Regularized state-constrained boundary optimal control of the Navier-Stokes equations

The numerical solution of the Dirichlet boundary optimal control problem of the Navier-Stokes equations in presence of pointwise state constraints is investigated. Two different regularization techniques are considered. First, a Moreau-Yosida regularization of the problem is studied. Optimality conditions are derived and the convergence of the regularized solutions towards the original one is proved. A source representation of the control combined with a Lavrentiev type regularization strategy is also presented. The analysis concerning optimality conditions and convergence of the regularized solutions is carried out. In the last part of the paper numerical experiments are presented. For the numerical solution of each regularized problem a semi-smooth Newton method is applied.     

An Infinite-Dimensional Semismooth Newton Method for Elasto-Plastic Contact Problems

A Fenchel dualization scheme for the one-step time-discretized elasto-plastic contact problem with kinematic or isotropic hardening is considered. The associated path is induced by a combined Moreau-Yosida / Tichonov regularization of the dual problem. The sequence of solutions to the regularized problems is shown to converge strongly to the solution of the original problem. This property relies on the density of the intersection of certain convex sets. The corresponding conditions are worked out and customary regularization approaches are shown to be valid in this context. It is also argued that without higher regularity assumptions on the data the resulting problems possess Newton differentiable optimality systems in infinite dimensions [2]. Consequently, each regularized subsystem can be solved mesh-independently at a local superlinear rate of convergence [6]. Numerically the problems are solved using conforming finite elements. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A combined BDF‐semismooth Newton approach for time‐dependent Bingham flow

This article is devoted to the numerical simulation of time‐dependent convective Bingham flow in cavities. Motivated by a primal‐dual regularization of the stationary model, a family of regularized time‐dependent problems is introduced. Well posedness of the regularized problems is proved, and convergence of the regularized solutions to a solution of the original multiplier system is verified. For the numerical solution of each regularized multiplier system, a fully discrete approach is studied. A stable finite element approximation in space together with a second‐order backward differentiation formula for the time discretization are proposed. The discretization scheme yields a system of Newton differentiable nonlinear equations in each time step, for which a semismooth Newton algorithm is used. We present two numerical experiments to verify the main properties of the proposed approach. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Path-following and augmented Lagrangian methods for contact problems in linear elasticity

A certain regularization technique for contact problems leads to a family of problems that can be solved efficiently using infinite-dimensional semismooth Newton methods, or in this case equivalently, primal–dual active set strategies. We present two procedures that use a sequence of regularized problems to obtain the solution of the original contact problem: first-order augmented Lagrangian, and path-following methods. The first strategy is based on a multiplier-update, while path-following with respect to the regularization parameter uses theoretical results about the path-value function to increase the regularization parameter appropriately. Comprehensive numerical tests investigate the performance of the proposed strategies for both a 2D as well as a 3D contact problem.     

A regularization smoothing method for second-order cone complementarity problem

In this paper, the second-order cone complementarity problem is studied. Based on the Fischer–Burmeister function with a perturbed parameter, which is also called smoothing parameter, a regularization smoothing Newton method is presented for solving the sequence of regularized problems of the second-order cone complementarity problem. Under proper conditions, the global convergence and local superlinear convergence of the proposed algorithm are obtained. Moreover, the local superlinear convergence is established without strict complementarity conditions. Preliminary numerical results suggest the effectiveness of the algorithm.     

Preconditioned Iterative Methods for Algebraic Systems from Multiplicative Half-Quadratic Regularization Image Restorations

<正>Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term.A regularized convex term can usually preserve the image edges well in the restored image.In this paper,we consider a class of convex and edge-preserving regularization functions,i.e.,multiplicative half-quadratic regularizations,and we use the Newton method to solve the correspondingly reduced systems of nonlinear equations.At each Newton iterate,the preconditioned conjugate gradient method,incorporated with a constraint preconditioner,is employed to solve the structured Newton equation that has a symmetric positive definite coefficient matrix. The eigenvalue bounds of the preconditioned matrix are deliberately derived,which can be used to estimate the convergence speed of the preconditioned conjugate gradient method.We use experimental results to demonstrate that this new approach is efficient, and the effect of image restoration is reasonably well.     

