Abstract
The purpose of this study is twofold. First, we revisit a shape optimization reformulation of a shape inverse problem and propose a simple and efficient numerical approach for realizing the minimization problem. Second, we examine the short-time existence and uniqueness of a classical solution to a Hele-Shaw-like system derived from the shape optimization formulation of the shape inverse problem. This is a joint work with Dr. Julius Fergy T. Rabago, Kanazawa University.

Abstract
In the theory of viscoelasticity, an important class of models admits a representation in terms of springs and dashpots. Widely used members of this class are the Maxwell model and its extended version. This talk concerns resolvent estimates for the system of equations for the anisotropic, extended Maxwell model, abbreviated as the EMM, and its marginal realization which includes an inertia term; special attention is paid to the introduction of augmented variables.

Abstract
Data assimilation (DA) is a computational technique that integrates numerical simulation models and observational data based on Bayesian statistics. The four-dimensional variational method (4DVar) is widely used in DA for large-scale models, such as weather forecasting, to optimize model parameters and/or initial conditions and to evaluate their uncertainties. We introduce the foundation of the 4DVar, including our algorithm that is capable of uncertainty quantification [1]. Then we show application examples of the 4DVar to the models in solid Earth science related to slow earthquakes [2] and mantle convection [3,4].

References
[1] Ito, S., H. Nagao, A. Yamanaka, Y. Tsukada, T. Koyama, M. Kano, and J. Inoue, Data assimilation for massive autonomous systems based on a second-order adjoint method, Phys. Rev. E 94 (2016) 043307.
[2] Ito, S., M. Kano, and H. Nagao, Adjoint-based uncertainty quantification for inhomogeneous friction on a slow-slipping fault, Geophys. J. Int. 232(1) (2023) pp. 671--683.
[3] Nakao, A., T. Kuwatani, S. Ito, and H. Nagao, Adjoint-based data assimilation for reconstruction of thermal convection in a highly viscous fluid from surface velocity and temperature snapshots, Geophys. J. Int. 236(1) (2024) pp. 379--394.
[4] Nakao, A., T. Kuwatani, S. Ito, and H. Nagao, Adjoint-based marker-in-cell data assimilation for constraining thermal and flow processes from Lagrangian particle records, J. Geophys. Res. Machine Learning and Computation in revision.

Abstract
For a large earthquake, it usually takes several tens second to a few minutes to complete the rupture in the scale of several to tens meters. Slow earthquake, however, needs several hours to days to complete the fault slip process. While a regular earthquake is a quick slip that causes strong ground shaking, a slow earthquake is a slow slip causing weak tremble on the surface. During recent decades, the discovery of slow earthquakes in the forms of slow slip events (detected geodetically) and tectonic tremors (detected seismologically) have established a broad spectrum of fault slip. Mounting evidences show that slow and megathrust earthquakes are spatially complementary in distribution, and slow earthquakes sometimes trigger large earthquakes in their vicinities. Therefore, improved understanding of the interplay between slow slip and regular earthquakes is essential to better understand the process of earthquake nucleation. However, the role of slow slip in the earthquake cycle remains controversial. Abundant ${\rm M} \ge 6$ earthquakes and aseismic slip in a double-vergence suture in eastern Taiwan provide a rare opportunity to better understand the role of aseismic slip in the earthquake cycle deformation and triggering relationships. In this talk I will walk you into the slow earthquake observations in Taiwan for their relationship with a ${\rm M7.3}$ earthquake earlier this year in April and seek for advices on how the mathematical approaches will help the better understanding of seismic-aseismic interaction.

Abstract
In this talk we present a novel numerical method to x-ray computed tomography where measurement data is given on an arc. It is reduced to the inverse source problem of the transport equation by employing A-analytic theory by A. L. Bukhgeim. A Cauchy type boundary integral formula plays an essential role in reconstruction, and by applying the jump relation a Cauchy type singular integral equation is introduced for treating the partial measurement setting. We also quantitatively estimate the numerical instability of the singular integral equation in terms of the condition number, and prove that it increases at most in polynomial order. This means the numerical feasibility of the x-ray tomography under the current situation. Some numerical experiments are exhibited in the presentation. We also show an application of our strategy to the initial value problem of elliptic equations.
This talk is based on a joint project with Kamran Sadiq (RICAM) and Alexandru Tamasan (University of Central Florida).

References
[1] H. Fujiwara, N. Oishi, K. Sadiq, and A. Tamasan, Numerical computation of x-ray computerized tomography from partial measurement, JASCOME 21 (2021) pp. 37--41.
[2] H. Fujiwara, K. Sadiq, and A. Tamasan, Numerical reconstruction of radiative sources from partial boundary measurements, SIAM J. Imaging Sci. 16 (2023) pp. 948--968.

Abstract
We often encounter matrices whose pattern (zero-nonzero, or sign) is known while the precise value of each entry is not clear. Thus, a natural question is what we can say about the spectral property of matrices of a given pattern. When the matrix is real and symmetric, one may use a simple graph to describe its off-diagonal nonzero support. For example, it is known that an irreducible tridiagonal matrix (whose pattern is described by a path) only allows eigenvalues of multiplicity one. In contrast, a periodic Jacobi matrix (whose pattern is described by a cycle) allows multiplicity two but no more. The inverse eigenvalue problem of a graph (IEPG) focuses on the matrices whose pattern is described by a given graph and studies their possible spectra. In this talk, we will go through some of the histories of the IEPG and see how combinatorial methods (zero forcing) and analytic methods (implicit function theorem) can come into play in modern-day research.

