Abstract: We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems.

Abstract: Our purpose in this paper is to provide a general approach to model selection via penalization for Gaussian regression and to develop our point of view about this subject. The advantage and importance of model selection come from the fact that it provides a suitable approach to many different types of problems, starting from model selection per se (among a family of parametric models, which one is more suitable for the data at hand), which includes for instance variable selection in regression models, to nonparametric estimation, for which it provides a very powerful tool that allows adaptation under quite general circumstances. Our approach to model selection also provides a natural connection between the parametric and nonparametric points of view and copes naturally with the fact that a model is not necessarily true. The method is based on the penalization of a least squares criterion which can be viewed as a generalization of Mallows’Cp. A large part of our efforts will be put on choosing properly the list of models and the penalty function for various estimation problems like classical variable selection or adaptive estimation for various types of lp-bodies.

Abstract: In this paper, we introduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced. The approach can apply readily to many other statistical contexts. Various new penalty functions are proposed. The hard-thresholding and soft-thresholding estimators of Donoho and Johnstone are specific members of nonlinear regularized wavelet estimators. They correspond to the lower and upper envelopes of a class of the penalized least squares estimators. Necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties. Oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions. The sampling properties of nonlinear regularized wavelet estimators are established and are shown to be adaptively minimax. To efficiently solve penalized least squares problems, nonlinear regularized Sobolev interpolators (NRSI) are proposed as initial estimators, wh...

Abstract: We analyze three discontinuous Galerkin approximations for solving elliptic problems in two or three dimensions. In each one, the basic bilinear form is nonsymmetric: the first one has a penalty term on edges, the second has one constraint per edge, and the third is totally unconstrained. For each of them we prove hp error estimates in the H1 norm, optimal with respect to h, the mesh size, and nearly optimal with respect to p, the degree of polynomial approximation. We establish these results for general elements in two and three dimensions. For the unconstrained method, we establish a new approximation result valid on simplicial elements. L2 estimates are also derived for the three methods.

Abstract: The convergence of a penalty method for solving the discrete regularized American option valuation problem is studied. Sufficient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution. These conditions also ensure that the solution of the penalty problem is an approximate solution to the discrete linear complementarity problem. The efficiency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the American constraint explicitly. Convergence rates are studied as the timestep and mesh size tend to zero. It is observed that an implicit treatment of the American constraint does not converge quadratically (as the timestep is reduced) if constant timesteps are used. A timestep selector is suggested which restores quadratic convergence.

Abstract: This paper presents an evolutionary algorithm for generic multiobjective design optimization problems. The algorithm is based on nondominance of solutions in the objective and constraint space and uses effective mating strategies to improve solutions that are weak in either. Since the methodology is based on nondominance, scaling and aggregation affecting conventional penalty function methods for constraint handling does not arise. The algorithm incorporates intelligent partner selection for cooperative mating. The diversification strategy is based on niching which results in a wide spread of solutions in the parametric space. Results of the algorithm for the design examples clearly illustrate the efficiency of the algorithm in solving multidisciplinary design optimization problems.

Abstract: An optimal algorithm for the reconstruction of a surface from its shading image is presented. The algorithm solves the 3D reconstruction from a single shading image problem. The shading image is treated as a penalty function and the height of the reconstructed surface is a weighted distance. A consistent numerical scheme based on Sethian's fast marching method is used to compute the reconstructed surface. The surface is a viscosity solution of an Eikonal equation for the vertical light source case. For the oblique light source case, the reconstructed surface is the viscosity solution to a different partial differential equation. A modification of the fast marching method yields a numerically consistent, computationally optimal, and practically fast algorithm for the classical shape from shading problem. Next, the fast marching method coupled with a back tracking via gradient descent along the reconstructed surface is shown to solve the path planning problem in robot navigation.

Abstract: We consider the discretized zero-one minimum compliance topology optimization problem of elastic continuum structures under multiple load conditions. The binary design variables indicate presence or absence of material in the finite elements. A common approach to solve these problems is to relax the binary constraints, i.e. allow the design variables to attain values between zero and one, and penalize intermediate values to obtain a "black and white" (zero-one) design. To avoid convergence to a local minimum, it has been suggested that a continuation method should be used, where the penalized problems are solved with increasing penalization. In this paper, the trajectories associated with optimal solutions to the penalized problems, for continuously increasing penalization, are studied on some carefully chosen examples. Two different penalization techniques are used. The global trajectory is defined as the path followed by the global optimal solutions to the penalized problems, and we present examples for which the global trajectory is discontinuous even though the original zero-one problem has a unique solution. Furthermore, we present examples where the penalization method combined with a continuation approach fails to produce a black and white design, no matter how large the penalization becomes.

