waveform seismic inversion that uses a Moreau-Yosida regularization to handle additional constraints on the material parameters. We establish results on the differentiability of the parameter-to-state operator and analyze the proposed optimization method in a function space setting. The elastic wave equation is discretized by a high-order continuous Galerkin method in space and an explicit Newmark time-stepping scheme. The matrixf-ree implementation relies on the adjoint-based computation of the gradient and Hessian-vector products and on an MPI-based parallelization. Numerical results are shown for an application in geophysical exploration at reservoir scale. [ABSTRACT FROM AUTHOR]

We consider the application of the globalized semismooth Newton method to the solution of (the KKT conditions of) quasi variational inequalities. We show that the method is globally and locally superlinearly convergent for some important classes of quasi variational inequality problems. We report numerical results to illustrate the practical behavior of the method. [ABSTRACT FROM AUTHOR]

In this paper, some characterizations of nonemptiness and boundedness of solution sets for vector variational inequalities are studied in finite and infinite dimensional spaces, respectively. By using a new proof method which is different from the one used in Huang et al. (J Optim Theory Appl 162:548-558 ), a sufficient and necessary condition for the nonemptiness and boundedness of solution sets is established. Basing on this result, some new characterizations of nonemptiness and boundedness of solution sets for vector variational inequalities are proved. Compared with the known results in Huang et al. (), the key assumption that $$K_\infty \cap (F(K))^{w\circ }_C=\{0\}$$ is not required in finite dimensional spaces. Furthermore, the corresponding result of Huang et al. () is extended to the case of infinite dimensional spaces. Some examples are also given to illustrated the main results. [ABSTRACT FROM AUTHOR]

In this paper, we consider using the inexact nonsmooth Newton method to efficiently solve the symmetric cone constrained variational inequality (VISCC) problem. It red provides a unified framework for dealing with the variational inequality with nonlinear constraints, variational inequality with the second-order cone constraints, and the variational inequality with semidefinite cone constraints. We get convergence of the above method and apply the results to three special types symmetric cones. [ABSTRACT FROM AUTHOR]

The piecewise linear system is a nonsmooth but semismooth equation. In this paper, a damped semismooth Newton method is presented for solving a class of piecewise linear systems. Under appropriate conditions, both monotone convergence and finite termination properties are investigated for the proposed method. [ABSTRACT FROM AUTHOR]