Dirichlet-Neumann Map Research Articles

Scattering by a periodic tube in : part i. The limiting absorption principle**Dedicated to the memory of my friend and colleague Armin Lechleiter.

Scattering of time-harmonic waves from periodic structures at some fixed real-valued wave number becomes analytically difficult whenever there arise surface waves: These non-zero solutions to the homogeneous scattering problem physically correspond to modes propagating along the periodic structure and clearly imply non-uniqueness of any solution to the scattering problem. In this paper, we consider a medium, described by a refractive index which is periodic along the axis of an infinite cylinder in and constant outside of the cylinder. We prove that there is a so-called limiting absorption solution to the associated scattering problem. By definition, such a solution is the limit of a sequence of unique solutions for artificial complex-valued wave numbers tending to the above-mentioned real-valued wave number. By the standard one-dimensional Floquet–Bloch transform and the introduction of the exterior Dirichlet–Neumann map we first reduce the scattering problem to a class of periodic problems in a bounded cell, depending on the wave number k and the Bloch parameter . We use a functional analytic singular perturbation result to study this problem in a neighborhood of a singular pair . This abstract result allows us to derive explicitly a representation for the limiting absorption solution as a sum of a decaying part (along the axis of the cylinder) and a finite sum of propagating modes.

Stability estimates for a magnetic Schrödinger operator with partial data

In this paper we study local stability estimates for a magnetic Schrodinger operator with partial data on an open bounded set in dimension \begin{document}$ n≥3$\end{document} . This is the corresponding stability estimates for the identifiability result obtained by Bukhgeim and Uhlmann [ 2 ] in the presence of a magnetic field and when the measurements for the Dirichlet-Neumann map are taken on a neighborhood of the illuminated region of the boundary for functions supported on a neighborhood of the shadow region. We obtain log log-estimates for the magnetic fields and log log log-estimates for the electric potentials.

Splitting scheme for the stability of strong shock profile

In this paper, we use the Laplace transform and Dirichlet–Neumann map to give a systematical scheme to study the small wave perturbation of general shock profile with general amplitude. Here we use certain non-classical shock waves for a rotationally invariant system of viscous conservation laws to demonstrate this scheme. We derive an explicit solution and show that it converges pointwise to another over-compressive profile exponentially, when the perturbations of the initial data to a given over-compressive shock profile are sufficiently small.

On domain of Poisson operators and factorization for divergence form elliptic operators

We consider second order uniformly elliptic operators of divergence form in \(\mathbb {R}^{d+1}\) whose coefficients are independent of one variable. For such a class of operators we establish a factorization into a product of first order operators related with Poisson operators and Dirichlet–Neumann maps. Consequently, we obtain a solution formula for the inhomogeneous elliptic boundary value problem in the half space, which is useful to show the existence of solutions in a wider class of inhomogeneous data. We also establish \(L^2\) solvability of boundary value problems for a new class of the elliptic operators.

On Poisson operators and Dirichlet-Neumann maps in $H^s$ for divergence form elliptic operators with Lipschitz coefficients

We consider second order uniformly elliptic operators of divergence form in $\R^{d+1}$ whose coefficients are independent of one variable. Under the Lipschitz condition on the coefficients we characterize the domain of the Poisson operators and the Dirichlet-Neumann maps in the Sobolev space $H^s(\R^d)$ for each $s\in [0,1]$. Moreover, we also show a factorization formula for the elliptic operator in terms of the Poisson operator.

Numerical analysis of Fokas' unified method for linear elliptic PDEs

In this paper we examine a numerical implementation of Fokas' unified method for elliptic boundary value problems on convex polygons. Within this setting the unified method provides a reconstruction of the unknown boundary values from the known boundary data for an important class of elliptic boundary value problems. We demonstrate that this approach can be used to study the classical Dirichlet eigenvalue problem for the Laplacian on convex polygons. We show that this problem is equivalent to a nonlinear eigenvalue problem for a semi-Fredholm operator which has holomorphic dependence on the eigenvalue itself. We also study the classical Dirichlet problem for the Helmholtz operator. We provide a new Galerkin scheme for the underlying Dirichlet–Neumann map, and prove that for sufficiently regular Dirichlet data this scheme converges with spectral accuracy.

