Yongzhi Xu

Generalized dual space indicator method for underwater imaging

This paper presents a generalized dual space indicator method for imaging an obstacle in ocean environments. The method is based on the observation that the combination (weighted integration) of the measured scattered field can approximate the Green function very well when the Green functions source point is inside the obstacle, but not so well when the source is outside the obstacle. We set up an integral equation whose right-hand side is the Green function with a source point from a searching region. From our numerical experiments, we notice that the norm of the solution of the integral equation has local extrema that lie inside the unknown obstacle. Plotting the norm as a function of the source point in the searching region, and filtering out the region with no local extrema of the norm, we obtain a good image of the unknown obstacle. Imaging algorithms and numerical examples are presented.

Acoustic response of a rigid-frame porous medium plate with a periodic set of inclusions.

The acoustic response of a rigid-frame porous plate with a periodic set of inclusions is investigated by a multipole method. The acoustic properties, in particular, the absorption, of such a structure are then derived and studied. Numerical results together with a modal analysis show that the addition of a periodic set of high-contrast inclusions leads to the excitation of the modes of the plate and to a large increase in the acoustic absorption.

The unidentified object problem in a shallow ocean

This work addresses the inverse problem of the identification of a passive three-dimensional impenetrable object in a shallow-water environment. The latter is assumed to have flat perfectly reflecting (sound-soft top and sound-hard bottom) boundaries and therefore acts as a guide for acoustic waves. These waves are employed to interrogate the object and the scattered acoustic wavefield is measured on the surface of a (virtual) vertically oriented cylinder (of finite or infinite radius, corresponding to near- or far-field measurements) fully enclosing the object. The direct scattering problem is resolved in approximate manner by employing, in a local manner, the known separated-variable solution for a scattering by a vertically oriented cylinder in a perfect waveguide. The inverse problem is resolved in the same manner (i.e., with the same approximate field ansatz) by least-squares matching of theoretical fields (for trial objects) to the measured field. Examples are given of successful shape reconstructio...

Direct and inverse problems of mathematical physics

Preface. 1. Algorithms of the Asymptotic Nonlinear Analysis A. Bruno. 2. Transmission Loss in a Depth-varying Ocean over a Poroelastic Seabed J. Buchanan, R.P. Gilbert. 3. Variation on (chi, psi) Duality R. Carroll. 4. Uniqueness of Continuation Theorems M. Eller. 5. Determination of a Distributed Inhomogeneity in a Two-layer Waveguide from Scattered Sound R.P. Gilbert, et al. 6. Differentiability with Respect to Parameters of Integrated Semigroups M. He. 7. Coundedness of Pseudo-differential Operators on Hormander Spaces G. Iancu. 8. Coefficient Identification in Elliptic Differential Equations I. Knowles. 9. Quasi-exponential Solutions for Some PDE with Coefficients of Limited Regularity A. Panchenko. 10. Analytically Smoothing Effect for Schrodinger Type Equations with Variable Coefficients K. Kajitani, W. Wakabayashi. 11. Initial Boundary Value Problem for the Viscous Incompressible Flows H. Kato. 12. Initial-boundary Value Problems for an Equation of Internal Waves in a Stratified Fluid P. Krutitskii. 13. Homogenization of the System Equations of High Frequency Nonlinear Acoustics E.A. Lapshin, G.P. Panasenko. 14. Identification of a Reflection Boundary Coefficient in an Acoustic Wave Equation by Optimal Control Techniques S. Lenhart, et al. 15. Numerical Solutions to Acoustic Scattering in Shallow Oceans by Periodic Wavelets W. Lin, X. Wang. 16. Solution of the Robin and Dirichlet Problem for the Laplace Equation D. Medkova. 17. Existence and Decay of Solutions of someNonlinear Degenerate Parabolic Equations T. Nanbu. 18. On Regularity Results for Variational-hemivariational Inequalities Z. Naniewicz, P.D. Panagiotoopoulos. 19. Quasi-exponential Solutions for some PDE with Coefficients of Limited Regularity A. Panchenko. 20. An Inverse Problem in Elastodynamics L. Rachele. 21. Denseness of Co (RN) in the Generalized Sobolev Spaces S. Samko. 22. Singularities of Reflected and Refracted Riemann Functions of Elastic Wave Propagation Problems in Stratified Media S. Shimizu. 23. Exact Boundary Controllability of a First Order, Non-linear Hyperbolic Equation with Non-local Integral Terms Arising in Epidemic Modeling I. Lasiecka, R. Triggiani. 24. Singularities of Solutions for Nonlinear Hyperbolic Equations of Second Order M. Tsuji. 25. Positive Solutions of Semilinear Elliptic Boundary Value Problems in Chemical Reactor Theory K. Umezu, K. Taira. 26. Fast Solvers of the Lippman-Schwinger Equation G. Vainikko. 27. Inverse Source Problems for the Stokes System M. Yamamota, O.Y. Imanuvilov.

