Variational Approach to Scattering by Inhomogeneous Layers Above Rough Surfaces

In this paper, we study, via variational methods, the problem of scattering of time harmonic acoustic waves by unbounded inhomogeneous layers above a sound soft rough surface. We first propose a variational formulation and exploit it as a theoretical tool to prove the well-posedness of this problem when the media is non-absorbing for arbitrary wave number and obtain an estimate about the solution, which exhibit explicitly dependence of bound on the wave number and on the geometry of the domain. Then, based on the non-absorbing results, we show that the variational problem remains uniquely solvable when the layer is absorbing by means of a priori estimate of the solution. Finally, we consider the finite element approximation of the problem and give an error estimate.     

An acoustic scattering problem for periodic,inhomogeneous media

We study the problem of the scattering by a periodic, inhomogeneous, penetrable medium. Using the Dirichlet-to-Neumann operator from the classical formulation of the problem we derive a variational equation and give regularity result to show the equivalence of both formulations. We present certain uniqueness results, which by the Fredholm alternative yield existence of the solution and its continuous dependence on the incoming wave. We prove existence of a solution for special incident waves even if there is no uniqueness. A result about analytical dependence of the solution on the wave number and the incident angle is given. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Instability of standing wave, global existence and blowup for the Klein-Gordon-Zakharov system with different-degree nonlinearities

This paper discusses the Klein-Gordon-Zakharov system with different-degree nonlinearities in two and three space dimensions. Firstly, we prove the existence of standing wave with ground state by applying an intricate variational argument. Next, by introducing an auxiliary functional and an equivalent minimization problem, we obtain two invariant manifolds under the solution flow generated by the Cauchy problem to the aforementioned Klein-Gordon-Zakharov system. Furthermore, by constructing a type of constrained variational problem, utilizing the above two invariant manifolds as well as applying potential well argument and concavity method, we derive a sharp threshold for global existence and blowup. Then, combining the above results, we obtain two conclusions of how small the initial data are for the solution to exist globally by using dilation transformation. Finally, we prove a modified instability of standing wave to the system under study.     

Bounds and estimates on the effective properties for nonlinear composites

In this paper we derive lower bounds and upper bounds on the effective properties for nonlinear heterogeneous systems. The key result to obtain these bounds is to derive a variational principle, which generalizes the variational principle by P. Ponte Castaneda from 1992. In general, when the Ponte Castaneda variational principle is used one only gets either a lower or an upper bound depending on the growth conditions. In this paper we overcome this problem by using our new variational principle together with the bounds presented by Lukkassen, Persson and Wall in 1995. Moreover, we also present some examples where the bounds are so tight that they may be used as a good estimate of the effective behavior.     

Finite convergence analysis and weak sharp solutions for variational inequalities

In this paper, we study the weak sharpness of the solution set of variational inequality problem (in short, VIP) and the finite convergence property of the sequence generated by some algorithm for finding the solutions of VIP. In particular, we give some characterizations of weak sharpness of the solution set of VIP without considering the primal or dual gap function. We establish an abstract result on the finite convergence property for a sequence generated by some iterative methods. We then apply such abstract result to discuss the finite termination property of the sequence generated by proximal point method, exact proximal point method and gradient projection method. We also give an estimate on the number of iterates by which the sequence converges to a solution of the VIP.     

An ultra-weak method for acoustic fluid–solid interaction

We introduce the ultra-weak variational formulation (UWVF) for fluid–solid vibration problems. In particular, we consider the scattering of time-harmonic acoustic pressure waves from solid, elastic objects. The problem is modeled using a coupled system of the Helmholtz and Navier equations. The transmission conditions on the fluid–solid interface are represented in an impedance-type form after which we can employ the well known ultra-weak formulations for the Helmholtz and Navier equations. The UWVF approximation for both equations is computed using a superposition of propagating plane waves. A condition number based criterion is used to define the plane wave basis dimension for each element. As a model problem we investigate the scattering of sound from an infinite elastic cylinder immersed in a fluid. A comparison of the UWVF approximation with the analytical solution shows that the method provides a means for solving wave problems on relatively coarse meshes. However, particular care is needed when the method is used for problems at frequencies near the resonance frequencies of the fluid–solid system.     

Evolution variational inequalities and projected dynamical systems with application to human migration

In this paper, we explore the relationship between projected dynamical systems and evolution variational inequalities (also sometimes referred to as parabolic variational inequalities). The methodology of evolution variational inequalities is then utilized for the first time to model the dynamic adjustment of a socio-economic process in the context of human migration. The questions of dynamics and convergence of algorithms in this framework are addressed and answered. In particular, we provide existence and uniqueness results for the solution path without assuming Lipschitz continuity and propose a finite-difference scheme for the solution of the human migration problem. The algorithm, an ordinary implicit scheme, is a discrete-time version of the model. Its convergence estimate is also established.     

