Numerical optimal control of the wave equation: optimal boundary control of a string to rest in finite time

In many real-life applications of optimal control problems with constraints in form of partial differential equations (PDEs), hyperbolic equations are involved which typically describe transport processes. Since hyperbolic equations usually propagate discontinuities of initial or boundary conditions into the domain on which the solution lives or can develop discontinuities even in the presence of smooth data, problems of this type constitute a severe challenge for both theory and numerics of PDE constrained optimization.     

Models for Image Interpolation Based on the Optical Flow

Optical flow is the 2D motion that needs to be recovered from a video sequence. In this paper we study variational principles for the generation of interpolating sequences between two images. The basic assumption is that there exists an underlying video sequence that solves the optic flow equation and interpolates the two images. The numerical solution of the interpolation problem is reduced to the solution of a system of coupled partial differential equations. Some numerical simulations are presented. Received June 29, 2000; revised November 29, 2000     

L ∞-error estimates of higher order mixed finite element approximations for elliptic optimal control problems

In this paper, we investigate the L ^∞-error estimates of the numerical solutions of linear-quadratic elliptic control problems by using higher order mixed finite element methods. The state and co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise polynomials of order k (k≥1). Optimal L ^∞-error estimates are derived for both the control and the state approximations. These results are seemed to be new in the literature of the mixed finite element methods for optimal control problems.     

A Semi-smooth Newton Method for Regularized State-constrained Optimal Control of the Navier-Stokes Equations

In this paper, we study semi-smooth Newton methods for the numerical solution of regularized pointwise state-constrained optimal control problems governed by the Navier-Stokes equations. After deriving an appropriate optimality system for the original problem, a class of Moreau-Yosida regularized problems is introduced and the convergence of their solutions to the original optimal one is proved. For each regularized problem a semi-smooth Newton method is applied and its local superlinear convergence verified. Finally, selected numerical results illustrate the behavior of the method and a comparison between the max-min and the Fischer-Burmeister as complementarity functionals is carried out.     

Variational Methods on the Space of Functions of Bounded Hessian for Convexification and Denoising

In this paper we investigate variational principles on the space of functions of bounded Hessian for denoising, for numerical calculation of convex envelopes and for approximation by convex functions.     

A variational approximation method for 2nd order elliptic eigenvalue problems in a composite structure with nonstandard boundary conditions

In this paper we deal with the finite element analysis of a class of eigenvalue problems (EVPs) in a composite structure in the plane, consisting of rectangular subdomains which enclose an intermediate region. Nonlocal boundary conditions (BCs) of Robin type are imposed on the inner boundaries, i.e. on the interfaces of the respective subdomains with the intermediate region. On the eventual interfaces between two subdomains we impose discontinuous transition conditions (TCs). Finally, we have classical local BCs at the outer boundaries. Such problems are related to some heat transfer problems e.g. in a horizontal cross section of a wall enclosing an air cave.     

Composite Finite Elements for Elliptic Boundary Value Problems with Discontinuous Coefficients

In this paper, we will introduce composite finite elements for solving elliptic boundary value problems with discontinuous coefficients. The focus is on problems where the geometry of the interfaces between the smooth regions of the coefficients is very complicated. On the other hand, efficient numerical methods such as, e.g., multigrid methods, wavelets, extrapolation, are based on a multi-scale discretization of the problem. In standard finite element methods, the grids have to resolve the structure of the discontinuous coefficients. Thus, straightforward coarse scale discretizations of problems with complicated coefficient jumps are not obvious. In this paper, we define composite finite elements for problems with discontinuous coefficients. These finite elements allow the coarsening of finite element spaces independently of the structure of the discontinuous coefficients. Thus, the multigrid method can be applied to solve the linear system on the fine scale. We focus on the construction of the composite finite elements and the efficient, hierarchical realization of the intergrid transfer operators. Finally, we present some numerical results for the multigrid method based on the composite finite elements (CFE–MG).     

On the Numerical Solution of Some Semilinear Elliptic Problems – II

In the earlier paper [6], a Galerkin method was proposed and analyzed for the numerical solution of a Dirichlet problem for a semi-linear elliptic boundary value problem of the form –U=F(·,U). This was converted to a problem on a standard domain and then converted to an equivalent integral equation. Galerkins method was used to solve the integral equation, with the eigenfunctions of the Laplacian operator on the standard domain D as the basis functions. In this paper we consider the implementing of this scheme, and we illustrate it for some standard domains D.     

Error analysis of thermal equation with moving ends

A mathematical model of the linear thermodynamic equations with moving ends, based on the Stefan Problem is considered. In this work, we are interested in obtaining existence, uniqueness and regularity using the Galerkin method and mainly to establish an error estimates of solutions in Sobolev spaces for the semi-discrete problem, with discretization of space variable, continuous time and for the completely discrete problem with discretization of space variable and time.     

2D-discontinuity detection from scattered data

We describe a numerical approach for the detection of discontinuities of a two dimensional function distorted by noise. This problem arises in many applications as computer vision, geology, signal processing. The method we propose is based on the two-dimensional continuous wavelet transform and follows partially the ideas developed in [2], [6] and [8]. It is well-known that the wavelet transform modulus maxima locate the discontinuity points and the sharp variation points as well. Here we propose a statistical test which, for a suitable scale value, allows us to decide if a wavelet transform modulus maximum corresponds to a function value discontinuity. Then we provide an algorithm to detect the discontinuity curves fromscattered and noisy data.     

