Abstract : The identification of unknown ingredients in wave, oscillation and flow phenomena governed by evolutionary PDEs from observational data is a central challenge in a variety of areas of science and engineering. This mini-symposium aims at gathering together international scientific researchers to share the latest progress on the related control and inverse problems. It provides a platform to discuss recent developments and emerging challenges, which include but are not limited to
1. Stability and controllability for inverse parabolic and hyperbolic problems;
2. Uniqueness of inverse problems for subdiffusion and viscoelasticity;
3. Data-driven inversion methods and optimal control;
4. Related numerical schemes for reconstruction.
[03795] Recovery in vivo viscoelasticity from elastography measured data
Format : Talk at Waseda University
Author(s) :
Yu Jiang (Shanghai University of Finance and Economics)
Abstract : This talk will briefly describe how to solve the inverse problem of recovering in vivo viscoelasticity from elastography (magnetic resonance elastography, ultrasound elastography) measurements. To solve it robustly, one need to have a proper partial differential equation model to describe the wave motion inside living body. And based on this PDE model and given interior measurements, theoretical and numerical inverse analyzes need to be performed. As the PDE model, we start with a dynamic viscoelastic model and several simplified models are given. For inversion analysis, we give several practical numerical inversion methods to identify viscoelasticity, such as regularized numerical differentiation method etc.
[03880] Acousto-electric tomography imaging model and algorithm based on two-point gradient method
Format : Talk at Waseda University
Author(s) :
Min Zhong (School of Mathematics, Southeast University )
Abstract : We study the numerical reconstruction problem in acousto-electric tomography of recovering the conductivity distribution in a bounded domain from interior power density data. We propose a numerical method for recovering discontinuous conductivity distributions, by utilizing the two point gradient method, the piecewise constant conductivity can be efficiently reconstructed. Extensive numerical experiments are presented to illustrate the feasibility of the proposed approach.
[04269] Numerical method for unique continuation of elliptic equations and applications
Format : Talk at Waseda University
Author(s) :
Yu Chen (Shanghai University of Finance and Economics)
Jin Cheng (Fudan University)
Abstract : Numerical method for unique continuation of elliptic equations is related to information reconstruction from interior local measurements. The conditional stability of unique continuation along analytic sub-manifolds in three dimensions will be provided. A stable numerical algorithm is constructed based on Tikhonov regularization according to the conditional stability, in order to deal with the ill-posedness. The evaluation of reliable region of a reconstruction and potential applications of the present method in atmospheric flows will be discussed.
[04089] Reconstruction of piecewise smooth diffusion coefficient and initial value with adaptive regularization
Format : Talk at Waseda University
Author(s) :
Shuli CHEN (Hokkaido University)
Haibing Wang (Southeast University)
Abstract : This talk will present an inverse problem of simultaneously reconstructing the piecewise smooth diffusion coefficient and initial value in a diffusion equation from extra measurements. The inverse problem is transformed into solving an optimization problem with a switchable regularization to simultaneously preserve multiple properties of unknown functions. The existence, stability and consistency result are rigorously analyzed by introducing a domain index. Then, we decompose the optimization problem into a SOLVE-MARK-REFINE-looping scheme and develop an adaptive iterative algorithm. Finally, we present several numerical examples to show the validity of the proposed algorithm.
[04229] Inverse problems for the Duffing equation in pediatrics
Format : Talk at Waseda University
Author(s) :
Manabu Machida (Hamamatsu University School of Medicine)
Abstract : The prognostic prediction of the neonatal hypoxic-ischemic encephalopathy (HIE) has been tried with near-infrared spectroscopy. In this talk, the Duffing equation is proposed as a model which governs the time-evolution of the cerebral blood volume. We will consider an inverse problem of determining coefficients and initial values of the equation.
[04402] Unique determination of source and Robin coefficient in fractional diffusion equation
Format : Talk at Waseda University
Author(s) :
Zhiyuan Li (Ningbo University)
Daijun Jiang (Central Central China Normal UniversityChina Normal University)
Abstract : In this talk, we consider an inverse problem of simultaneously determining the spatially dependent source term and the Robin boundary coefficient in a time fractional diffusion equation, with the aid of extra measurement data at a subdomain near the accessible boundary. Firstly, the spatially varying source is uniquely determined in view of the unique continuation principle and Duhamel principle for the fractional diffusion equation. The Hopf lemma for a homogeneous time-fractional diffusion equation is proved and then used to prove the uniqueness of recovering the Robin boundary coefficient.