Abstract
Stommel (1950) considered an Oberbeck-Boussinesq model of thermal convection in a liquid layer with non-uniform heat supply under the gravity. The stress free surface is maintained at the temperature $T = t \cos (\pi l x), 0 < x < 1/l$. He obtained approximate solutions by an asymptotic expansion and showed those pictures of contour line of the stream function and isothermal lines. It may be considered as a simplest model of thermal effect to the ocean current.
Here we have an existence theorem of stationary solutions of the system under stress free boundary condition on the top surface and fixed boundary condition on the bottom, and we show some flow patterns by the numerical computations, which include some cases out of the assumptions of the theorem.

References
[1] Henry Stommel, `` An example of thermal convection '', Transactions, American Geophysical Union, 31, No.4, (1950).
[2] Hiroshi Fujiwara and Takaaki Nishida, ``Heat convection in the horizontal layer with non-uniform heat supply'', Japan Journal of Industrial and Applied Mathematics, (2024). https://doi.org/10.1007/s13160-024-00655-5.
[3] Paul C. Fife, `` The Bénard problem for general fluid dynamicl equations and remarks on the Boussinesq approximation '', Indiana University Mathematics Journal, 20, p. 303, (1970).

Abstract
In this talk I will consider the inverse boundary value problem for a quasilinear, anisotropic, elliptic equation of the form $\nabla\cdot(\gamma\nabla u+|\nabla u|^{p-2}\nabla u)=0$, where $\gamma$ is a smooth, matrix valued, function with a uniform lower bound. I will show that the Dirichlet-to-Neumann map determines the coefficient matrix uniquely, in dimension 3 and higher. This stands in contrast to the classical linear anisotropic Calderón problem where there is a known obstruction to uniqueness due to the invariance of the boundary data under transformations of the equation by boundary fixing diffeomorphisms.

Abstract
We present an approach to accurately determine the Lagrange multiplier for volume constraints in topology optimization based on the level-set method. In the level-set method, the material domain subject to optimization is modeled as the region where the level-set function is positive, and the optimal shape is found by evolving the distribution of the level-set function such that it satisfies the reaction-diffusion equation. The reaction term in the reaction-diffusion equation includes the topological derivative and the Lagrange multiplier derived from the volume constraint. In conventional level-set methods, it has been difficult to fully satisfy the volume constraint during the iterative optimization process. In this study, we separate the level-set function into a part related to the topological derivative and a part related to the Lagrange multiplier for the volume constraint. By solving the reaction-diffusion equations for each part, we are able to satisfy the volume constraint accurately from the beginning of the iterative process.

References
[1] Y. Cui, T. Takahashi, K. Taji, and T. Matsumoto, The influence of volume constraint method on achieving the exact boundary representation in FEM-based topology optimization: Case studies, Transactions of JASCOME, 22, Paper No. 22-221216, pp. 189--198.
[2] Y. Cui , T. Takahashi, and T. Matsumoto, An exact volume constraint method for topology optimization via reaction-diffusion equation, Computers and Structures 280 (2023) 106986, https://doi.org/10.1016/j.compstruc.2023.106986.
[3] Y. Cui, S. Yoon, S. Guit, T. Takahashi, and T. Matsumoto, A generalized exact volume constraint method for the topology optimization based on the nonlinear reaction-diffusion equation, Transactions of JASCOME, 23, Paper No. 23-231124, pp. 81--92.

Abstract
We employ the enclosure method to reconstruct unknown inclusions within an object that is governed by a semilinear elliptic equation with power-type nonlinearity. Motivated by [1], we tried to solve the problem without using special solutions such as complex geometrical optics solutions to the equation. Instead, we construct approximate solutions obtained from Taylor approximation of the solution operator. By incorporating such approximate solutions into the definition of an indicator functional, we are able to use the classical Calderón-type harmonic functions to accomplish the reconstruction task.

References
[1] Matti Lassas, Tony Liimatainen, Yi-Hsuan Lin, and Mikko Salo. Inverse problems for elliptic equations with power type nonlinearities. Journal de mathématiques pures et appliquées, 145:44-82, 2021.

Abstract
In this work, we consider the inverse scattering problem of determining an unknown refractive index from the far-field measurements using the nonparametric Bayesian approach. We use a collection of large ``samples'', which are noisy discrete measurements taking from the scattering amplitude. We will study the frequentist property of the posterior distribution as the sample size tends to infinity. Our aim is to establish the consistency of the posterior distribution with an explicit contraction rate in terms of the sample size. We will consider two different priors on the space of parameters. The proof relies on the stability estimates of the forward and inverse problems. Due to the ill-posedness of the inverse scattering problem, the contraction rate is of a logarithmic type. We also show that such contraction rate is optimal in the statistical minimax sense.
This is based on joint work [1] with Pu-Zhao Kow (National Chengchi University), Jenn-Nan Wang ( National Taiwan University).

References
[1] Takashi Furuya, Pu-Zhao Kow, Jenn-Nan Wang, Consistency of the Bayes method for the inverse scattering problem, Inverse Problems, 40, 055001.

Abstract
In this talk, we consider the problem of recovering the density of a Herglotz wave function. The Herglotz wave function can be expressed in terms of Fourier transform. It is shown that the number of features that can be stably recovered (stable region) becomes larger as the frequency increases, whereas one has strong instability for the rest of the features (unstable region). We prove that the singular values of the forward operator stay roughly constant in the stable region and decay exponentially in the unstable region.
This talk is prepared based on our work (https://arxiv.org/abs/2404.18482).