Abstract : Anomalous diffusion has received a lot of attention recently due to its extraordinary capability for describing nonstandard diffusion processes arising in multiple physical sciences and engineering. The relevant mathematical models often involve a fractional-order derivative in time or space. The nonlocality of the model substantially changes the analytical behaviour of the mathematical models when compared with the standard counterpart. This has also big impact on the behaviour of related inverse problems, which has witnessed many exciting and important developments in the last few years. In this mini-symposium, we aim at gathering researchers working on the topic to discuss recent advances on mathematical and numerical analysis of inverse problems for anomalous diffusion, in order to further promote the developments of the topic.
[01356] Coefficient identification space-fractional equation with Abel type operators
Format : Online Talk on Zoom
Author(s) :
Barbara Kaltenbacher (University of Klagenfurt)
Abstract : We consider the inverse problem of recovering an unknown, spatially-dependent coefficient $q(x)$ from the fractional order equation $\mathbb{L}_\alpha u = f$ defined in a two-dimensional spatial domain from boundary information.
Here $\mathbb{L_\alpha} ={D}^{\alpha_x}_x +{D}^{\alpha_y}_y +q(x)$ contains fractional derivative operators based on the Abel fractional integral.
We develop uniqueness and reconstruction results and show how the ill-conditioning of this inverse problem depends on the geometry of the region and the fractional powers $\alpha_x$ and $\alpha_y$.
[04481] Inverse problems for simultaneous determination of several scalar parameters and source factors in anomalous diffusion equations
Format : Online Talk on Zoom
Author(s) :
Jaan Janno (Tallinn University of Technology)
Abstract : We consider inverse problems for anomalous diffusion equations where unknowns are orders of multiterm fractional time derivatives/multiterm fractional Laplacians or kernels of distributed fractional time derivatives/distributed fractional Laplacians. Along with these quantities an unknown is also a space-dependent or time-dependent source factor. We prove uniqueness for simultaneous determination of these quantities form final or boundary data.
[03427] Parameter inverse problem for coupled time-fractional diffusion systems
Format : Talk at Waseda University
Author(s) :
Yikan Liu (Hokkaido University)
Abstract : This talk is concerned with determining fractional orders in a coupled system of time-fractional diffusion equations. Defining the mild solution, we first establish fundamental unique existence of the solution, which mostly inherit those of a single equation. Owing to the coupling effect, we obtain the uniqueness for determining all orders by the single point observation of a single component of the solution.
[04183] Identification of potential in diffusion equations from terminal observation
Format : Online Talk on Zoom
Author(s) :
Zhidong Zhang (Sun Yat-sen University)
Abstract : We consider an inverse potential problem in a (sub)diffusion equation. We construct a monotone operator, one of whose fixed points is the unknown potential. The uniqueness of the identification is proved. Next, a fixed point iteration is applied to reconstruct the potential. We prove the linear convergence of the iterative algorithm and present a thorough error analysis for the reconstructed potential. Numerical experiments are provided to illustrate and complement our theoretical analysis.
[04736] Classical Unique Continuation Property for Time Fractional Evolution Equations
Format : Talk at Waseda University
Author(s) :
Gen Nakamura (Hokkaido University)
Ching-Lung LIn (National Cheng Kung University)
Abstract : Let $q_j=q_j(x),\,\,2\le j\le m$ with $q_1=1$ and let $2>\alpha=\alpha_1>\alpha_2>\cdots>\alpha_m>0$.
Then, the classical unique continuation property of solutions in $H^{\alpha,2}((0,T)\times\Omega)$ holds for the time
fractional evolution equation (tfEE) whose leading part given as
$\sum_{j=1}^mq_j\partial_t^{\alpha_j} u(t,y)-{\it L}\, u(t,y)$ over a domain $\Omega\subset{\mathbb R}^n$
with a time dependent strongly eilliptic operator $L$ of oder $2$, where $\partial_t^{\alpha_j}$ is the Caputo derivative whenever $\alpha_j\not\in{\mathbb Z}$.