[05327] Fine-tuning neural-operator architectures for training and generalization
Format : Talk at Waseda University
Author(s) :
Takashi Furuya (Shimane University)
Abstract : We investigate the generalization error of Neural Operators (NOs), which aim to approximate the forward operator in PDEs, and propose modifications of NOs, referred to as ${\textit{s}}{\text{NO}}+\varepsilon$. We establish generalization error bounds for both NOs and ${\textit{s}}{\text{NO}}+\varepsilon$, and we observe that our bound is sharper than that for NOs. Additionally, our experiments, particularly the wave equation experiment, demonstrate that our proposed network exhibits remarkable generalization capabilities, whereas NOs perform poorly in out-of-distribution scenarios.
[03684] Reconstruction of location for a single point target in time-domain fluorescence diffuse optical tomography
Format : Talk at Waseda University
Author(s) :
Junyong Eom
Gen Nakamura (Hokkaido University)
Goro Nishimura (Hokkaido University)
Chunlong Sun (Nanjing University of Aeronautics and Astronautics)
Abstract : The time-domain fluorescence diffuse optical tomography (FDOT) problem is to recover the distribution of fluorophores in biological tissue from the time-domain measurement on the boundary. The measurement is conducted by several pairs (S-D pairs) of a point source and a point detector. In this paper, we identify the location of the distribution of fluorophores over a point, refer as a point target. We first express a solution for the forward problem in a dimensionless form and consider its asymptotic expansion. Then, we theoretically investigate the minimal number of S-D pairs to determine the point target location, analyzing the sensitivity matrix. Finally, we numerically verify the invertibility of the matrix and demonstrate the local solvability for locating well-separated multiple point targets.
[04623] Carleman estimates and some inverse problems for the coupled quantitative thermoacoustic equations
Format : Online Talk on Zoom
Author(s) :
Michel Cristofol (Aix-Marseille Université)
Shumin Li (University of Science and Technology of China)
Yunxia Shang (Shanghai Normal University)
Abstract : We consider the determination of a coefficient or the source term in a strong coupled quantitative thermoacoustic system of equations. For this purpose, we establish a Carleman estimate for the coupled quantitative thermoacoustic equations. Applying this Carleman estimate, we prove stability estimates of Hölder type for inverse problems involving the observation of only one component: the temperature or the pressure.
[05434] Inverse problems for transport equations
Format : Online Talk on Zoom
Author(s) :
Giuseppe Floridia (Sapienza Università di Roma)
Abstract : In this talk we present several inverse problems, introduced in recent papers, for transport equations via Carleman
estimates. We approach also the general case of first-order hyperbolic equations with time-dependent coefficients.
[04572] Unique continuation for wave equations in asymptotically anti-de Sitter spaces
Format : Talk at Waseda University
Author(s) :
Hiroshi Takase (Kyushu University)
Abstract : An asymptotic anti-de Sitter space is a Lorentzian manifold with a Lorentz metric diverging at the boundary of the manifold. It is also used as a model of outer space, and in particular, the problem of determining the inner structure from the data at the boundary has attracted attention in theoretical physics as AdS/CFT correspondence. The wave equation in this space is a degenerate equation. In this talk, I will present results on the fundamental property of unique continuation for this wave equation.
[05416] STABILITY ESTIMATES FOR AN INVERSE PROBLEM FOR SCHRODINGER OPERATORS AT HIGH FREQUENCIES FROM ARBITRARY PARTIAL BOUNDARY MEASUREMENTS
Format : Online Talk on Zoom
Author(s) :
Ganghua Yuan
Xiaomeng Zhao (Northeast Normal University)
Abstract : In this talk, we consider the partial data inverse boundary value problem for the Schrodinger operator at a high frequency in a bounded domain in $\mathbb R^n$, $n\geq3$. Assuming that the potential is known in a neighborhood of the boundary, we obtain the logarithmic stability when both Dirichlet and Neumann data are taken on arbitrary open subsets of the boundary . We used a method combining the CGO solution, Runge approximation and Carleman estimate.
