Neumann Data Research Articles

Revisiting the probe and enclosure methods

This paper is concerned with reconstruction issue of inverse obstacle problems governed by partial differential equations and consists of two parts. (i) The first part considers the foundation of the probe and enclosure methods for an impenetrable obstacle embedded in a medium governed by the stationary Schrödinger equation. Under a general framework, some natural estimates for a quantity computed from a pair of the Dirichlet and Neumann data on the outer surface of the body occupied by the medium are given. The estimates enables us to derive almost immediately the necessary asymptotic behaviour of indicator functions for both methods. (ii) The second one considers the realization of the enclosure method for a penetrable obstacle embedded in an absorbing medium governed by the stationary Schrödinger equation. The unknown obstacle considered here is modeled by a perturbation term added to the background complex-valued L ∞-potential. Under a jump condition on the term across the boundary of the obstacle and some kind of regularity for the obstacle surface including Lipschitz one as a special case, the enclosure method using infinitely many pairs of the Dirichlet and Neumann data is established.

Solutions for non-homogeneous wave equations subject to unusual and Neumann boundary conditions

This paper examines the accuracy, stability, and convergence of the forward and inverse solutions for non-homogeneous wave equations in different cases. The first case is when the wave governing equation is subject to the so-called unusual (non-local/non-classical) boundary condition. Where there is a non-local displacement and flux tension boundary data are placed at the very left end of the examined domain (string or rob), combined with Dirichlet and Neumann data at blue the left end of the domain. We explore the ideal value for the rational stiffness coefficient to be used for the second case which concerns moving Dirichlet data to the right and adding over-determination Neumann boundary conditions to the right or left. The forward solutions for the given problems, represented in the flux tension values, are investigated using the finite difference method (FDM) combined with the separation variable method. We also test the effect of the rotational stiffness coefficient on the effectiveness and convergence of those direct solutions. On the other hand, to obtain stable numerical results for the corresponding inverse problems (concerning retrieval of the force source function), the zeroth, the first, and the second-order of the Tikhonov regularization parameters are tested and the minimum error norm is calculated accordingly to find the best estimate for that space-dependent force source. Because such inverse problems are ill-posed, the existence and uniqueness of their solutions are tolerated and verified based on existing theorems in literature. This work includes numerical examples which illustrate the usefulness of the approach for solving non-homogeneous wave equations under novel kinds of boundary conditions.

The Korteweg–de Vries equation on the half-line with Robin and Neumann data in low regularity spaces

The well-posedness of the initial–boundary value problem (ibvp) for the Korteweg–de Vries equation on the half-line is studied for initial data u0(x) in spatial Sobolev spaces Hs(0,∞), s>−3/4, and Robin and Neumann boundary data φ(t) in the temporal Sobolev spaces suggested by the time regularity of the Cauchy problem for the corresponding linear equation. First, linear estimates in Bourgain spaces are derived by utilizing the Fokas solution formula of the ibvp for the forced linear equation. Then, using these and the needed bilinear estimates, it is shown that the iteration map defined by the Fokas solution formula is a contraction in an appropriate solution space.

Maximal L^1-regularity for parabolic initial-boundary value problems with inhomogeneous data

End-point maximal L^1-regularity for parabolic initial-boundary value problems is considered. For the inhomogeneous Dirichlet and Neumann data, maximal L^1-regularity for initial-boundary value problems is established in time end-point case upon the homogeneous Besov space {dot{B}}_{p,1}^s({mathbb {R}}^n_+) with 1< p< infty and -1+1/p<sle 0 as well as optimal trace estimates. The main estimates obtained here are sharp in the sense of trace estimates and it is not available by known theory on the class of UMD Banach spaces. We utilize a method of harmonic analysis, in particular, the almost orthogonal properties between the boundary potentials of the Dirichlet and the Neumann boundary data and the Littlewood-Paley dyadic decomposition of unity in the Besov and the Lizorkin–Triebel spaces.

Non-iterative domain decomposition for the Helmholtz equation with strong material discontinuities

Many wave propagation problems involve discontinuous material properties. We propose to solve such problems by non-overlapping domain decomposition combined with the method of difference potentials (MDP). The MDP reduces the Helmholtz equation on each subdomain to a Calderon's boundary equation with projection on the boundary. The unknowns for the Calderon's equation are the Dirichlet and Neumann data. Coupling between neighboring subdomains is rendered by applying their respective Calderon's equations to the same data at the common interface. Solutions on individual subdomains are computed concurrently using a direct solver. Our method proves to be insensitive to large jumps in the wavenumber for transmission problems, as well as interior cross-points and mixed boundary conditions, which may be a challenge to many other domain decomposition methods.

