Penalty Functionals Research Articles

A New Approach to Source Conditions in Regularization with General Residual Term

This article addresses Tikhonov-like regularization methods with convex penalty functionals for solving nonlinear ill-posed operator equations formulated in Banach or, more general, topological spaces. We present an approach for proving convergence rates that combines advantages of approximate source conditions and variational inequalities. Precisely, our technique provides both a wide range of convergence rates and the capability to handle general and not necessarily convex residual terms as well as nonsmooth operators. Initially formulated for topological spaces, the approach is extensively discussed for Banach and Hilbert space situations, showing that it generalizes some well-known convergence rates results.

Topological optimization of structures subject to Von Mises stress constraints

The topological asymptotic analysis provides the sensitivity of a given shape functional with respect to an infinitesimal domain perturbation. Therefore, this sensitivity can be naturally used as a descent direction in a structural topology design problem. However, according to the literature concerning the topological derivative, only the classical approach based on flexibility minimization for a given amount of material, without control on the stress level supported by the structural device, has been considered. In this paper, therefore, we introduce a class of penalty functionals that mimic a pointwise constraint on the Von Mises stress field. The associated topological derivative is obtained for plane stress linear elasticity. Only the formal asymptotic expansion procedure is presented, but full justifications can be deduced from existing works. Then, a topology optimization algorithm based on these concepts is proposed, that allows for treating local stress criteria. Finally, this feature is shown through some numerical examples.

A penalty method for topology optimization subject to a pointwise state constraint

This paper deals with topology optimization of domains subject to a pointwise constraint on the gradient of the state. To realize this constraint, a class of penalty functionals is introduced and the expression of the corresponding topological derivative is obtained for the Laplace equation in two space dimensions. An algorithm based on these concepts is proposed. It is illustrated by some numerical applications.

(087). Interferon monotherapy treatment in haemodialysis patients with chronic hepatitis C virus

A simple parametrization, built from the definition of cubic splines, is shown to facilitate the implementation and interpretation of penalized spline models, whatever configuration of knots is used. The parametrization is termed value-first derivative parametrization. Inference is Bayesian and explores the natural link between quadratic penalties and Gaussian priors. However, a full Bayesian analysis seems feasible only for some penalty functionals. Alternatives include empirical Bayes inference methods involving model selection type criteria. The proposed methodology is illustrated by an application to survival analysis where the usual Cox model is extended to allow for time-varying regression coefficients.

Parametrization and penalties in spline models with an application to survival analysis

A simple parametrization, built from the definition of cubic splines, is shown to facilitate the implementation and interpretation of penalized spline models, whatever configuration of knots is used. The parametrization is termed value-first derivative parametrization. Inference is Bayesian and explores the natural link between quadratic penalties and Gaussian priors. However, a full Bayesian analysis seems feasible only for some penalty functionals. Alternatives include empirical Bayes inference methods involving model selection type criteria. The proposed methodology is illustrated by an application to survival analysis where the usual Cox model is extended to allow for time-varying regression coefficients.

On Variational Based Scale‐Bridging of Inelastic Composites

AbstractThe paper discusses formulations for the theoretical and numerical analysis of inelastic composites with scale separation. The basic underlying structure is a canonical variational setting in the fully nonlinear range based on incremental energy minimization. We focus on formulations of strain–driven homogenization for representative composite aggregates with emphasis on the development of canonical families of algorithms based on Lagrange and penalty functionals to cover alternative boundary constraints of (i.) linear deformations, (ii.) periodic deformations and (iii.) uniform tractions. As a key result, we present a compact matrix formulation for homogenization covering introduced alternative boundary constraints. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Least squares and bounded variation regularization with nondifferentiable functionals

Let A be an operator from a real Banach space into a real Hilbert space. In this paper we study least squares regularization methods for the ill-posed operator equation A(u) = f using nonlinear nondifferentiable penalty functionals. We introduce a notion of distributional approximation, and use constructs of distributional approximations to establish convergence and stability of approximations of bounded variation solutions of the operator equation. We also show that the results provide a framework for a rigorous analysis of numerical methods based on Euler-Lagrange equations to solve the minimization problem. This justifies many of the numerical implementation schemes of bounded variation minimization that have been recently proposed.

