Special Session 20: Stochastic analysis, inverse problems and related topics

Solving the phaseless inverse source problem of the biharmonic equation

Yukun Guo Harbin Institute of Technology Peoples Rep of China Co-Author(s):    Yan Chang, Yue Zhao

Abstract: This talk concerns an inverse source problem for the biharmonic wave equation. A two-stage numerical method is proposed to identify the unknown source from the multifrequency phaseless data. In the first stage, we introduce some artificial auxiliary point sources to the inverse source system and establish a phase retrieval formula. Theoretically, we point out that the phase can be uniquely determined and estimate the stability of this phase retrieval approach. Once the phase information is retrieved, the Fourier method is adopted to reconstruct the source function from the phased multifrequency data. Numerical examples will be presented to verify the performance of the proposed method.

Monotonicity and Convexity in inverse coefficient problems

Bastian Harrach Goethe University Frankfurt Germany Co-Author(s):

Abstract: Several applications in medical imaging and non-destructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its solutions. This is usually a highly non-linear ill posed inverse problem, for which unique reconstructability results, stability and resolution estimates and global convergence of numerical methods are very hard to achieve. In this talk we will review some recent results on Loewner Monotonicity and Convexity that may help in overcoming these issues.

Aspects of the ill-posed inverse problem of deautoconvolution

Bernd Hofmann TU Chemnitz Germany Co-Author(s):    Yu Deng and Frank Werner

Abstract: There is extensive mathematical and physical literature on the ill-posed inverse problem of deautoconvolution for the reconstruction of real-valued as well as complex-valued functions $x$ with support on the unit interval $[0,1] \subset \mathbb{R}$ from its autoconvolution $y=x*x$, and we mention some application in laser optics. However, little is known about the reconstruction of functions with support on the $d$-dimensional unit cube $[0,1]^d \subset \mathbb{R}^d$ from autoconvolution data. This talk presents recent analytical and numerical results for deautoconvolution in two and more dimensions with different types of data. In particular, there are new assertions on uniqueness or twofoldness of solutions to the deautoconvolution problem in the multidimensional case, which are based on extensions of the Titchmarsh convolution theorem published by Lions and Mikusi\`{n}ski.

Inverse problems in population models

Catharine WK Lo City University of Hong Kong Hong Kong Co-Author(s):    Yuhan Li, Hongyu Liu

Abstract: I will discuss several recent works focusing on inverse problems in population ecological models, such as Lotka-Volterra models, chemotaxis models, and aggregation models. These works delve into the unique identifiability of multiple parameters in these models, including interaction terms, compression, prey attack, crowding, carrying capacity, diffusion coefficients, environmental factors such as gravitational potential and the oxygen carrying-capacity. These studies contribute unique insights and novel methodologies to advance the understanding of inverse problems for complex systems across diverse scientific disciplines.

Time Behavior of Acoustic Resonators and Applications to Inverse Problems

Mourad Sini Austrian Academy of Sciences Austria Co-Author(s):    Long Li and Soumen Senapati

Abstract: We deal with the time-domain acoustic wave propagation in the presence of subwavelength resonators given by small scaled bubbles enjoying high contrasting mass density and bulk modulus. It is well known that such bubbles generate a single subwavelength resonance called Minnaert resonance. We derive the point-approximation expansion of the wave field in terms of the contrasting scales. The dominant part is a sum of two terms. The first one, i.e. the primary wave, is the one generated in the absence of the bubble. The second one, i.e. the resonant wave, is generated by the interaction between the bubble and the background. 1. We estimate the birth-time of the resonant wave. This is nothing but the travel time needed by any wave to reach the location of the bubble. 2. We show that the life-time of the resonant wave is inversely proportional to the imaginary part of the resonance. 3. In addition, the period of the resonant wave is characterized by the real part of this resonance. The birth time, life-time and period have signatures of the background where the bubble is located. They have important applications in inverse problems for imaging modalities using contrast agents.

Generating customized field concentration via surface transmission resonance

HU YUEGUANG City University of HONG KONG Hong Kong Co-Author(s):    YUEGUANG HU, HONGYU LIU, XIANCHAO WANG, AND DEYUE ZHANG

Abstract: In this paper, we develop a mathematical framework for generating strong customized field concentration locally around the inhomogeneous medium inclusion via surface transmission resonance. The purpose of this paper is twofold. Firstly, we show that for a given inclusion embedded in an otherwise uniformly homogeneous background space, we can design an incident field to generate strong localized field concentration at any specified places around the inclusion. The aforementioned customized field concentration is crucially reliant on the peculiar spectral and geometric patterns of certain transmission eigenfunctions. Secondly, we prove the existence of a sequence of transmission eigenfunctions for a specific wavenumber and they exhibit distinct surface resonant behaviors, accompanying strong surface-localization and surface-oscillation properties. These eigenfunctions as the surface transmission resonant modes fulfill the requirement for generating the field concentration.

Propagation of chaos rate across dimensions and the L^p convergence rate of the numerical approximation for super-linear MV-SDEs

Yuhang Zhang Harbin Institute of Technology Zhengzhou Research Institute Peoples Rep of China Co-Author(s):    Minghui Song

Abstract: In this paper, we study the $\mathcal{L}^p$ convergence rate of the numerical approximation to the solution of the McKean-Vlasov stochastic differential equations (MV-SDEs) with super-linear growth in the spatial component in the drift. In contrast to standard SDEs, MV-SDEs require an approximation of the distribution law, and here we adopt the stochastic particle method to approximate the true measure using the empirical measure, the time-stepping scheme adopted here is the tamed Euler method. First, we show the strong convergence rate of the propagation of chaos (PoC) is of order $O(N^{-1/{2}})$ under the $\mathcal{L}^p$-norm for any $p\ge 2$, where N is the number of weakly interacting particles. This order is not only better than the existing results, but it is also across dimensions. In the second part we prove that the tamed Euler method is strongly convergent with the order of $O(\Delta^{1/2})$, which is consistent with the classical results. Finally, we present numerical experiments which confirm our theoretical estimates.