Abstract: Fair resource allocation in high-speed networks such as the Internet can be viewed as a constrained optimization program. Kelly and his co-workers have shown that an unconstrained penalty function formulation of this problem can be used to design congestion controllers that are stable. In this paper, we examine the question of providing feedback from the network such that the congestion controllers derived from the penalty function formulation lead to the solution of the original unconstrained problem. This can be viewed as the decentralized design of early congestion notification (ECN) marking rates at each node in the Internet to ensure global loss-free operation of a fluid model of the network. We then look at the stability of such a scheme using a time-scale decomposition of the system. This results in two separate systems which are stable individually and we show that under certain assumptions the entire system is semi-globally stable and converges to the equilibrium point exponentially fast.

Abstract: In this paper, we propose a smoothing method for minimax problem. The method is based on the exponential penalty function of Kort and Bertsekas for constrained optimization. Under suitable condition, the method is globally convergent. Preliminary numerical experiments indicate the promising of the algorithm.

Abstract: In this paper, we focus on a useful modification of the decomposition method by He et al. (Ref. 1). Experience on applications has shown that the number of iterations of the original method depends significantly on the penalty parameter. The main contribution of our method is that we allow the penalty parameter to vary automatically according to some self-adaptive rules. As our numerical simulations indicate, the modified method is more flexible and efficient in practice. A detailed convergence analysis of our method is also included.

Abstract: With the operation of the European Remote Sensing (ERS) satellites and RADARSAT, radar images are now readily available. One of the new applications of radar images is their use for bathymetric mapping in shallow seas. The Bathymetry Assessment System (BAS), described in detail in this paper, constructs accurate depth maps from radar images and a limited number of echo soundings. The BAS consists of a forward imaging model and an inversion part. The system needs a first guess depth map that may be derived from echo soundings or an old map of the area. The forward model calculates a simulated radar image. This is compared with the actual radar image by evaluating a penalty function. The penalty function also contains a term that compares model depths with measured depths and a term that contains a smoothness criterion, prohibiting speckle noise to be interpreted as depth variations. The inversion part of the system consists of optimization of the penalty function. This leads to an iterative procedure in wh...

Abstract: A single input, single output active noise control system using the time-domain Filtered-X LMS algorithm with output constraint is investigated. The constraint on the output of the control filter is applied by three different methods: the leakage algorithm based on the transformation method using a penalty function; the re-scaling algorithm based on the active set method; and the simple practical (clipping) algorithm which just clips the output if a constraint is encountered. A comparison of the three algorithms shows that the re-scaling algorithm can usually work successfully under the constraint, while the leakage algorithm usually needs a large leakage coefficient to satisfy the constraint with a resulting performance loss. The clipping algorithm has potential problems both with the stability and convergence speed.

Abstract: In this paper we introduced and analyzed the Log-Sigmoid (LS) multipliers method for con- strained optimization. The LS method is to the recently developed smoothing technique as augmented Lagrangian to the penalty method or modified barrier to classical barrier methods. At the same time the LS method has some specific properties, which make it substantially different from other nonquadratic augmented Lagrangian techniques. We established convergence of the LS type penalty method under very mild assumptions on the input data and estimated the rate of convergence of the LS multipliers method under the standard second order optimality condition for both exact and nonexact minimization. Some important properties of the dual function and the dual problem, which are based on the LS Lagrangian, were discovered and the primal-dual LS method was introduced.

Abstract: Maintaining metro ridership in the future will require an increasing focus on customer perceptions of the service quality. This work outlines a process whereby passenger expectations may be encapsulated into a mathematical evaluation function that can then be used in an on-line optimization procedure. The function proposed penalizes excess waiting time, travelling time and congestion following a disturbance such as a platform delay. With the function defined, the problem resolves into finding a set of optimum arrival and departure times (the decision variables) that minimizes this penalty function. Ultimately, the concept is intended to form part of an on-line, real-time traffic controller residing on the despatching computer in a control centre. An embedded simulation is used to perform the optimization calculations, and several techniques are described that reduce the computing time needed to a level feasible for on-line use. Comparisons are made with some previous control algorithms. Using data...

Abstract: We have recently shown that an energy penalty for the incorporation of residual tensorial constraints into molecular structure calculations can be formulated without the explicit knowledge of the Saupe orientation tensor (Moltke and Grzesiek, J. Biomol. NMR, 1999, 15, 77–82). Here we report the implementation of such an algorithm into the program X-PLOR. The new algorithm is easy to use and has good convergence properties. The algorithm is used for the structure refinement of the HIV-1 Nef protein using 252 dipolar coupling restraints. The approach is compared to the conventional penalty function with explicit knowledge of the orientation tensor's amplitude and rhombicity. No significant differences are found with respect to speed, Ramachandran core quality or coordinate precision.

