Stability estimates of solutions to extremal problems for a nonlinear convection-diffusion-reaction equation

We consider an identification problem for a stationary nonlinear convection–diffusion–reaction equation in which the reaction coefficient depends nonlinearly on the concentration of the substance. This problem is reduced to an inverse extremal problem by an optimization method. The solvability of the boundary value problem and the extremal problem is proved. In the case that the reaction coefficient is quadratic, when the equation acquires cubic nonlinearity, we deduce an optimality system. Analyzing it, we establish some estimates of the local stability of solutions to the extremal problem under small perturbations both of the quality functional and the given velocity vector which occurs multiplicatively in the convection–diffusion–reaction equation.     

On stability of solutions of the coefficient inverse extremal problems for the stationary convection-diffusion equation

The coefficient inverse extremal problems are studied for the stationary convectiondiffusion equation in a bounded domain under mixed boundary conditions on the boundary of the domain. The role of control is played by the velocity vector of a medium and the functions that are involved in the boundary conditions for temperature. The solvability of the extremal problems is proven both for an arbitrary weakly lower semicontinuous quality functional and for the particular quality functionals. On the basis of analysis of the optimality system some sufficient conditions are established on the initial data providing the uniqueness and stability of optimal solutions under sufficiently small perturbations of both the quality functional and one of the functions involved in the original boundary value problem.     

An inverse problem for the Helmholtz equation

The paper considers the problem of optimal determination of linear functionals of the source intensity under various assumptions. Some theorems on optimal estimates are proved and estimation errors are determined.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 66, pp. 10–17, 1988.     

Stability estimates in identification problems for the convection-diffusion-reaction equation

Identification problems for the stationary convection-diffusion-reaction equation in a bounded domain with a Dirichlet condition imposed on the boundary of the domain are studied. By applying an optimization method, these problems are reduced to inverse extremum problems in which the variable diffusivity and the volume density of substance sources are used as control functions. Their solvability is proved for an arbitrary weakly lower semicontinuous cost functional and particular cost functionals. An analysis of the optimality system is used to establish sufficient conditions on the input data under which the solutions of particular extremum problems are unique and stable with respect to small perturbations in the cost functional and in one of the functions involved in the boundary value problem.     

Interpolation for solutions of the Helmholtz equation

We study the interpolation problem for solutions of the two-dimensional Helmholtz equation, which are sampled along a line. The data are the function values and the normal derivatives at a discrete set of point sensors. A wave transform is used, analogous to the common Fourier transform. The inverse wave transform defines the Hilbert space for oscillatory Helmholtz solutions. We thereby introduce an interpolant that has some advantages over the usual sinc x in the Whittaker–Shannon sampling in one dimension; in particular, coefficients of the two-dimensional solution are invariant under translations and rotations of the sampling line. The analysis is relevant for the optical sampling problem by sensors on a screen. © 1995 John Wiley & Sons, Inc.     

Stability of solutions to extremal problems of boundary control for stationary heat convection equations

We study extremal problems of boundary control for stationary heat convection equations with Dirichlet boundary conditions on velocity and temperature. As the cost functional we choose the mean square integral deviation of the required temperature field from a given temperature field measured in some part of the flow region. The controls are functions appearing in the Dirichlet conditions on velocity and temperature. We prove the stability of solutions to these problems with respect to certain perturbations of both the quality functional and one of the known functions appearing in the original equations of the model.     

Two canonical wedge problems for the Helmholtz equation

Boundary-transmission problems for two-dimensional Helmholtz equations in a quadrant and its complement, respectively, are considered in a Sobolev space setting. The first problem of a quadrant with Dirichlet condition on one face and transmission condition on the other is solved in closed form for the case where all the quadrants are occupied by the same medium. Unique solvability can also be shown in the case of two different media up to exceptional cases of wave numbers, while the Fredholm property holds in general. In the second problem, transmission conditions are prescribed on both faces. Similar results are obtained in the one-medium case, but the two-media case turns out to be more complicated and the equivalent system of boundary pseudodifferential equations cannot be completely analysed by this reasoning.     

Maximal estimates for solutions to the nonelliptic Schrodinger equation

Maximal estimates are studied for solutions to an initial valueproblem for the nonelliptic Schrödinger equation. A resultof Rogers, Vargas and Vega is extended.     

On extremal solutions to stochastic control problems

We identify two solutions of a controlled diffusion if the corresponding one-dimensional marginals of the state and control process agree. The extreme points of the set of such equivalence classes are shown to correspond to Markov controls.     

On the extremal solutions for capacitated network problems

In this note we give a unifying approach to the problem of characterizing the extreme points of those convex matrix sets which correspond to the domains of various types of capacitated network problems. It is shown that we can determine whether a matrix is an extreme point of the sets by examining the pattern of a certain graph associated with it. We also study the extreme points of the convex matrix sets which are related to network problems free from capacity constraints by linking them up with certain capacitated network problem.     

Some special solutions of the Helmholtz equation

Functions are constructed which describe a cylindrical wave diverging from the origin and tangent to a semi-infinite plane wave. The use of such functions is illustrated by the example of the problem of diffraction of a plane wave by a semi-infinite screen.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 51, pp. 197–202, 1975.     

Stability estimates for the solutions of control problems for the stationary magnetohydrodynamic equations

We study control problems for the stationary magnetohydrodynamic equations. In these problems, one has to find an unknown vector function occurring in the boundary condition for the magnetic field and the solution of the boundary value problem in question by minimizing a performance functional depending on the velocity and pressure. We derive new a priori estimates for the solutions of the original boundary value problem and the extremal problem and prove theorems on the local uniqueness and stability of solutions for specific performance functionals.     

Stability estimates in inverse scattering

An algorithm is given for calculating the solution to the 3D inverse scattering problem with noisy discrete fixed energy data. The error estimates for the calculated solution are derived. The methods developed are of a general nature and can be used in many applications: in nondestructive evaluation and remote sensing, in geophysical exploration, medical diagnostics, and technology.     

Hessian estimates for convex solutions to quadratic Hessian equation

We derive Hessian estimates for convex solutions to quadratic Hessian equation by compactness argument.     

Potential-based numerical solution of Dirichlet problems for the Helmholtz equation

Three-dimensional Dirichlet problems for the Helmholtz equation are considered in generalized formulations. By applying single-layer potentials, they are reduced to Fredholm boundary integral equations of the first kind. The equations are discretized using a special averaging method for integral operators with weak singularities in the kernels. As a result, the integral equations are approximated by systems of linear algebraic equations with easy-to-compute coefficients, which are solved numerically by applying the generalized minimal residual method. A modification of the method is proposed that yields solutions in the spectra of interior Dirichlet problems and integral operators when the integral equations are not equivalent to the original differential problems and are not well-posed. Numerical results are presented for assessing the capabilities of the approach.     

Generalized exterior boundary-value problems and optimization for the Helmholtz equation

A class of radiation problems is considered for the Helmholtz equation in exterior domains bounded by a smooth surface on which Dirichlet, Neumann, or Robin boundary conditions are imposed. The problem of finding the boundary data which maximizes far field power in a restricted subset of far field directions is formulated as a constrained maximization problem. Existence of an optimal solution in a variety of control domains is established. The particular case when the boundary is circular and the control domain is the unit ball inL ² is treated in detail. An algorithm for constructing the optimal solution is derived and used to obtain explicit numerical results.This work was supported by the US Air Force under Grant No. AFOSR 81-0156. The work was completed while the first author was on leave to the Institut für Numerische und Angewandte Mathematik, Universität Göttingen, Göttingen, BRD.