Adaptive optimal control of Signorini’s problem

In this article, we present a-posteriori error estimations in context of optimal control of contact problems; in particular of Signorini’s problem. Due to the contact side-condition, the solution operator of the underlying variational inequality is not differentiable, yet we want to apply Newton’s method. Therefore, the non-smooth problem is regularized by penalization and afterwards discretized by finite elements. We derive optimality systems for the regularized formulation in the continuous as well as in the discrete case. This is done explicitly for Signorini’s contact problem, which covers linear elasticity and linearized surface contact conditions. The latter creates the need for treating trace-operations carefully, especially in contrast to obstacle contact conditions, which exert in the domain. Based on the dual weighted residual method and these optimality systems, we deduce error representations for the regularization, discretization and numerical errors. Those representations are further developed into error estimators. The resulting error estimator for regularization error is defined only in the contact area. Therefore its computational cost is especially low for Signorini’s contact problem. Finally, we utilize the estimators in an adaptive refinement strategy balancing regularization and discretization errors. Numerical results substantiate the theoretical findings. We present different examples concerning Signorini’s problem in two and three dimensions.     

Multigrid Solution of a Lavrentiev-Regularized State-Constrained Parabolic Control Problem

A mesh-independent, robust, and accurate multigrid scheme to solve a linear state-constrained parabolic optimal control problem is presented. We first consider a Lavrentiev regularization of the state-constrained optimization problem. Then, a multigrid scheme is designed for the numerical solution of the regularized optimality system. Central to this scheme is the construction of an iterative pointwise smoother which can be formulated as a local semismooth Newton iteration. Results of numerical experiments and theoretical two-grid local Fourier analysis estimates demonstrate that the proposed scheme is able to solve parabolic state-constrained optimality systems with textbook multigrid efficiency.     

Global Convergence of a Closed-Loop Regularized Newton Method for Solving Monotone Inclusions in Hilbert Spaces

We analyze the global convergence properties of some variants of regularized continuous Newton methods for convex optimization and monotone inclusions in Hilbert spaces. The regularization term is of Levenberg–Marquardt type and acts in an open-loop or closed-loop form. In the open-loop case the regularization term may be of bounded variation.     

Optimal control for systems described by hyperbolic equations with strong nonlinearity

An optimization control problem for a hyperbolic equation is considered. The system is nonlinear with respect to the state derivative. The regularization technique for the state equation is applied. The necessary conditions of optimality for the regularized control problem are proved. It uses the extended differentiability of the control-state mapping for the regularized equation. The convergence of the regularization method is proved. Thus the optimal control for the regularized problem with a small enough regularization parameter can be chosen as an approximate solution of the initial optimization problem.     

Lagrange multipliers, (exact) regularization and error bounds for monotone variational inequalities

We examine two central regularization strategies for monotone variational inequalities, the first a direct regularization of the operative monotone mapping, and the second via regularization of the associated dual gap function. A key link in the relationship between the solution sets to these various regularized problems is the idea of exact regularization, which, in turn, is fundamentally associated with the existence of Lagrange multipliers for the regularized variational inequality. A regularization is said to be exact if a solution to the regularized problem is a solution to the unregularized problem for all parameters beyond a certain value. The Lagrange multipliers corresponding to a particular regularization of a variational inequality, on the other hand, are defined via the dual gap function. Our analysis suggests various conceptual, iteratively regularized numerical schemes, for which we provide error bounds, and hence stopping criteria, under the additional assumption that the solution set to the unregularized problem is what we call weakly sharp of order greater than one.     

A semismooth Newton method for a class of semilinear optimal control problems with box and volume constraints

In this paper we consider optimal control problems subject to a semilinear elliptic state equation together with the control constraints 0≤u≤1 and ∫u=m. Optimality conditions for this problem are derived and reformulated as a nonlinear, nonsmooth equation which is solved using a semismooth Newton method. A regularization of the nonsmooth equation is necessary to obtain the superlinear convergence of the semismooth Newton method. We prove that the solutions of the regularized problems converge to a solution of the original problem and a path-following technique is used to ensure a constant decrease rate of the residual. We show that, in certain situations, the optimal controls take 0–1 values, which amounts to solving a topology optimization problem with volume constraint.     

A minimum effort optimal control problem for the wave equation

A minimum effort optimal control problem for the undamped wave equation is considered which involves L ^∞-control costs. Since the problem is non-differentiable a regularized problem is introduced. Uniqueness of the solution of the regularized problem is proven and the convergence of the regularized solutions is analyzed. Further, a semi-smooth Newton method is formulated to solve the regularized problems and its superlinear convergence is shown. Thereby special attention has to be paid to the well-posedness of the Newton iteration. Numerical examples confirm the theoretical results.     

An operator Newton method for the Stefan problem based on smoothing: A local perspective

The initial/Neumann boundary-value enthalpy formulation for the two-phase Stefan problem is regularized by smoothing. Known estimates predict a convergence rate of ^1/2, and this result is extended in this paper to include the case of a (nonzero) residual in the regularized problem. A modified Newton Kantorovich framework is established, whereby the exact solution of the regularized problem is replaced by one Newton iteration. It is shown that a consistent theory requires measure-theoretic hypotheses on the starting guess and the Newton iterate, otherwise residual decrease is not expected. The circle closes in one spatial dimension, where it is shown that the residual decrease of Newton's method correlates precisely with the ^1/2 convergence theory.     

