Joyce R. McLaughlin

Research

Several sets of inverse problems are being considered. One of theseis the inverse problem of elastography. There thegoal is to create images of the variations of shear wavespeed inbiological tissue; the aim is to develop a medical diagnostic tool.These images extend the doctor's palpation exam where the doctorpresses against the skin to feel the presence of abnormal tissuewhich is stiffer than normal tissue.

In one elastography experiment the tissue is initiallyat rest when a low frequency pulse is applied to the boundary or along a line in the interior - (supersonic imaging). An elastic wave propagates into the body and the downward displacement of this propagatingwave as a function of position and time is measured remotely on a grid of points interior to the body using Dopplerultrasound. The inverse problem is to determine the shear wavespeedfrom these interior measurements. We develop well-posedness results andfast algorithms for this wavespeed recovery. Both synthetic data anddata measured in the laboratory of Mathias Fink, Laboratoire Ondeset Acoustique, ESPCI, Universite Paris VII, are tested in ourwavespeed recovery algorithms. In one algorithm we exploit thefeature that the data has a propagating front and develop the Arrival Time algorithm.

In a second elastography experiment a traveling wave in tissue is produced by exciting the tissue at one frequency and combining the ultrasound data with another frequency, generated in the ultrasound receiver. This second frequency nearly the same, but not equal to, the excitation frequency. This produces a traveling wave whose phase is altered by inhomogeneities. We recover the location size, and amplitude of the inhomogeneities using an adaptation of the Arrival Time algorithm.

A second set of problems are inverse problems and wave propagationalgorithms in waveguides. There we develop exact one way algorithmsfor calculating the solution of Helmholtz equation in a range anddepth dependent ocean. For inverse problems we utilize our knowledgeof waveguides to develop efficient methods for identificationof objects in waveguides and for time reversal problems.

Inverse spectral problems are also addressed. There the data is naturalfrequencies and eigenmode measurements. One set of data is the levelsets of eigenmodes. The inverse problem solution is material parameterssuch as density, sound speed, or elasticity coefficients. Mathematicalmodels are second or fourth order partial and ordinary differential equations.Well-posedness results are obtained and algorithms are developedand tested. he functional relationship between spectral data and material parameters is highly nonlinear. Nonlinear functional analytic techniques are applied to show uniqueness results and to develop algorithms. In one dimension, algebraic and differential geometry techniques yield exact solutions and global existence of solutions. Furthermore, results when material properties arevery rough are obtained and it is shown that new phenomena occur as the material coefficients become rougher. In higher dimensions, asymptotic forms for the spectral data are obtained and so far these asymptotic resultsare established only when the material properties are quite smooth. Application of variational methods yields algorithms. Solution techniques are aimed at maintaining the full nonlinearity of the inverse problem without employing linearization methods.

Recent Publications

"Inverse Spectral Theory Using Nodal Points as Data - A Uniqueness Result," J. Diff. Eq., Vol. 73, 1988, pp. 354-362.

" Solutions to Inverse Nodal Problems" (with O.H. Hald), Inverse Problems, Vol. 5, 1989, pp. 307-347.

"Examples of Inverse Nodal Problems" (with O.H. Hald), Inverse Problems in Action, ed. P. Sabatier, Springer - Verlag, 1990, pp. 147-151.

"Extremal Eigenvalue Problems for Composite Membranes, I," (with S. Cox) Appl. Math and Op., Vol. 22, 1990, pp. 153-167.

"Extremal Eigenvalue Problems for Composite Membranes, II," (with S. Cox) Appl. Math and Op., Vol. 22, 1990, pp.169-187.

"Solution of the inverse spectral problem for an impedance with integrable derivative," Part I(with Carol Coleman), Comm. Pure and Appl. Math, Vol. XLVI (1993), pp. 145-184.

"Solution of the inverse spectral problem for an impedance with integrable derivative," Part II (with Carol Coleman), Comm. Pure and Appl. Math, Vol. XLVI (1993), pp. 185-212.

"A formula for finding a potential from nodal lines," (with Ole H. Hald) Bulletin ofthe AMS 32 (1995),pp. 241-247.

"Reconstruction of a spherically symmetric speed of sound," (with Peter Polykov and PaulSacks) SIAM J. Appl. Math 54(1994),pp. 1203-1223.

"The effect of structural damping on nodes for the Euler-Bernoulli Beam: a specific case study," (with B. Geist), Appl. Math Lett., Vol. 7, 1994, pp. 51-55.

"Formulas for Finding Coefficients from Nodes/Nodal Lines", Proceedings ofthe International Congress of Mathematicians, Zurich, Switzerland, 1994,Birkhauser Verlag, Basel, Switzerland, 1995, pp. 1494-1501.(Text of Paper(without figures))

"Inverse Nodal Problems: Finding the Potential from Nodal Lines," (with Ole H. Hald) AMS Memoir, Vol. 119, No. 572, January 1996 (148 pages).(Introduction)

"The Riccati method for the Helmholtz equation", (with Ya Yan Lu), J.Acoust.Soc. Am., Vol. 100 (3), (1996), pp. 1432-1446.