Estimation in a linear multivariate measurement error model with a change point in the data

A linear multivariate measurement error model AX=B is considered. The errors in are row-wise finite dependent, and within each row, the errors may be correlated. Some of the columns may be observed without errors, and in addition the error covariance matrix may differ from row to row. The columns of the error matrix are united into two uncorrelated blocks, and in each block, the total covariance structure is supposed to be known up to a corresponding scalar factor. Moreover the row data are clustered into two groups, according to the behavior of the rows of true A matrix. The change point is unknown and estimated in the paper. After that, based on the method of corrected objective function, strongly consistent estimators of the scalar factors and X are constructed, as the numbers of rows in the clusters tend to infinity. Since Toeplitz/Hankel structure is allowed, the results are applicable to system identification, with a change point in the input data.     

An Inverse Transmission Scattering Problem and the Enclosure Method

We consider an inverse scattering problem in two-dimensions for a penetrable polygonal obstacle having different density from the back ground medium, however, the speed of sound is constant in the whole space. Using a single set of the Cauchy data of the response for a single incident plane wave with a fixed wave number on a circle surrounding the obstacle, we give an extraction formula of the convex hull of the obstacle. An algorithm based on the formula is also described.     

Identifying a control function in parabolic partial differential equations from overspecified boundary data

Determination of an unknown time-dependent function in parabolic partial differential equations, plays a very important role in many branches of science and engineering. In the current investigation, the Adomian decomposition method is used for finding a control parameter p(t) in the quasilinear parabolic equation u_t=u_xx+p(t)u+, in [0,1]×(0,T] with known initial and boundary conditions and subject to an additional condition in the form of which is called the boundary integral overspecification. The main approach is to change this inverse problem to a direct problem and then solve the resulting equation using the well known Adomian decomposition method. The decomposition procedure of Adomian provides the solution in a rapidly convergent series where the series may lead to the solution in a closed form. Furthermore due to the rapid convergence of Adomian’s method, a truncation of the series solution with sufficiently large number of implemented components can be considered as an accurate approximation of the exact solution. This method provides a reliable algorithm that requires less work if compared with the traditional techniques. Some illustrative examples are presented to show the efficiency of the presented method.     

Regularization methods for a problem of analytic continuation

In this paper, we prove a sharp stability estimate for the problem of analytic continuation. Based on the obtained stability estimate, a generalized Tikhonov regularization is provided and the corresponding error estimate is obtained. Moreover, we give many other regularization methods. For illustration, a numerical experiment is constructed to demonstrate the feasibility and efficiency of the proposed method.     

On the linear advection equation subject to random velocity fields

This paper deals with the random linear advection equation for which the time-dependent velocity and the initial condition are independent random functions. Expressions for the density and joint density functions of the solution are given. We also verify that in the Gaussian time-dependent velocity case the probability density function of the solution satisfies a convection-diffusion equation with a time-dependent diffusion coefficient. Some exact examples are presented.     

A Hermitian Box-scheme for One-dimensional Elliptic Equations – Application to Problems with High Contrasts in the Ellipticity

We introduce a new box-scheme, called ``hermitian box-scheme' on the model of the one-dimensional Poisson problem. The scheme combines features of the box-scheme of Keller, [20], [13], with the hermitian approximation of the gradient on a compact stencil, which is characteristic of compact schemes, [9], [21]. The resulting scheme is proved to be 4th order accurate for the primitive unknown u and its gradient p. The proved convergence rate is 1.5 for (u,p) in the discrete L ² norm. The connection with a non standard mixed finite element method is given. Finally, numerical results are displayed on pertinent 1-D elliptic problems with high contrasts in the ellipticity, showing in practice convergence rates ranging from 1 to 2.5 in the discrete H ¹ norm. This work has been performed with the support of the GDR MOMAS, (ANDRA, CEA, EDF, BRGM and CNRS): Modélisation pour le stockage des déchets radioactifs. The author thanks especially A. Bourgeat for his encouragements and his interest in this work.     

Error estimation in the approximation of the solution of nonlinear fuzzy Fredholm integral equations

Using a fixed point technique, the sequence of successive approximations and a recent quadrature formula for fuzzy-number-valued functions, it is constructed a numerical method for the solution of nonlinear fuzzy Fredholm integral equations. Moreover, the error estimate of the method and a criterion to stop the corresponding algorithm are given.     

A Posteriori Error Estimation for the Dirichlet Problem with Account of the Error in the Approximation of Boundary Conditions

H ¹, independently of the discretization method chosen. In particular, our error estimator can be applied also to problems and discretizations where the Galerkin orthogonality is not available. We will present different strategies for the evaluation of the error estimator. Only one constant appears in its definition which is the one from Friedrichs' inequality; that constant depends solely on the domain geometry, and the estimator is quite non-sensitive to the error in the constant evaluation. Finally, we show how accurately the estimator captures the local error distribution, thus, creating a base for a justified adaptivity of an approximation. Received April 15, 2002; revised March 10, 2003 Published online: June 23, 2003     

Optimal design and operation of a wastewater purification system

Due to the importance of coastal areas, is of the highest interest to implement purification systems that with minimum cost are able to assure water quality standards in agreement with the regional legislations. This work addresses the optimal design (outfall locations) and optimal operation (level of oxygen discharges) of a wastewater treatment system. This problem can be mathematically formulated as a two-objective mixed design and optimal control problem with constraints on the states and the design and control variables.