Modular solvers for image restoration problems using the discrepancy principle

Many problems in image restoration can be formulated as either an unconstrained non‐linear minimization problem, usually with a Tikhonov‐like regularization, where the regularization parameter has to be determined; or as a fully constrained problem, where an estimate of the noise level, either the variance or the signal‐to‐noise ratio, is available. The formulations are mathematically equivalent. However, in practice, it is much easier to develop algorithms for the unconstrained problem, and not always obvious how to adapt such methods to solve the corresponding constrained problem. In this paper, we present a new method which can make use of any existing convergent method for the unconstrained problem to solve the constrained one. The new method is based on a Newton iteration applied to an extended system of non‐linear equations, which couples the constraint and the regularized problem, but it does not require knowledge of the Jacobian of the irregularity functional. The existing solver is only used as a black box solver, which for a fixed regularization parameter returns an improved solution to the unconstrained minimization problem given an initial guess. The new modular solver enables us to easily solve the constrained image restoration problem; the solver automatically identifies the regularization parameter, during the iterative solution process. We present some numerical results. The results indicate that even in the worst case the constrained solver requires only about twice as much work as the unconstrained one, and in some instances the constrained solver can be even faster. Copyright © 2002 John Wiley & Sons, Ltd.     

Preconditioners for state‐constrained optimal control problems with Moreau–Yosida penalty function

Optimal control problems with partial differential equations as constraints play an important role in many applications. The inclusion of bound constraints for the state variable poses a significant challenge for optimization methods. Our focus here is on the incorporation of the constraints via the Moreau–Yosida regularization technique. This method has been studied recently and has proven to be advantageous compared with other approaches. In this paper, we develop robust preconditioners for the efficient solution of the Newton steps associated with the fast solution of the Moreau–Yosida regularized problem. Numerical results illustrate the efficiency of our approach. Copyright © 2012 John Wiley & Sons, Ltd.     

The wavelet multiscale method for inversion of porosity in the fluid-saturated porous media

In this paper, the wavelet multiscale method is applied to the inversion of porosity in the fluid-saturated porous media. The inverse problem is decomposed to multiple scales with wavelet transform and hence the original inverse problem is re-formulated to be a set of sub-inverse problem corresponding to different scales and is solved successively according to the size of scale from the smallest to the largest. On each scale, regularization Gauss–Newton method is carried out, which is stable and fast, until the optimum solution of original inverse problem is found. The results of numerical simulations demonstrate that the method is a widely convergent optimization method and exhibits the advantages of conventional regularization Gauss–Newton method methods on computational efficiency and precision.     

A new smoothing and regularization Newton method for the symmetric cone complementarity problem

Based on a regularized Chen–Harker–Kanzow–Smale (CHKS) smoothing function, we propose a new smoothing and regularization Newton method for solving the symmetric cone complementarity problem. By using the theory of Euclidean Jordan algebras, we establish the global and local quadratic convergence of the method on certain assumptions. The proposed method uses a nonmonotone line search technique which includes the usual monotone line search as a special case. In addition, our method treats both the smoothing parameter \(\mu \) and the regularization parameter \(\varepsilon \) as independent variables. Preliminary numerical results are reported which indicate that the proposed method is effective.     

Robust Solution to Fuzzy Identification Problem with Uncertain Data by Regularization

This study considers the robust identification of the parameters describing a Sugeno type fuzzy inference system with uncertain data. The objective is to minimize the worst-case residual error using a numerically efficient algorithm. The Sugeno type fuzzy systems are linear in consequent parameters but nonlinear in antecedent parameters. The robust consequent parameters identification problem can be formulated as second-order cone programming problem. The optimal solution of this second-order cone problem can be interpreted as solution of a Tikhonov regularization problem with a special choice of regularization parameter which is optimal for robustness (Ghaoui and Lebret (1997). SAIM Journal of Matrix Analysis and Applications 18, 1035–1064). The final regularized nonlinear optimization problem allowing simultaneous identification of antecedent and consequent parameters is solved iteratively using a generalized Gauss–Newton like method. To illustrate the approach, several simulation studies on numerical examples including the modelling of a spectral data function (one-dimensional benchmark example) is provided. The proposed robust fuzzy identification scheme has been applied to approximate the physical fitness of patients with a fuzzy expert system. The identified fuzzy expert system is shown to be capable of capturing the decisions (experiences) of a medical expert.