"Finding the density of a membrane from nodal lines", (with C.J. Lee), InverseProblems in Wave Propagation, eds. G. Chavent, G. Papanicolaou, P. Sacks,W.W. Symes, Springer, 1997, pp. 325-345.

"Perturbation expansions for eigenvalues and eigenvectors for a rectangularmembrane subject to a restorative force", (with A. Portnoy), ElectronicResearch Announcements of the AMS, Vol. 3 (1997), pp. 72-77.

"Double Eigenvalues for the Uniform Timoshenko Beam," (with B. Geist), Appl. Math. Letters, Vol. 10 (1997), pp. 129-134.

"Perturbing a rectangular membrane with a restorative force: effects on eigenvalues and eigenfunctions", (with A. Portnoy), Comm. P.D.E., Vol. 23(1&2), 1998, pp. 243-285 .

"Inverse Problems: Recovery of BV Coefficients from Nodes", (with O.H. Hald),Inverse Problems, Vol. 14, No. 2, 1998, pp. 245-273.(Text of Paper(without figures))

"Eigenvalue Formulas for the Uniform Timoshenko Beam: The Free-Free Problem",(with B. Geist) Electronic Research Ann., AMS, Vol. 4, 1998.

"Recovery of a vertically stratified seabed in shallow water", (with S. Wang)Mathematical and Numerical Aspects of Wave Propagation, ed. John A. DeSanto,SIAM, Philadelphia, 1998, pp. 232-236.
(Text of Paper(with figures))

"Solving Inverse Problems with Spectral Data", Surveys on Solution Methodsfor Inverse Problems, eds. D. Colton, H. Engl, A. Louis, J. McLaughlin,W. Rundell, Springer, New York, 2000, pp. 169-194.(Text of Paper(with figures))

"Asymptotic Formulas for the Eigenvalues of the Timoshenko Beam", (with B. Geist) JMAA, Vol. 253, 2001, pp. 341-380.
(Text of Paper)

"Local orthogenal transformations and one-way methods for acoustic waveguides," (with Y. Y. Lu and J. Huang), Wave Motion, Vol. 34, 2001, pp. 193-207.

"Using a Hankel function expansion to identify stiffness forthe boundary impulse input experiment", (with Lin Ji) , AMS Contemporary Mathematics (CONM) Book Series: Proc. of the Conf. on Inverse Problems and Applications (Pisa, Italy) 2003 ed. G. Allessandrini and G. Uhlman.
(Text of Paper(with figures))

"Interior Elastodynamics Inverse Problems: Shear Wave Speed Reconstruction in Transient Elastography", (with Lin Ji,Daniel Renzi, and Jeong-Rock Yoon),Inverse Problems, Vol. 19, No. 6, December 2003, pp. s1-s29.
(Text of Paper(with figures))

"Propagation in Helmholtz Waveguides using DtN, NtD and R_atd Maps: Part I, a Second Order Method", (with Ya Yan Lu), submitted.
(Text of Paper(with figures))

"Recovery of the Lamè parameter in biological tissues", (with Lin Ji), Inverse Problems, Vol. 20, No. 1, February 2004, pp. 1-24.
(Text of Paper(with figures))

"Unique identifiability of elastic parameters from time-dependent interior displacement measurement", (with Jeong-Rock Yoon), Inverse Problems, Vol. 20, No. 1, February 2004, pp. 25-46.
(Text of Paper(with figures))

"Shear Wave Speed Recovery in Transient Elastography And Supersonic Imaging Using Propagating Fronts", (with Daniel Renzi), accepted, Inverse Problems.
(Text of Paper(with figures))

" Using Level Set Based Inversion of Arrival Times To Recover Shear Wavespeed In Transient Elastography And Supersonic Imaging ", (with Daniel Renzi), accepted, Inverse Problems.
(Text of Paper(with figures))

"Shear Wavespeed recovery using moving interference patterns obtained in sonoelastography experiments", (with K. Parker, D. Renzi, C. Wu), preprint.(Text of Paper(with figures))

"Recovering inhomogeneities in a waveguide using eigensystem decomposition", (with Sava Dediu), preprint.(Text of paper (with figures))

Tutorials

"Interior Elastodynamic Inverse Problems," University of Washington, August, 2005.

"Interior Elastodynamic Inverse Problems," IPAM, Fall, 2003;

"Inverse Spectral Problems," IPAM, Fall, 2003;

"Using spectral data to solve inverse problems," December 2001, CBMS Lectures, University of Texas, Pan American;

"Solving inverse problems using frequencies and nodes," Womens Mentoring Program, IAS, June, 1995; lectures for undergraduates.

Ph.D. Thesis

"Recovering Inhomogeneities In a WaveguideUsing Eigensystem Decomposition", Sava Dediu

Coordinates

110 8th Street
Sciences
Rensselaer Polytechnic Institute
Troy, New York 12180