Lipschitz stability for the inverse conductivity problem for a conformal class of anisotropic conductivities

We consider the stability issue of the inverse conductivity problem for a conformal class of anisotropic conductivities in terms of the local Dirichlet–Neumann map. We extend here the stability result obtained by Alessandrini and Vessella (Alessandrini G and Vessella S 2005 Lipschitz stability for the inverse conductivity problem Adv. Appl. Math. 35 207–241), where the authors considered the piecewise constant isotropic case.

A partial data result for the magnetic Schrödinger inverse problem

This article shows that knowledge of the Dirichlet-Neumann map on certain subsets of the boundary for input functions supported roughly on the rest of the boundary can be used to determine a magnetic Schr\{o}dinger operator. With some geometric conditions on the domain, either the subset on which the DN map is measured or the subset on which the input functions have support may be made arbitrarily small. This is an improvement on the partial data result in a paper by Dos Santos Ferreira, Kenig, Sj\{o}strand, and Uhlmann. The method involves modifying the Carleman estimate in that paper by conjugation with operators built from pseudodifferential pieces.

Efficient numerical solution of the generalized Dirichlet–Neumann map for linear elliptic PDEs in regular polygon domains

A new and novel approach for analyzing boundary value problems for linear and for integrable nonlinear PDEs was recently introduced. For linear elliptic PDEs, an important aspect of this approach is the characterization of a generalized Dirichlet–Neumann map: given the derivative of the solution along a direction of an arbitrary angle to the boundary, the derivative of the solution perpendicularly to this direction is computed without solving on the interior of the domain. For this computation, a collocation-type numerical method has been recently developed. Here, we study the collocation’s coefficient matrix properties. We prove that, for the Laplace’s equation on regular polygon domains with the same type of boundary conditions on each side, the collocation matrix is block circulant, independently of the choice of basis functions. This leads to the deployment of the FFT for the solution of the associated collocation linear system, yielding significant computational savings. Numerical experiments are included to demonstrate the efficiency of the whole computation.

Fast reconstruction of harmonic functions from Cauchy data using the Dirichlet-to-Neumann map and integral equations

We propose and investigate a method for the stable determination of a harmonic function from knowledge of its value and its normal derivative on a part of the boundary of the (bounded) solution domain (Cauchy problem). We reformulate the Cauchy problem as an operator equation on the boundary using the Dirichlet-to-Neumann map. To discretize the obtained operator, we modify and employ a method denoted as Classic II given in [J. Helsing, Faster convergence and higher accuracy for the Dirichlet–Neumann map, J. Comput. Phys. 228 (2009), pp. 2578–2576, Section 3], which is based on Fredholm integral equations and Nyström discretization schemes. Then, for stability reasons, to solve the discretized integral equation we use the method of smoothing projection introduced in [J. Helsing and B.T. Johansson, Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques, Inverse Probl. Sci. Eng. 18 (2010), pp. 381–399, Section 7], which makes it possible to solve the discretized operator equation in a stable way with minor computational cost and high accuracy. With this approach, for sufficiently smooth Cauchy data, the normal derivative can also be accurately computed on the part of the boundary where no data is initially given.

A two level domain decomposition preconditioner based on local Dirichlet-to-Neumann maps

Coarse grid correction is a key ingredient in order to have scalable domain decomposition methods. In this Note we construct the coarse grid space using the low frequency modes of the subdomain DtN (Dirichlet–Neumann) maps, and apply the obtained two-level preconditioner to the linear system arising from an overlapping domain decomposition. Our method is suitable for the parallel implementation and its efficiency is demonstrated by numerical examples on problems with high heterogeneities.

