Local Boundary Data Research Articles

On increasing stability in an inverse source problem with local boundary data at many wave numbers

ABSTRACTTo derive the increasing stability of the source term in the Helmholtz equation from local boundary data, we utilize sharp bounds of the analytic continuation for higher wave numbers, the Huygens' principle, and bounds in the lateral Cauchy problem for the wave equation.

Local uniqueness for an inverse boundary value problem with partial data

In dimension n ≥ 3 n\geq 3 , we prove a local uniqueness result for the potentials q q of the Schrödinger equation − Δ u + q u = 0 -\Delta u+qu=0 from partial boundary data. More precisely, we show that potentials q 1 , q 2 ∈ L ∞ q_1,q_2\in L^\infty with positive essential infima can be distinguished by local boundary data if there is a neighborhood of a boundary part where q 1 ≥ q 2 q_1\geq q_2 and q 1 ≢ q 2 q_1\not \equiv q_2 .

A simultaneous approach to inverse source problem by Green's function

AbstractIn this paper, we consider an inverse source problem of identification of F(t) function in the linear parabolic equation ut = uxx + F(t) and u0(x) function as the initial condition from the measured final data and local boundary data. Based on the optimal control framework by Green's function, we construct Fréchet derivative of Tikhonov functional. The stability of the minimizer is established from the necessary condition. The CG algorithm based on the Fréchet derivative is applied to the inverse problem, and results are presented for a test example. Copyright © 2014 John Wiley & Sons, Ltd.

What do non-relativistic CFTs tell us about Lifshitz spacetimes?

Abstract We study the reconstructability of (d + 2)-dimensional bulk spacetime from (d + 1)-dimensional boundary data, particularly concentrating on backgrounds which break (d + 1)-dimensional Lorentz invariance. For a large class of such spacetimes, there exist null geodesics which do not reach the boundary. Therefore classically one might guess some information is trapped in the bulk and thus invisible at the boundary. We show that this classical intuition correctly predicts the quantum situation: whenever there are null geodesics which do not reach the boundary, there are also “trapped scalar modes” whose boundary imprint is exponentially suppressed. We use these modes to show that no smearing function exists for pure Lifshitz spacetime, nor for any flow which includes a Lifshitz region. Indeed, for any (planar) spacetime which breaks (d + 1)-dimensional Lorentz invariance at any radius, we show that local boundary data cannot reconstruct complete local bulk data.

On an inverse problem in electromagnetism with local data: stability and uniqueness

In this paper we prove a stable determination of the coefficients of the time-harmonic Maxwell equations from local boundary data. The argument --due to Isakov-- requires some restrictions on the domain.

Lipschitz stability for a hyperbolic inverse problem by finite local boundary data

In this article we consider the inverse problem of determining the potential q in a wave equation in a bounded smooth domain Ω in from a finite number of data of the hyperbolic Dirichlet to Neumann map and we prove the Lipschitz stability in determining q. Our main result is stated as follows. Let T> diam Ω and Γ0⊂∂Ω. For any k-dimensional space X in L ∞(Ω), there exist 2k-functions f 1, …, f 2k on (0,T)×∂Ω such that , provided that q1, q2∈X are uniformly bounded in a suitable Sobolev space. Here Λq is the Dirichlet to Neumann map for the coefficient q(x).

Calderóns' Inverse Problem for Anisotropic Conductivity in the Plane

We study the inverse conductivity problem for an anisotropic conductivity σ ∈ L ∞ in bounded and unbounded domains. Also, we give applications of the results in the case when two sets of local boundary data are given.

Microstructural Decomposition Error Estimates

AbstractComputational simulations of interacting microstructure in solid structures, with methods such as the finite element method, require solutions to numerically enormous boundary value problems. The primary objective of this work is to introduce a‐posteriori error bounds for a domain decomposition which can be used to reduce the computational complexity of boundary value problems associated with such simulations. The approach is to partition and decouple the heterogeneous body into more computationally tractable, nonoverlapping, subdomains whose union forms the entire domain under analysis. This is achieved by computing a relatively inexpensive auxiliary „decoupling problem”︁ with regularized coefficients but with the same external geometry and loading as the original body. The solution to the decoupling problem is then used to construct local boundary data for the subdomains, which can thereafter be solved independently, possibly in parallel.

On the Uniqueness of Inverse Problems from Incomplete Boundary Data

We present a generalization of our previous results on the uniqueness of inverse coefficient problems for elliptic equations. The theorem proved here is formulated for piecewise-constant coefficients with piecewise smooth interfaces of discontinuities which require local boundary data. We consider problems with coefficients unknown at one side as well as at both sides of the boundary. The uniqueness of the former is proven; the latter can yield up to two solutions. The proofs are based on a simple analysis of singularities of the Green functions. The presented approach exactly conforms to some important inverse problems arising in applications of electromagnetic geophysics.

Stability estimates for hyperbolic inverse problems with local boundary data

The authors obtain stability estimates of recovery of two coefficients of a hyperbolic partial differential equation from all possible measurements implemented at a part of the lateral boundary. These estimates are of logarithmic type in the plane case and of Holder type in the three-dimensional case. As an important auxiliary result they have proved stability estimates in (attenuated) integral geometry with incomplete data.

Local boundary data of eigenfunctions on a Riemannian symmetric space

Let X = G/K be a Riemannian symmetric space of non-compact type, and let D(X) be the algebra of invariant differential operators on X. In a previous paper [1] we developed a theory of asymptotic expansions for joint eigenfunctions of D(X) of at most exponential growth. In the present paper we show that local asymptotic data determine the eigenfunctions completely. We also develop a theory of asymptotic expansions "along walls". Both results are of importance for the theory of the discrete series of a semisimple symmetric space. Let a be a maximal abelian split subspace of the Lie algebra 9 of G, and denote by a* its complexified dual. Let E be the restricted root system of a in g, and W the associated Weyl group. Then Harish-Chandra's isomorphism D ( X ) ~ S(a) w determines a bijection 2~--~Z~ from a*/W to the set D(X) ^ of algebra homomorphisms D ( X ) ~ C . Given 2ca* , let ga(X) denote the joint eigenspace of functions f e C o~ (X) satisfying: