Results 1 
6 of
6
Nonparametric Methods for Inference in the Presence of Instrumental Variables
 Annals of Statistics
, 2005
"... We suggest two nonparametric approaches, based on kernel methods and orthogonal series to estimating regression functions in the presence of instrumental variables. For the first time in this class of problems, we derive optimal convergence rates, and show that they are attained by particular estima ..."
Abstract

Cited by 28 (7 self)
 Add to MetaCart
We suggest two nonparametric approaches, based on kernel methods and orthogonal series to estimating regression functions in the presence of instrumental variables. For the first time in this class of problems, we derive optimal convergence rates, and show that they are attained by particular estimators. In the presence of instrumental variables the relation that identifies the regression function also defines an illposed inverse problem, the “difficulty ” of which depends on eigenvalues of a certain integral operator which is determined by the joint density of endogenous and instrumental variables. We delineate the role played by problem difficulty in determining both the optimal convergence rate and the appropriate choice of smoothing parameter. 1. Introduction. Data (Xi,Yi
Unification of the probe and singular sources methods for the inverse boundary value problem by the noresponse test
 Comm. Partial Differential Equations
"... In this article, we use the noresponse test idea, introduced in Luke and Potthast (2003) and Potthast (Preprint) and the inverse obstacle problem, to identify the interface of the discontinuity of the coefficient � of the equation � · ��x� � + c�x� with piecewise regular � and bounded function c�x ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
In this article, we use the noresponse test idea, introduced in Luke and Potthast (2003) and Potthast (Preprint) and the inverse obstacle problem, to identify the interface of the discontinuity of the coefficient � of the equation � · ��x� � + c�x� with piecewise regular � and bounded function c�x�. We use infinitely many Cauchy data as measurement and give a reconstructive method to localize the interface. We will base this multiwave version of the noresponse test on two different proofs. The first one contains a pointwise estimate as used by the singular sources method. The second one is built on an energy (or an integral) estimate which is the basis of the probe method. As a conclusion of this, the probe and the singular sources methods are equivalent regarding their convergence and the noresponse test can be seen as a unified framework for these methods. As a further contribution, we provide a formula to reconstruct the values of the jump of ��x�, x ∈ �D at the boundary. A second consequence of this formula is that the blowup rate of the indicator functions of the probe and singular sources methods at the interface is given by the order of the singularity of the fundamental solution.
Data Analysis and Representation on a General Domain using Eigenfunctions of Laplacian
, 2007
"... We propose a new method to analyze and represent data recorded on a domain of general shape in R d by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
We propose a new method to analyze and represent data recorded on a domain of general shape in R d by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz equation (which can be quite complicated and costly), we find the integral operator commuting with the Laplacian and diagonalize that operator. Although our eigenfunctions satisfy neither the Dirichlet nor the Neumann boundary condition, computing our eigenfunctions via the integral operator is simple and has a potential to utilize modern fast algorithms to accelerate the computation. We also show that our method is better suited for small sample data than the KarhunenLoève Transform/Principal Component Analysis. In fact, our eigenfunctions depend only on the shape of the domain, not the statistics of the data. As a further application, we demonstrate the use of our Laplacian eigenfunctions for solving the heat equation on a complicated domain.
DISCRETE APPROXIMATION OF NONCOMPACT OPERATORS DESCRIBING CONTINUUMOFALLELES MODELS
, 2004
"... Abstract We consider the eigenvalue equation for the largest eigenvalue of certain kinds of noncompact linear operators given as the sum of a multiplication and a kernel operator. It is shown that, under moderate conditions, such operators can be approximated arbitrarily well by operators of finite ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Abstract We consider the eigenvalue equation for the largest eigenvalue of certain kinds of noncompact linear operators given as the sum of a multiplication and a kernel operator. It is shown that, under moderate conditions, such operators can be approximated arbitrarily well by operators of finite rank, which constitutes a discretization procedure. For this purpose, two standard methods of approximation theory, the Nyström and the Galerkin method, are generalized. The operators considered describe models for mutation and selection of an infinitely large population of individuals that are labeled by real numbers, commonly called continuumofalleles (COA) models.
A FreeSpace Adaptive FMMBased PDE Solver in Three Dimensions
, 2008
"... We present a kernelindependent, adaptive fast multipole method (FMM) of arbitrary order accuracy for solving elliptic PDEs in three dimensions with radiation boundary conditions. The algorithm requires only a Green’s function evaluation routine for the governing equation and a representation of the ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We present a kernelindependent, adaptive fast multipole method (FMM) of arbitrary order accuracy for solving elliptic PDEs in three dimensions with radiation boundary conditions. The algorithm requires only a Green’s function evaluation routine for the governing equation and a representation of the source distribution (the righthand side) that can be evaluated at arbitrary points. The performance of the FMM is accelerated in two ways. First, we construct a piecewise polynomial approximation of the righthand side and compute farfield expansions in the FMM from the coefficients of this approximation. Second, we precompute tables of quadratures to handle the nearfield interactions on adaptive octree data structures, keeping the total storage requirements in check through the exploitation of symmetries. We present numerical examples for the Laplace, modified Helmholtz and Stokes equations. 1
The No Response Test for the Reconstruction of Polyhedral Objects in Electromagnetics
"... We develope a No Response Test for the reconstruction of some polyhedral obstacle from one or few timeharmonic electromagnetic incident waves in electromagnetics. The basic idea of the test is to probe some region in space with waves which are small on some test domain and, thus, do not generate a ..."
Abstract
 Add to MetaCart
We develope a No Response Test for the reconstruction of some polyhedral obstacle from one or few timeharmonic electromagnetic incident waves in electromagnetics. The basic idea of the test is to probe some region in space with waves which are small on some test domain and, thus, do not generate a response when the scatterer is inside of this test domain. This is the first formulation of the No Response Test for electromagnetics. We will prove convergence of the method for testing a nonvibrating domain B whether the far field pattern of some scattered timeharmonic field is analytically extendable into the interior of B. We will describe algorithmical realizations of the No Response Test. Finally, we will show the feasibility of the method by reconstruction of polygonal objects in three dimensions.