Faster convergence and higher accuracy for the Dirichlet–Neumann map

New techniques allow for more efficient boundary integral algorithms to compute the Dirichlet–Neumann map for Laplace’s equation in two-dimensional exterior domains. Novelties include a new post-processor which reduces the need for discretization points with 50%, a new integral equation which reduces the error for resolved geometries with a factor equal to the system size, systematic use of regularization which reduces the error even further, and adaptive mesh generation based on kernel resolution.

Direct and iterative solution of the generalized Dirichlet–Neumann map for elliptic PDEs on square domains

In this work we derive the structural properties of the Collocation coefficient matrix associated with the Dirichlet–Neumann map for Laplace’s equation on a square domain. The analysis is independent of the choice of basis functions and includes the case involving the same type of boundary conditions on all sides, as well as the case where different boundary conditions are used on each side of the square domain. Taking advantage of said properties, we present efficient implementations of direct factorization and iterative methods, including classical SOR-type and Krylov subspace (Bi-CGSTAB and GMRES) methods appropriately preconditioned, for both Sine and Chebyshev basis functions. Numerical experimentation, to verify our results, is also included.

Conformal Dirichlet-Neumann Maps and Poincaré-Einstein Manifolds

A conformal description of Poincare-Einstein manifolds is developed: these structures are seen to be a special case of a natural weakening of the Einstein condition termed an almost Einstein structure. This is used for two purposes: to shed light on the relationship between the scattering construction of Graham-Zworski and the higher order conformal Dirichlet-Neumann maps of Branson and the author; to sketch a new construction of non-local (Dirichlet-to-Neumann type) conformal operators between tensor bundles.

The generalized Dirichlet–Neumann map for linear elliptic PDEs and its numerical implementation

A new approach for analyzing boundary value problems for linear and for integrable nonlinear PDEs was introduced in Fokas [A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A 53 (1997) 1411–1443]. For linear elliptic PDEs, an important aspect of this approach is the characterization of a generalized Dirichlet to Neumann map: given the derivative of the solution along a direction of an arbitrary angle to the boundary, the derivative of the solution perpendicularly to this direction is computed without solving on the interior of the domain. This is based on the analysis of the so-called global relation, an equation which couples known and unknown components of the derivative on the boundary and which is valid for all values of a complex parameter k. A collocation-type numerical method for solving the global relation for the Laplace equation in an arbitrary bounded convex polygon was introduced in Fulton et al. [An analytical method for linear elliptic PDEs and its numerical implementation, J. Comput. Appl. Math. 167 (2004) 465–483]. Here, by choosing a different set of the “ collocation points” (values for k), we present a significant improvement of the results in Fulton et al. [An analytical method for linear elliptic PDEs and its numerical implementation, J. Comput. Appl. Math. 167 (2004) 465–483]. The new collocation points lead to well-conditioned collocation methods. Their combination with sine basis functions leads to a collocation matrix whose diagonal blocks are point diagonal matrices yielding efficient implementation of iterative methods; numerical experimentation suggests quadratic convergence. The choice of Chebyshev basis functions leads to higher order convergence, which for regular polygons appear to be exponential.

Nonlinear three-dimensional simulation of solid tumor growth

We present a new, adaptive boundary integral method to simulate solid tumor growth in 3-d. We use a reformulation of a classical model that accounts for cell-proliferation, apoptosis, cell-to-cell and cell-to-matrix adhesion. The 3-d method relies on accurate discretizations of singular surface integrals, a spatial rescaling and the use of an adaptive surface mesh. The discretized boundary integral equations are solved iteratively using GMRES and a discretized version of the Dirichlet-Neumann map, formulated in terms of a vector potential, is used to determine the normal velocity of the tumor surface. Explicit time stepping is used to update the tumor surface. We present simulations of the nonlinear evolution of growing tumors. At early times, good agreement is obtained between the results of a linear stability analysis and nonlinear simulations. At later times, linear theory is found to overpredict the growth of perturbations. Nonlinearity results in mode creation and interaction that leads to the formation of dimples and the tumor surface buckles inwards. The morphologic instability allows the tumor to increase its surface area, relative to its volume, thereby allowing the cells in the tumor bulk greater access to nutrient. This in turn allows the tumor to overcome the diffusional limitations on growth and to grow to larger sizes than would be possible if the tumor were spherical. Consequently, instability provides a means for avascular tumor invasion.