See C-L. Lin and G. Nakamura, Math. Ann. 385 (2023), pp. 551–574 for the details.
[02704] The Calderón problem for nonlocal parabolic operators
Format : Talk at Waseda University
Author(s) :
Yi-Hsuan Lin (Department of Applied Mathematics, National Yang Ming Chiao Tung University)
Abstract : We investigate inverse problems in the determination of leading coefficients for nonlocal parabolic operators, by knowing the corresponding Cauchy data in the exterior space-time domain. The key contribution is that
we reduce nonlocal parabolic inverse problems to the corresponding local inverse problems with the lateral boundary Cauchy data. In addition, we derive a new equation and offer a novel proof of the unique continuation property for this new equation. We also build both uniqueness and non-uniqueness results for both nonlocal isotropic and anisotropic parabolic Calder´on problems, respectively.
[02699] Inverse Problems for Subdiffusion from Observation at an Unknown Terminal Time
Format : Online Talk on Zoom
Author(s) :
Bangti Jin (The Chinese University of Hong Kong)
Yavar Kian (Aix Marseille University)
Zhi Zhou (The Hong Kong Polytechnic University)
Abstract : Time-fractional subdiffusion equations represent an important class of mathematical models with a broad range of applications. The related inverse problems of recovering space-dependent parameters, e.g., initial condition, space dependent source or potential coefficient, from the terminal observation have been extensively studied in recent years. However, all existing studies have assumed that the terminal time at which one takes the observation is exactly known. In this talk, we present uniqueness and stability results for three canonical inverse problems, e.g., backward problem, inverse source and inverse potential problems, from the terminal observation at an unknown time. The subdiffusive nature of the problem indicates that one can simultaneously determine the terminal time and space-dependent parameter.
[04403] Numerical Recovery of Multiple Parameters from One Lateral Boundary Measurement
Format : Talk at Waseda University
Author(s) :
Siyu Cen (The Hong Kong Polytechnic University)
Bangti Jin (Chinese University of Hong Kong)
Yikan Liu (Hokkaido University)
Zhi Zhou (The Hong Kong Polytechnic University)
Abstract : This talk is concerned with numerically recovering multiple parameters in a partly unknown subdiffusion model from one lateral measurement on the boundary. We prove that the boundary measurement uniquely determines the fractional order and the polygonal support of the diffusion coefficient, without knowing either the initial condition or the source. We present an algorithm for recovering the fractional order and diffusion coefficient which combines small-time asymptotic expansion, analytic continuation and the level set method.
[04435] Long-Short time asymptotic estimates for time-fractional diffusion-wave equation
Format : Talk at Waseda University
Author(s) :
Xinchi HUANG (Tokyo Institute of Technology)
Xinchi Huang (Tokyo Institute of Technology)
Yikan Liu (Hokkaido University)
Abstract : In this talk, we consider the time-fractional diffusion-wave equations and show the long-time asymptotic estimate of the solution, which can be used to prove the long-time strict positivity of the solution and the uniqueness for an inverse source problem of determining the time-varying factor. Besides, we discuss the short-time asymptotic behavior and provide an application to the determination of the spatial varying factor in the source.
[04612] Numerical identification of conductivity in (sub)diffusion equations from terminal measurement
Format : Online Talk on Zoom
Author(s) :
Bangti Jin (Chinese University of Hong Kong)
Xiliang Lu (Wuhan University)
Qimeng Quan (Wuhan University)
Zhi Zhou (The Hong Kong Polytechnic University)
Abstract : This talk focuses on the inverse diffusion problems in (sub)diffusion equation. Typically, we employ Tikhonov strategy and discretize the regularized model by finite element methods. One critical issue is to establish a priori error estimate on the concerned parameter. In this talk, the speaker will discuss their recent study of recovering a space-dependent diffusion coefficient from terminal observation by a novel conditional stability.