On the coupling of regularization techniques and the boundary element method for a hemivariational inequality modelling a delamination problem

In this paper, we couple regularization techniques of nondifferentiable optimization with the h‐version of the boundary element method (h‐BEM) to solve nonsmooth variational problems arising in contact mechanics. As a model example, we consider the delamination problem. The variational formulation of this problem leads to a hemivariational inequality with a nonsmooth functional defined on the contact boundary. This problem is first regularized and then discretized by an h‐BEM. We prove convergence of the h‐BEM Galerkin solution of the regularized problem in the energy norm, provide an a priori error estimate and give a numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

On finite convergence of proximal point algorithms for variational inequalities

In this paper, we first characterize finite convergence of an arbitrary iterative algorithm for solving the variational inequality problem (VIP), where the finite convergence means that the algorithm can find an exact solution of the problem in a finite number of iterations. By using this result, we obtain that the well-known proximal point algorithm possesses finite convergence if the solution set of VIP is weakly sharp. As an extension, we show finite convergence of the inertial proximal method for solving the general variational inequality problem under the condition of weak g-sharpness.     

An overfilled cavity problem for Maxwell's equations

This paper is concerned with the mathematical analysis of the scattering of a time‐harmonic electromagnetic plane wave by an open and overfilled cavity that is embedded in a perfect electrically conducting infinite ground plane, where the electromagnetic wave propagation is governed by the Maxwell equations. Above the flat ground surface and the open aperture of the cavity, the space is assumed to be filled with a homogeneous medium with a constant permittivity and permeability, whereas the interior of the cavity is filled with some inhomogeneous medium with a variable permittivity and permeability. The scattering problem is modeled as a boundary value problem over a bounded domain, with transparent boundary condition proposed on the hemisphere enclosing the inhomogeneity represented by the cavity. The existence and uniqueness of the weak solution for the model problem are established by using a variational approach. The perfectly matched layer (PML) method is investigated to truncate the unbounded electromagnetic cavity scattering problem. It is shown that the truncated PML problem attains a unique solution. An explicit error estimate is given between the solution of the original scattering problem and that of the truncated PML problem. The error estimate implies that the PML solution converges exponentially to the original cavity scattering problem by increasing either the PML medium parameter or the PML layer thickness. The convergence result is expected to be useful for determining the PML medium parameter in the computational electromagnetic scattering problem. Copyright © 2012 John Wiley & Sons, Ltd.     

Diffraction of a plane wave by a circular disk

In this paper we solve the problem of diffraction of a normally incident plane wave by a circular disk. We treat both the hard and soft disk. In each case we obtain the solution as a series which converges when the product of the wave number and the radius of the disk is large. Our construction leads directly to asymptotic approximations to the solution for large wave number.     

An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping

In this paper, we introduce and study an iterative method to approximate a common solution of split variational inclusion problem and fixed point problem for a nonexpansive mapping in real Hilbert spaces. Further, we prove that the sequences generated by the proposed iterative method converge strongly to a common solution of split variational inclusion problem and fixed point problem for a nonexpansive mapping which is the unique solution of the variational inequality problem. The results presented in this paper are the supplement, extension and generalization of the previously known results in this area.     

The mathematics of scattering by unbounded,rough, inhomogeneous layers

In this paper we study, via variational methods, a boundary value problem for the Helmholtz equation modelling scattering of time harmonic waves by a layer of spatially varying refractive index above an unbounded rough surface on which the field vanishes. In particular, in the 2D case with TE polarization, the boundary value problem models the scattering of time harmonic electromagnetic waves by an inhomogeneous conducting or dielectric layer above a perfectly conducting unbounded rough surface, with the magnetic permeability a fixed positive constant in the medium. Via analysis of an equivalent variational formulation, we show that this problem is well-posed in two important cases: when the frequency is small enough; and when the medium in the layer has some energy absorption. In this latter case we also establish exponential decay of the solution with depth in the layer. An attractive feature is that all constants in our estimates are bounded by explicit functions of the index of refraction and the geometry of the scatterer.     

Error bound results for generalized D-gap functions of nonsmooth variational inequality problems

Solving a variational inequality problem can be equivalently reformulated into solving a unconstraint optimization problem where the corresponding objective function is called a merit function. An important class of merit function is the generalized D-gap function introduced in [N. Yamashita, K. Taji, M. Fukushima, Unconstrained optimization reformulations of variational inequality problems, J. Optim. Theory Appl. 92 (1997) 439-456] and Yamashita and Fukushima (1997) [17]. In this paper, we present new fractional local/global error bound results for the generalized D-gap functions of nonsmooth variational inequality problems, which gives an effective estimate on the distance between a specific point to the solution set, in terms of the corresponding function value of the generalized D-gap function. Numerical examples and a simple application to the free boundary problem are also presented to illustrate the significance of our error bound results.     