An iterative method for electrostatic object reconstruction in a half space

Sensing electrodes arranged in or around a display can provide input function for interactive displays. Commercially this is interesting because the sensing electrodes and electronics can be made in the same manufacturing process as that of the display itself thus reducing cost. In engineering terms, the electrodes measure capacitance changes resulting from the presence and movement of objects such as hands and fingers in front of the display. At the quasi-static frequencies used (100 kHz) the human body is conductive and the hands or fingers provide a screen between the capacitive electrodes. There is no need to touch the actual display and the overall system constitutes a touchless gesture input system. Determining the shape of the hand or fingers is a boundary condition reconstruction problem of finding the boundary of an earthed conductive object D from electrostatic measurements. This is the ill-posed problem of recovering the zero surface of a solution to Laplace's equation from Cauchy data on part of the boundary of a domain. The problem has similarities with object reconstruction in EIT or inverse scattering but is complicated because only a partial Dirichlet–Neumann map is available as experimental data. We suggest an algorithm where at each iteration we have an approximation ∂Dk to ∂D on which we calculate approximate Cauchy data by solving a Tikhonov regularized linear system. These data are used to modify ∂Dk by extrapolation towards the zero surface giving the next approximation ∂Dk+1. We implemented the algorithm in two and three space dimensions using the boundary element method for discretization. Numerical results using simulated data with added noise show that simply connected but not necessarily convex objects can be reconstructed with reasonable positional accuracy and approximate shape, but as might be expected the shape is more accurately determined near the plane of measurements.

Unique identifiability of the common support of coefficients of a second order anisotropic elliptic system by the Dirichlet–Neumann map

We consider the system of second order elliptic equations - ∑ k ∇ ( E i , k ∇ u k ) + P i , k u k = 0 , 1 ⩽ i ⩽ n in a bounded simply-connected domain B. Using a factorisation method ansatz, we show that the difference of two Neumann-to-Dirichlet maps Λ - Λ 0 is sufficient to uniquely determine the support of the tensor E - E 0 and the matrix P - P 0 , where E 0 and P 0 are known and - ∑ k ∇ ( E i , k 0 ∇ u k 0 ) + P i , k 0 u k 0 = 0 , 1 ⩽ i ⩽ n

A boundary integral formulation of quasi-steady fluid wetting

This paper considers the motion of a liquid droplet on a solid surface. When capillary relaxation is much faster than the motion of the contact line, the fluid geometry and its dynamical evolution can be characterized in terms of the contact line alone. This problem can be cast in terms of boundary integral equations involving a Dirichlet–Neumann map coupled to a volume conservation constraint. A computational method for this formulation is described which has two principal advantages over approaches which track the entire free surface: (1) only the curve which describes the contact line is computed and (2) the resulting method exhibits only mild numerical stiffness, obviating the need for implicit timestepping. Effects of both capillary and body forces are considered. Computational examples include surface inhomogeneities, topological transitions and cusp formation.

A natural domain decomposition method with non-matching grids

In this paper a natural domain decomposition method based on local Dirichlet–Neumann maps is considered. The global variational problem is defined on the skeleton of the domain decomposition only. For the approximation of the Dirichlet–Neumann maps Dirichlet boundary value problems need to be solved locally. The local finite element spaces within the subdomains can be chosen independently of the global trial space on the skeleton. In particular, this approach can be used to couple non-matching triangulations across the interfaces without an additional framework such as introducing Lagrange multipliers. Numerical results for two model problems confirm the stability and error estimates given here.