Regularized Reduced Order Lippman-Schwinger-Lanczos Method for Inverse Scattering Problems in the Frequency Domain

Inverse scattering has a broad applicability in quantum mechanics, remote sensing, geophysical, and medical imaging. This paper presents a robust direct reduced order model (ROM) method for solving inverse scattering problems based on an efficient approximation of the resolvent operator regularizing the Lippmann-Schwinger-Lanczos (LSL) algorithm. We show that the efficiency of the method relies upon the weak dependence of the orthogonalized basis on the unknown potential in the Schr\"odinger equation by demonstrating that the Lanczos orthogonalization is equivalent to performing Gram-Schmidt on the ROM time snapshots. We then develop the LSL algorithm in the frequency domain with two levels of regularization. We show that the same procedure can be extended beyond the Schr\"odinger formulation to the Helmholtz equation, e.g., to imaging the conductivity using diffusive electromagnetic fields in conductive media with localized positive conductivity perturbations. Numerical experiments for Helmholtz and Schr\"odinger problems show that the proposed bi-level regularization scheme significantly improves the performance of the LSL algorithm, allowing for good reconstructions with noisy data and large data sets

Spectral analysis and connectivity of porous microstructures in bone

a b s t r a c t Cancellous bone is a porous composite of calcified tissue interspersed with soft marrow. Sea ice is also a porous composite, consisting of pure ice with brine, air, and salt inclusions. Interestingly, the microstructures of bone and sea ice exhibit notable similarities. In recent years, we have developed mathematical and experimental techniques for imaging and characterizing the brine microstructure of sea ice, such as its volume fraction and connectivity, as well as a range of theoretical approaches for studying fluid, thermal, and electromagnetic transport in sea ice. Here we explore the application of our sea ice techniques to investigate trabecular bone. For example, percolation theory that quantifies brine connectivity and its thermal evolution can also help assess the impact of osteoporosis on trabecular structure. Central to our approach is the spectral measure of a composite material, which contains detailed information about the mixture geometry, and can be used in powerful integral representations to compute the effective properties. The spectral measure is obtained from the eigenvalues and eigenvectors of a self-adjoint operator determined exclusively by the composite microgeometry. Here we compute the spectral measures for discretizations of images of healthy and osteoporotic bone. The measures are used to compute the effective electromagnetic properties of the bone specimens. These data are then inverted to reconstruct the porosity of the original specimens, with excellent agreement

Pade approximation of Herglotz functions and applications to composite materials

Non UBCUnreviewedAuthor affiliation: University of UtahFacult