Fourier Moment Method with Regularization for the Cauchy Problem of Helmholtz Equation

In this paper, we consider the reconstruction of the wave field in a bounded domain. By choosing a special family of functions, the Cauchy problem can be transformed into a Fourier moment problem. This problem is ill-posed. We propose a regularization method for obtaining an approximate solution to the wave field on the unspecified boundary. We also give the convergence analysis and error estimate of the numerical algorithm. Finally, we present some numerical examples to show the effectiveness of this method.     

A one-sweep method for linear elliptic equations over irregular domains

The determination of the configuration of equilibrium in a number of problems in mechanics and structures such as torsion, deflection of elastic membranes,etc., involve the solution of variational problems defined over irregular regions. This problem, in turn, may be reduced to the solution of elliptic differential equations subject to boundary conditions. In this paper, we study a method for the solution of such a problem when the region is of irregular shape. The method consists in solving the problem over a larger, imbedding, rectangular domain subject to appropriate constraints such as to satisfy the conditions of the original problem at the boundary. In this paper, we introduce the constraints by considering appropriate factors on the Green's function of the auxiliary problem. A conveniently discretized version of the problem is then treated by invariant imbedding, yielding some earlier results plus some new ones, namely, a direct one-sweep procedure that minimizes storage requirements. In addition, the present solution appears to be very convenient when the solution is required at a limited number of points. The derivations are specialized to Laplace's equation, but the method can be applied readily to general systems of second-order elliptic equations with no essential modifications. Finally, the existence of the necessary matrices in the imbedding equations is established.     

On two-coefficient identification in elliptic variational inequalities

Abstract

This paper is concerned with elliptic variational inequalities that depend on two parameters. First, we investigate the dependence of the solution of the forward problem on these parameters and prove a Lipschitz estimate. Then, we study the inverse problem of identification of these two parameters and formulate two optimization approaches to this parameter identification problem. We extend the output least-squares approach, provide an existence result and establish a convergence result for finite-dimensional approximation. Further, we investigate the modified output least-squares approach which is based on energy functionals. This latter approach can be related to vector approximation.     

A mountain pass method for the numerical solution of semilinear wave equations

Summary It is well-known that periodic solutions of semilinear wave equations can be obtained as critical points of related functionals. In the situation that we studied, there is usually an obvious solution obtained as a solution of linear problem. We formulate a dual variational problem in such a way that the obvious solution is a local minimum. We then find additional non-obvious solutions via a numerical mountain pass algorithm, based on the theorems of Ambrosetti, Rabinowitz and Ekeland. Numerical results are presented.Research supported in part by grant DMS-9208636 from the National Science FoundationResearch supported in part by grant DMS-9102632 from the National Science Foundation     

The role of conditioning in mesh selection algorithms for first order systems of linear two point boundary value problems

Codes for the numerical solution of two-point boundary value problems can now handle quite general problems in a fairly routine and reliable manner. When faced with particularly challenging equations, such as singular perturbation problems, the most efficient codes use a highly non-uniform grid in order to resolve the non-smooth parts of the solution trajectory. This grid is usually constructed using either a pointwise local error estimate defined at the grid points or else by using a local residual control. Similar error estimates are used to decide whether or not to accept a solution. Such an approach is very effective in general providing that the problem to be solved is well conditioned. However, if the problem is ill conditioned then such grid refinement algorithms may be inefficient because many iterations may be required to reach a suitable mesh on which to compute the solution. Even worse, for ill conditioned problems an inaccurate solution may be accepted even though the local error estimates may be perfectly satisfactory in that they are less than a prescribed tolerance. The primary reason for this is, of course, that for ill conditioned problems a small local error at each grid point may not produce a correspondingly small global error in the solution. In view of this it could be argued that, when solving a two-point boundary value problem in cases where we have no idea of its conditioning, we should provide an estimate of the condition number of the problem as well as the numerical solution. In this paper we consider some algorithms for estimating the condition number of boundary value problems and show how this estimate can be used in the grid refinement algorithm.     

A BOUNDARY ELEMENT PROCEDURE FOR THE SIGNORINI PROBLEM IN THREE-DIMENSIONAL ELASTICITY

In this paper, a new numerical method for the Signorini problem in three-dimensional elasticity is presented. The problem is reduced to a boundary variational inequality based on a new representation of the derivative of the doublelayer potential. Furthermore, a boundary element procedure is described for the numerical approximation of its solution and an abstract error estimate is given.