Jenn-Nan Wang

Stability estimates for the inverse boundary value problem by partial Cauchy data

In this paper we study the inverse boundary value problem for the Schrodinger equation with a potential and the conductivity equation using partial Cauchy data. We derive stability estimates for these inverse problems.

Reconstructing Discontinuities Using Complex Geometrical Optics Solutions

In this paper we provide a framework for constructing general complex geometrical optics solutions for several systems of two variables that can be reduced to a system with the Laplacian as the leading order term. We apply these special solutions to the problem of reconstructing inclusions inside a domain filled with known conductivity from local boundary measurements. Computational results demonstrate the versatility of these solutions to determine electrical inclusions.

Complex spherical waves and inverse problems in unbounded domains

This work is motivated by the inverse conductivity problem of identifying an embedded object in an infinite slab. The novelty of our approach is that we use complex spherical waves rather than classical Calderon-type functions. For Calderon-type functions, they grow exponentially on one side of a hyperplane and decay exponentially on the other side. Without extra modifications, they are inadequate for treating inverse problems in unbounded domains such as the infinite slab. The obvious reason for this is that Calderon-type functions are not integrable on hyperplanes. So they cannot be used as measurements on infinite boundaries. For complex spherical waves used here, they blow up faster than any given positive polynomial order on the inner side of the unit sphere and decay to zero faster than any given negative polynomial order on the outer side of the unit sphere. We shall construct these special solutions for the conductivity equation in the unbounded domain by a Carleman estimate. Using complex spherical waves, we can treat the inverse problem of determining the object in the infinite slab like the problem in the bounded domain. Most importantly, we can easily localize the boundary measurement, which is of great value in practice. On the other hand, since the probing fronts are spheres, it is possible to detect some concave parts of the object.

Robust partial pole assignment for vibrating systems with aerodynamic effects

This note proposes a novel algorithm for robust partial eigenvalue assignment (RPEVA) problem for a cubic matrix pencil arising from modeling of vibrating systems with aerodynamic effects. The RPEVA problem for a cubic pencil is the one of choosing suitable feedback matrices to reassign a few (say k<3n) unwanted eigenvalues while leaving the remaining large number (3n-k) of them unchanged, in such a way that the the eigenvalues of the closed-loop matrix are as insensitive as possible to small perturbation of the data. The latter amounts to minimizing the condition number of the closed-loop eigenvector matrix. The problem is solved directly in the cubic matrix polynomial setting without making any transformation to a standard first-order state-space system. This allows us to take advantage of the exploitable structures such as the sparsity, definiteness, bandness, etc., very often offered by large practical problems. The major computational requirements are: i) solution of a small Sylvester equation, ii) QR factorizations, and iii) solution of a standard least squares problem. The least-squares problem result from matrix rank-two update techniques used in the algorithm for reassigning complex eigenvalues. The practical effectiveness of the method is demonstrated by implementational results on simulated data provided by the Boeing company

Partial pole assignment for the quadratic pencil by output feedback control with feedback designs

In this paper we study the partial pole assignment problem for the quadratic pencil by output feedback control where the output matrix is also a designing parameter. In addition, the input matrix is set to be the transpose of the output matrix. Under certain assumption, we give a solution to this partial pole assignment problem in which the unwanted eigenvalues are moved to desired values and all other eigenpairs remain unchanged. Copyright

Increasing stability in an inverse problem for the acoustic equation

In this work, we study the inverse boundary value problem of determining the refractive index in the acoustic equation. It is known that this inverse problem is ill-posed. Nonetheless, we show that the ill-posedness decreases when we increase the frequency and the stability estimate changes from logarithmic type for low frequencies to a Lipschitz estimate for large frequencies.

A depth-dependent stability estimate in electrical impedance tomography

We study the inverse problem of determining an electrical inclusion from boundary measurements. We derive a stability estimate for the linearized map with explicit formulae on generic constants that shows that the problem becomes more ill-posed as the inclusion is farther from the boundary. We also show that this estimate is optimal.

Complex spherical waves for the elasticity system and probing of inclusions

We construct complex geometrical optics solutions for the isotropic elasticity system concentrated near spheres. We then use these special solutions, called complex spherical waves, to identify inclusions embedded in an isotropic, inhomogeneous, elastic background.

Partial pole assignment for the vibrating system with aerodynamic effect

The partial pole assignment (PPA) problem is the one of reassigning a few unwanted eigenvalues of a control system by feedback to suitably chosen ones, while keeping the remaining large number of eigenvalues unchanged. The problem naturally arises in modifying dynamical behaviour of the system. The PPA has been considered by several authors in the past for standard state–space systems and for quadratic matrix polynomials associated with second-order systems. In this paper, we consider the PPA for a cubic matrix polynomial arising from modelling of a vibrating system with aerodynamics effects and derive explicit formulas for feedback matrices in terms of the coefficient matrices of the polynomial. Our results generalize those of a quadratic matrix polynomial by Datta et al. (Linear Algebra Appl. 1997;257: 29) and is based on some new orthogonality relations for eigenvectors of the cubic matrix polynomial, which also generalize the similar ones reported in Datta et al. (Linear Algebra Appl. 1997;257: 29) for the symmetric definite quadratic pencil. Besides playing an important role in our solution for the PPA, these orthogonality relations are of independent interests, and believed to be an important contribution to linear algebra in its own right. Copyright

Optimal stability estimate of the inverse boundary value problem by partial measurements

In this work we establish log-type stability estimates for the inverse potential and conductivity problems with partial Dirichlet-to-Neumann map, where the Dirichlet data is homogeneous on the inaccessible part. This result, to some extent, improves our former result on the partial data problem in which log-log-type estimates were derived.