Data-driven basis for reconstructing the contrast in inverse scattering: Picard criterion, regularity, regularization, and stability

We consider the inverse medium scattering of reconstructing the medium contrast using Born data, including the full aperture, limited-aperture, and multi-frequency data. We propose data-driven basis functions for these inverse problems based on the generalized prolate spheroidal wave functions and related eigenfunctions. Such data-driven eigenfunctions are eigenfunctions of a Fourier integral operator; they remarkably extend analytically to the whole space, are doubly orthogonal, and are complete in the class of band-limited functions. We first establish a Picard criterion for reconstructing the contrast using the data-driven basis, where the reconstruction formula can also be understood from the viewpoint of data processing and analytic extrapolation. Another salient feature associated with the generalized prolate spheroidal wave functions is that the data-driven basis for a disk is also a basis for a Sturm-Liouville differential operator. With the help of Sturm-Liouville theory, we estimate the $L^2$ approximation error for a spectral cutoff approximation of $H^s$ functions. This yields a spectral cutoff regularization strategy for noisy data and an explicit stability estimate for contrast in $H^s$ ($0<s<1/2$) in the full aperture case. In the limited-aperture and multi-frequency cases, we also obtain spectral cutoff regularization strategies for noisy data and stability estimates for a class of contrast

Single Mode Multi-frequency Factorization Method for the Inverse Source Problem in Acoustic Waveguides

This paper investigates the inverse source problem with a single propagating mode at multiple frequencies in an acoustic waveguide. The goal is to provide both theoretical justifications and efficient algorithms for imaging extended sources using the sampling methods. In contrast to the existing far/near field operator based on the integral over the space variable in the sampling methods, a multi-frequency far-field operator is introduced based on the integral over the frequency variable. This far-field operator is defined in a way to incorporate the possibly non-linear dispersion relation, a unique feature in waveguides. The factorization method is deployed to establish a rigorous characterization of the range support which is the support of source in the direction of wave propagation. A related factorization-based sampling method is also discussed. These sampling methods are shown to be capable of imaging the range support of the source. Numerical examples are provided to illustrate the performance of the sampling methods, including an example to image a complete sound-soft block.Comment: 23 page

讀殷商無四時說

頃得友人自平郵寄清華周刊文史專號一冊，內有吾友商錫永先生殷商無四時說一文。錫永先生與吾雖初交，然其治甲骨之學，實較吾先，其所刊布諸箸作，吾嘗一一取而讀之，蓋久已心折之也。其學之謹慎，卽就此文之結論『此問題目前雖不易解决，然地下之材料，層出不已，信必有證明之一日；但處現今之地步，與其必之，無或疑之也』；以觀，亦可以窺見一二。然而不能無疑者，請為錫永先生及吾諸同志言之，或亦愚者盡其千慮，而大雅所許乎！ 原文大意，可分四部：一辯吾友葉葓漁（玉森）先生殷契鈞沈及揅契枝譚釋甲骨文春夏秋冬四字之誤，二辯董作賓先生卜辭中所見之殷曆之說，三辯吾友束世澂先生『殷以三月至五月爲春，六月至八月爲夏，九月至十一月爲秋，十二月至二月爲冬』。之說，四引申唐蘭先生釋作甫之說，以徵實葉氏秋字之誤釋

Boundary Integral Equations for the Transmission Eigenvalue Problem for Maxwell’s Equations

International audienceIn this paper we consider the transmission eigenvalue problem for Maxwell’s equations corresponding to non-magnetic inhomogeneities with contrast in electric permittivity that changes sign inside its support. We formulate the transmission eigenvalue problem as an equivalent homogeneous system of boundary integral equa- tion, and assuming that the contrast is constant near the boundary of the support of the inhomogeneity, we prove that the operator associated with this system is Fredholm of index zero and depends analytically on the wave number. Then we show the existence of wave numbers that are not transmission eigenvalues which by an application of the analytic Fredholm theory implies that the set of transmission eigenvalues is discrete with positive infinity as the only accumulation point