Some inverse problems raised from a mathematical model of ductal carcinoma in situ

The growth of avascular tumors has been studied by many authors. In this paper we study a special kind of cancer, ductal carcinoma in situ (DCIS) in order to investigate possible procedures to connect free boundary model of DCIS with clinical data. This paper is to present some results of our research on mathematical modeling, analysis, and numerical simulations of ductal carcinoma in situ. In particular, we formulate a number of inverse problems for the well-posed free boundary valued problem related to clinical diagnose of cancer.

Use of specific Green's functions for solving direct problems involving a heterogeneous rigid frame porous medium slab solicited by acoustic waves

A domain integral method employing a specific Greens function (i.e. incorporating some features of the global problem of wave propagation in an inhomogeneous medium) is developed for solving direct and inverse scattering problems relative to slab-like macroscopically inhomogeneous porous obstacles. It is shown how to numerically solve such problems, involving both spatially-varying density and compressibility, by means of an iterative scheme initialized with a Born approximation. A numerical solution is obtained for a canonical problem involving a two-layer slab. Copyright

Transient reflection and transmission of ultrasonic wave in cancellous bone

Cancellous bone is known to be poroelastic in structure. Ultrasonic wave propagation in cancellous bone can be formulated by using Biots equations. In this paper we present some results in our ongoing research on the reflection and transmission of ultrasonic waves in cancellous bone. We investigate the relations among reflected waves, transmitted waves and Biot coefficients. We present an algorithm for the determination of the porosity of cancellous bone.

An injective far-field pattern operator and inverse scattering problem in a finite depth ocean

The inverse scattering problem for acoustic waves in shallow oceans are different from that in the spaces of R 2 and R 3 in the way that the “propagating” far-field pattern can only carry the information from the N +1 propagating modes. This loss of information leads to the fact that the far-field pattern operator is not injective. In this paper, we will present some properties of the far-field pattern operator and use this information to construct an injective far-field pattern operator in a suitable subspace of L 2 (∂Ω). Based on this construction an optimal scheme for solving the inverse scattering problem is presented using the minimizing Tikhonov functional.

Computing porosity of cancellous bone using ultrasonic waves, II: The muscle, cortical, cancellous bone system

Based on a model and an algorithm proposed previously by Buchanan, Gilbert, Wirgin and Xu [J.L. Buchanan, R.P. Gilbert, A. Wirgin, Y. Xu, Transient reflection and transmission of ultrasonic waves in cancellous bones, Appl. Math. Comp. 142 (2003), 561-573, R.P. Gilbert, Y. Xu, Shangyou Zhang, Computing porosity of cancellous bone using ultrasonic waves] we numerically computed the dependence of the porosity of cancellous bone encased in cortical bone and muscle on the measured ultrasonic wave. The computation shows the bone porosity can be recovered very accurately. The numerical data show that the ultrasonic wave of higher frequency would produce a more accurate bone porosity.

Decoupling of the quasistatic system of thermoelasticity on the unit disk

The quasi-static, linearized thermoclastic system on the unit disk is decoupied, such that the temperature satisfies an integro-differential equation. The result, based on the function theoretic method, is of both theoretical and numerical interest.