Subsurface target illumination in elastic hosts: a comparison of time-reversal and inverse-source approaches

The need for focusing wave energy to selected targets embedded within a host is commonly shared among various engineering fields, whether for stimulation and imaging in exploration geophysics, for contaminant removal in geo-environmental engineering, or for diagnostic/therapeutic purposes in medicine. In this presentation, we discuss the problem of focusing wave energy to multiple subsurface targets embedded within a semi-infinite heterogeneous elastic host. We review the advantages and disadvantages of two distinct approaches: first, using the apparatus of PDE-constrained optimization, we describe an inverse source (IS) approach, where we seek to design optimal excitations (both in space and time) in order to maximize target illumination. The second approach is rooted on the time-reversibility (TR) of the lossless wave equation; however the unboundness of the host, and equipment limitations that necessitate that recorded Dirichlet data be time reversed as Neumann data, create particularly adverse conditions for time-reversal. To overcome, we discuss a switching time-reversing mirror, and report on numerical experiments that show illumination resolution improvement. We define target illumination metrics to assess the relative performance of the IS and TR methods, and, using numerical experiments, we show that both approaches have similar efficiency.

Serrin’s overdetermined problem for fully nonlinear nonelliptic equations

Let $u$ denote a solution to a rotationally invariant Hessian equation $F(D^2u)=0$ on a bounded simply connected domain $\Omega\subset R^2$, with constant Dirichlet and Neumann data on $\partial \Omega$. In this paper we prove that if $u$ is real analytic and not identically zero, then $u$ is radial and $\Omega$ is a disk. The fully nonlinear operator $F\not\equiv 0$ is of general type, and in particular, not assumed to be elliptic. We also show that the result is sharp, in the sense that it is not true if $\Omega$ is not simply connected, or if $u$ is $C^{\infty}$ but not real analytic.

Uniqueness in determining fractional orders of derivatives and initial values

For initial boundary value problems of time-fractional diffusion-wave equations with the zero Dirichlet boundary value, we consider an inverse problem of determining orders of the fractional derivatives and initial values by three kinds of measurement data on a time interval: (i) solution in a subdomain (ii) Neumann data on a subboundary (iii) spatial averaged values. We prove that the order and initial values are uniquely determined by the above data if the solutions do not identically vanish. Our main results show that the uniqueness for the fractional order holds even if the coefficients of the fractional diffusion-wave equations and source terms within some class are unknown. Moreover we consider pointwise data for the uniqueness in the inverse problem. The proof is based on the eigenfunction expansions and the asymptotic expansions of the Mittag–Leffler functions for large time.

Homogenization of a boundary optimal control problem governed by Stokes equations

This article considers an optimal control problem for the stationary Stokes system in a three-dimensional domain with a highly oscillating boundary. The controls are acting on the state through the Neumann data on the oscillating part of the boundary with appropriate scaling parameters with . The periodic unfolding operators are used to characterize the optimal controls. Using the unfolding operators, we analyse the asymptotic behaviour of the optimal control problem under consideration. For , the limit optimal control problem has both boundary and interior controls. For , the limit optimal control problem has only boundary controls.

An inverse problem for an elliptic equation in a spherical domain

This paper deals with the problem of determining a radially dependent coefficient in the equation in the unit sphere Ω from the Dirichlet–Neumann data pair . We discuss the uniqueness of this determination.

Convergence for a planar elliptic problem with large exponent Neumann data

We study positive solutions up of the nonlinear Neumann elliptic problem Δu=u in Ω, ∂u/∂ν=|u|p−1u on ∂Ω, where Ω is a bounded open smooth domain in R2. We investigate the asymptotic behavior of families of solutions up satisfying an energy bound condition when the exponent p is getting large. Inspired by the work of Davila-del Pino-Musso [8], we prove that up is developing m peaks xi∈∂Ω, in the sense upp/∫∂Ωupp approaches the sum of m Dirac masses at the boundary and we determine the localization of these concentration points.

A novel numerical method for the hydrodynamic analysis of floating bodies over a sloping bottom

A novel numerical method combining Eigenfunction matching method (EMM) and 3D Rankine source method is developed to investigate wave-body interaction over a sloping bottom. The extended EMM is proposed to create an incident wave model over the sloping bottom, thereby obtaining the Froude-Krylov force and Neumann data on wet surfaces of the floating body for the diffraction problem. A 3D Rankine source method concerning the sloping bottom is developed, in which the free surface and seabed are both divided into the inner domain and outer domain. Source panels are placed in an exponential manner in the latter domain, by which the far field radiation condition is well satisfied. To verify the proposed method, comparisons with other mathematical models involving added mass and damping coefficients, wave exciting forces and motion RAOs by a floating hemisphere and a LNG carrier are carried out.

On the Numerical Solution of a Time-Dependent Shape Optimization Problem for the Heat Equation

This article is concerned with the solution of a time-dependent shape identification problem. Specifically, we consider the heat equation in a domain which contains a time-dependent inclusion of zero temperature. The objective is to detect this inclusion from the given temperature and heat flux at the exterior boundary of the domain. To this end, for a given temperature at the exterior boundary, the mismatch of the Neumann data is minimized. This time-dependent shape optimization problem is then solved by a gradient-based optimization method. Numerical results are presented which validate the present approach.