Numerical Methods of Determination of Optimum Control of Electric Traction Vehicle

This paper is presenting a numerical optimization method of the drive control of an electric traction vehicle. The problem of the control optimization of the vehicle includes some basic restrictions such as the non-linear state equations, and additionally some non-linear restrictions of state and control variables. The proposed method of the optimization is based upon the conjugate gradient method, and upon the method of penalty functionals. There have been three varsions of the conjugate gradient method examined of the control problem of a commuting traction vehicle. It allowed the conclusion that the best version for this type of the task is the Polak-Rebiere version.

On a Class of Nonparametric Density and Regression Estimators

A class of maximum penalized likelihood estimators (MPLE) of the density function $f$ is constructed, through the use of a rather general roughness-penalty functional. This class contains all the density estimates in the literature that arise as solutions to MPLE problems with penalties on $f^{1/2}$. In addition, the flexibility of the penalty functional permits the construction of new spline estimates with improved performance at the peaks and valleys of the density curves. The consistency of the estimators in probability and a.s., in the $L_p(\mathbb{R}) -$ norms, $p = 1, 2, \infty$, in the Hellinger metric and Sobolev norms is established in a unified manner. A class of penalty functionals is identified which leads to estimators which approach the optimal rates of convergence predicted in Farrell (1972). Based on the above estimates, a class of MPLE regression estimators is introduced which has the appealing property of reducing to the classical nonparametric regression estimates when a smoothing parameter goes to zero. Finally, a theoretically justifiable and numerically efficient method for a data based choice of the smoothing parameter is proposed for further study. A number of numerical examples and graphs are presented.

Projection on a Cone, Penalty Functionals and Duality Theory for Problems with Inequaltity Constraints in Hilbert Space

Each element p of a real Hilbert space H can be uniquely decomposed into two orthogonal components, $p = p^D + p^{ - D^ * } $ where $p^D \in D$ is the projection of p on a closed convex cone D and $p^{ - D^ * } $ is the projection of p on the minus dual cone $ - D^ * $. Hence, if the cone D generates a partial order in H, then the positive part $p^D $ and the negative part $p^{ - D^ * } $ of each $p \in H$ can be distinguished. For a general optimization problem: minimize $Q(y)$ over $Y_p = \{ y \in E:p - P(y) \in D \subset H\} $, where $Q:E \to R$, $P:E \to H$, E is Banach, H is Hilbert: the violation of the constraint can be determined by ($(p - P(y))^{ - D^ * } $. Hence a generalized penalty functional and an augmented Lagrange functional can be defined for this problem. The paper presents a short review of known penalty techniques, some properties of the projection on a cone, basic properties of penalty functionals for a general optimization problem and duality theory for nonconvex problems in infinit...

Projection on a cone and generalized penalty functionals for problems with constraints in Hilbert space

Projection on a cone and generalized penalty functionals for problems with constraints in Hilbert space

Extensions of Lagrange Multipliers in Nonlinear Programming

defined for x > 0, u > 0. A saddle point of (1.2) is defined to be a point (x*, u*), x* > Q, u* > 0 such that +(x, u*) < +(x*, u*) _ +(x*, u) for all x _ 0, u ? 0. One of the Kuhn-Tucker results is that a saddle point of(1.2) provides a solution to (1.1). Under cer-eain additional regularity assumptions the equivalence of the saddle point problem and the program (1.1) has been shown [1], [10]. In particular, if the feasible set in (1.1) has an interior point (a nonnegative x such that g(x) < b),3 and if the objective function is concave and the constraint functions are convex, then x* is a solution to (1.1) if and only if there is a nonnegative u* such that (x*, u*) is a saddle point of (1.2). This is the so-called Kuhn-Tucker equivalence theorem of nonlinear programming. In this paper Lagrange multipliers are generalized from the usual constants to (possibly nonlinear) multiplier functions. This leads to the statement of several general equivalence results under quite weak assumptions. From a somewhat different point of view, Everett [4] drops differentiability assumptions and discusses a procedure for solving nonlinear programs in terms of