Abstract: The problem of pavement maintenance management at the network level is one of maintaining as high a level of serviceability as possible for a pavement network system through reactive and proactive repair actions, whilst optimising the use of available resources. This problem has traditionally been solved using techniques like mathematical programming and heuristic methods. Lately, the use of genetic algorithms (GAs) to solve resource allocation problems like the network pavement maintenance problem has received increased attention from researchers. GAs have been demonstrated to be better than traditional techniques in terms of solution quality and diversity. However, the performance of the GAs is affected by the method used to handle the many constraints present in the formulation of such resource allocation methods. Penalty as well as generate and repair methods are the usual techniques used to handle constraints, but these have their drawbacks in terms of computational efficiency and tendency to get trapped in sub-optimal solution spaces. The paper proposes a third method that is computationally more efficient than the previous methods. The method is based on prioritised allocation of resources to maintenance activities and the maximum utilisation of resources. Constraints on maximum resource availability are no longer used passively to check on solution feasibility (as in the previous methods) but are used to help generate feasible solutions during the resource allocation phase of the algorithm itself. It is demonstrated that the GA with the prioritised resource allocation method (PRAM) outperforms the traditional GA with repair or penalty methods. PRAM was able to consistently outperform the other two GA based methods, both in terms of solution quality as well as computational time. It is concluded that PRAM can be used as the basis of more efficient resource allocation procedures in the area of pavement maintenance management.

Abstract: This paper addresses the problem of reconstructing the location, shape, and dielectric permittivity distribution of an inhomogeneous dielectric object from measurements of the field scattered by the object. The object is an inhomogeneous infinite cylinder of arbitrary cross section illuminated by a transverse magnetic incident electric field. The approach is based on the Lippmann-Schuringer integral equation for the electromagnetic inverse scattering problem, approximated by applying the second-order Born approximation, which allows an extension of the range of contrast values that can be accurately imaged. The numerical approach is developed in the spatial domain and makes use of a multi-illumination multiview processing. In particular, the inverse problem is recast in a global nonlinear optimization problem (including a penalty function), solved by a stochastic method based on a genetic algorithm. In this paper, the mathematical formulation of the approach is described and the results of several dielectric reconstructions are reported, including comparisons with analogous reconstructions performed within the linearized (first-order) Born approximation.

Abstract: A die shape design sensitivity analysis (DSA) and optimization for a sheet metal stamping process is proposed based on a Lagrangian formulation. A hyperelasticity-based elastoplastic material model is used for the constitutive relation that includes a large deformation effect. The contact condition between a workpiece and a rigid die is imposed through the penalty method with a modified Coulomb friction model. The domain of the workpiece is discretized by a meshfree method. A continuum-based DSA with respect to the rigid die shape parameter is formulated using a design velocity concept. The die shape perturbation has an effect on structural performance through the contact variational form. The effect of the deformation-dependent pressure load to the design sensitivity is discussed. It is shown that the design sensitivity equation uses the same tangent stiffness matrix as the response analysis. The linear design sensitivity equation is solved at each converged load step without the need of iteration, which is quite efficient in computation. The accuracy of sensitivity information is compared to that of the finite difference method with an excellent agreement. A die shape design optimization problem is solved to obtain the desired shape of the workpiece to minimize spring-back effect and to show the feasibility of the proposed method. Copyright © 2001 John Wiley & Sons, Ltd.

Abstract: This paper examines the possibility of using a stochastic optimisation method (genetic algorithm) and penalty selection methods for computer-aided design of gears. The examples show that using genetic algorithms in order to optimise gears is a very efficient method.

Abstract: One of major difficulties in the implementation of meshless methods is the imposition of essential boundary conditions as the approximations do not pass through the nodal parameter values. As a consequence, the imposition of essential boundary conditions in meshless methods is quite awkward. In this paper, a displacement constraint equations method (DCEM) is proposed for the imposition of the essential boundary conditions, in which the essential boundary conditions is treated as a constraint to the discrete equations obtained from the Galerkin methods. Instead of using the methods of Lagrange multipliers and the penalty method, a procedure is proposed in which unknowns are partitioned into two subvectors, one consisting of unknowns on boundary Γu, and one consisting of the remaining unknowns. A simplified displacement constraint equations method (SDCEM) is also proposed, which results in a efficient scheme with sufficient accuracy for the imposition of the essential boundary conditions in meshless methods. The present method results in a symmetric, positive and banded stiffness matrix. Numerical results show that the accuracy of the present method is higher than that of the modified variational principles. The present method is a exact method for imposing essential boundary conditions in meshless methods, and can be used in Galerkin-based meshless method, such as element-free Galerkin methods, reproducing kernel particle method, meshless local Petrov–Galerkin method. Copyright © 2001 John Wiley & Sons, Ltd.