Piecewise Smooth Stationary Euler Flows with Compact Support Via Overdetermined Boundary Problems

We construct new stationary weak solutions of the 3D Euler equation with compact support. The solutions, which are piecewise smooth and discontinuous across a surface, are axisymmetric with swirl. The range of solutions we find is different from, and larger than, the family of smooth stationary solutions recently obtained by Gavrilov and Constantin–La–Vicol; in particular, these solutions are not localizable. A key step in the proof is the construction of solutions to an overdetermined elliptic boundary value problem where one prescribes both Dirichlet and (nonconstant) Neumann data.

The stability for an inverse problem of bottom recovering in water-waves

In this article we deal with a class of geometric inverse problem for bottom detection by one single measurement on the free surface in water-waves. We found upper and lower bounds for the size of the region enclosed between two different bottoms, in terms of Neumann and/or Dirichlet data on the free surface. Starting from the general water-waves system in bounded domains with side walls, we manage to formulate the problem in terms of the Dirichlet to Neumann operator and thus, as an elliptic problem in a bounded domain with Neumann homogeneous condition on the rigid boundary. Then we study the properties of the Dirichlet to Neumann map and analyze the called method of size estimation.

A homogenization function technique to solve the 3D inverse Cauchy problem of elliptic type equations in a closed walled shell

We solve the inverse Cauchy problem of elliptic type partial differential equations in an arbitrary 3D closed walled shell for recovering unknown data on an inner surface, with the over-specified Cauchy boundary conditions given on an outer surface. We first derive a homogenization function in the 3D domain to annihilate the Dirichlet as well as the Neumann data on the outer surface. Then, we can transform the inverse Cauchy problem to solve a direct problem inside the closed walled shell, using the homogenization technique and the domain type collocation method. The boundary functions are constructed from the 3D Pascal polynomials multiplied by an elementary boundary function, which are adopted as the bases to expand the numerical solution of the transformed elliptic equation. A simple scaling regularization is employed to reduce the condition number of the linear system to determine the expansion coefficients. Several numerical examples are presented to show that the novel method can overcome the highly ill-posed property of the inverse Cauchy problem in the 3D closed walled shell. The proposed algorithm is robust against large noise up to and very time saving to obtain accurate solution.

Active manipulation of Helmholtz scalar fields: near-field synthesis with directional far-field control

In this article, we propose a strategy for the active manipulation of scalar Helmholtz fields in bounded near-field regions of an active source while maintaining desired radiation patterns in prescribed far-field directions. This control problem is considered in two environments: free space and homogeneous ocean of constant depth, respectively. In both media, we proved the existence of and characterized the surface input, modeled as Neumann data (normal velocity) or Dirichlet data (surface pressure) such that the radiated field satisfies the control constraints. We also provide a numerical strategy to construct this predicted surface input by using a method of moments-approach with a Morozov discrepancy principle-based Tikhonov regularization. Several numerical simulations are presented to demonstrate the proposed scheme in scenarios relevant to practical applications.

On the mixed boundary value problem for semilinear elliptic equations

We investigate the existence of weak solutions of a mixed boundary value problem for second order semilinear elliptic equation. The result is obtained by using regularity estimates for mixed linear elliptic problems and an appropriate fixed point theorem. For the homogeneous problem, we also get a result on the dependence of the solution on the small perturbations of the boundary with the Dirichlet and the Neumann data. Based on the fixed point iteration from the proof of the main result, we propose a numerical scheme and provide numerical examples. Parallelisation via the domain decomposition method is also given.

Corrosion shape reconstruction of the mixed boundary in electrostatic imaging

We consider the corrosion detection problem in terms of the Laplace equation and study a simply connected bounded domain with Wentzell-type GIBC boundary condition. We derive the systems of integral equations and establish the equivalence to the inverse shape problem in a Sobolev space setting. For the direct problem, we use potential theory to simulate the Neumann data from Dirichlet data on Dirichlet boundary. Then we propose a Newton iterative approach based on the boundary integral equations derived from Green’s representation theorem. After describing the linearization and the iteration scheme for the inverse shape, we compute the Fréchet derivatives with respect to the unknowns. We conclude by presenting several numerical examples for shape reconstructions to show the validity of the proposed method.

Regularity of homogenized boundary data in periodic homogenization of elliptic systems

This paper is concerned with the periodic homogenization of second-order elliptic systems in divergence form with oscillating Dirichlet data or Neumann data of first order. We prove that the homogenized boundary data belongs to W^{1, p} for any 1 &lt; p &lt; \infty . In particular, this implies that the boundary layer tails are Hölder continuous of order \alpha for any \alpha \in (0,1) .