arXiv subjects

Gen Nakamura

Publications and source records attributed to Gen Nakamura.

At least 19 recordsLinked to original sources

arXiv2026

Reconstruction of time-dependent coefficients in a semilinear dynamical Schr{\"o}dinger equation

In the present manuscript, we study an inverse problem related to a semilinear dynamical Schr{\"o}dinger equation with lower order terms, in a bounded domain of $\Rb^{1+n},n\geq 2$. Our focus is on determination of the time-dependent coefficients appearing in the aforementioned equation, from the boundary measurements of the solutions. More precisely, we establish the {pointwise reconstruction} formulae for determining the time-dependent coefficients of linear and nonlinear terms from the knowledge of Dirichlet-to-Neumann map. Since the concerned non-linear Schr\"odinger equation possesses a trivial solution, we linearize the equation around the trivial solution and use the asymptotic solutions (\textit{with concentrated amplitudes}) of the linearized problem for reconstructing the aforementioned coefficients. To be more specific, we use first-order linearization to reconstruct vector and scalar potentials associated with the coefficients of linear terms and the higher-order linearization technique is used to reconstruct coefficients of nonlinearity. The nonlinear equation considered in this manuscript can be seen as a generalization of the Gross-Pitaevskii equation (GPE), which is employed to describe the dynamics of dilute Bose-Einstein condensates (BEC).

math.AP↗

arXiv2026

Conditonal Lipschitz stability for the Inverse Problem of the 2D Navier-Stokes System in a Bounded Domain

This paper concerns an inverse problem for the initial boundary value problem of the two-dimensional Navier-Stokes system defined in a bounded simply connected domain with slip, vorticity boundary conditions, and a global vorticity invariant constraint. We establish conditional Lipschitz stability and a local recovery for this inverse problem, where the velocity field and space-independent boundary vorticity are locally recovered from the given initial velocity field and the global vorticity invariant. Our analysis is based on well-posedness estimates and energy methods for the vorticity transport equation.

math.AP↗

arXiv2026

Uniqueness of dynamic elastography for isotropic standard linear solid viscoelastic media

Dynamic elastography is a widely used, safe, convenient, and cost-effective method to aid in medical diagnosis. It visualizes the wave field propagating through living tissues and quantitatively determines the wave propagation speed from the acquired data, thereby enabling the extraction of the viscoelastic properties of in vivo tissues. Notably, this identification process relies on the mathematical modeling of the viscoelastic characteristics of living tissues. When living tissues are simply modeled as isotropic elastic media, J. McLaughlin and J. Yoon established the uniqueness of the identification in \cite{MY} by reasoning that they called the ``shrink and spread argument". Given the realistic viscoelastic nature of biological tissues, generalizing their results by adopting viscoelastic models is of great significance. In this paper, using their reasoning, we prove the uniqueness of identification for two typical viscoelastic media: the isotropic extended Maxwell model and the isotropic extended standard linear solid model. More precisely, we demonstrate that the shear wave speed within a region of interest $\Omega$ can be uniquely determined from a single measurement of the wave field in $\Omega$.

math.AP↗

arXiv2026

Explicit form of relaxation tensor for isotropic extended Burgers model and its spectral inversion

Concerning the anelastic nature of Earth, the quasi-static extended Burgers model (abbreviated by q-EBM), an integro-differential system, is used to study the free oscillation of Earth (abbreviated by FOE). In this paper, we first provide a general method to obtain an explicit form of the relaxation tensor for inhomogeneous isotropic q-EBM. Then, we apply it to compute the eigenvalues of the free oscillation of Earth, assuming that Earth is a unit ball modeled as a homogeneous and isotropic q-EBM. So far, an analytical and systematic way to compute the eigenvalues of the FOE has been missing when modeling Earth as a q-EBM. In particular, we compute some clusters of eigenvalues (abbreviated by C-ev's). To be more precise, integrating by parts with respect to time of the q-EBM under the assumption that the initial strain is zero, the q-EBM becomes the sum of two terms. The first term, called the instantaneous term, doesn't have any integration with respect to time, but the second term, called the memory term, has such an integration. Then, consider the eigenvalues of the instantaneous part of the q-EBM. The eigenfunctions of C-ev's share the same eigenfunctions of the instantaneous part. However, the C-ev's may be shifted from the eigenvalues of the instantaneous part. Further, we analyze the structure of C-ev's and provide an inversion formula identifying the q-EBM from the C-ev's.

math.AP↗

arXiv2025

Inverse spectral problem for glassy state relaxation approximated by Prony series

The stretched exponential relaxation function is used to analyze the relaxation of the glassy state data. Due to the singularity of this function at the origin, this function is inconvenient for data analysis. Concerning this, a Prony series approximation of the stretched exponential relaxation function (J. Mauro, Y. Mauro, 2018), which is the extended Burgers model (abbreviated by EBM) known for viscoelasticity equations, was introduced. In our previous paper [arXiv:2509.16714], we gave an inversion method to identify the relaxation tensor of the EBM using clustered eigenvalues of the quasi-static EBM. As a next important research subject of this study, we numerically examine the performance of the inversion method. The performance reveals that it is a powerful method of data analysis, analyzing the relaxation of the glassy state data.

math.NA↗

arXiv2025

Clustered eigenvalue problem for glassy state relaxation and its inverse problem

For computational convenience, a Prony series approximation of the stretched exponential relaxation function of homogeneous glasses has been proposed (J. Mauro, Y. Mauro, 2018), which is the extended Burgers model known for viscoelasticity equations. The authors of [P. Loreti and D. Sforza, 2019] initiated a spectral analysis of glass relaxation along this line, and gave some numerical results on clusters of eigenvalues. A theoretical justification of the results and development of further numerical studies were left open. In this paper, we provide a complete theoretical justification of their results and their numerical verification. Besides these, we solve an inverse spectral problem for clusters of eigenvalues associated with the glass relaxation.

math.AP↗

arXiv2025

On the convergence of the no-response test for the heat equation

Domain sampling methods called the range test (RT) and no-response test (NRT), and their duality are known for several inverse scattering problems and an inverse boundary value problem for the Laplace operator (see Section 1 for more details). In our previous work [21], we established the duality between the NRT and RT, and demonstrated the convergence of the RT for the heat equation. We also provided numerical studies for both methods. However, we did not address the convergence for the NRT. As a continuation of this work, we prove the convergence of the NRT without using the duality. Specifically, assuming there exists a cavity $D$ inside a heat conductor $\Omega$, we define an indicator function $I_{NRT}(G)$ for a prescribed test domain $G$, where $\overline G\subset\Omega$ (i.e., $G\Subset\Omega$). By using the analytical extension property of solutions to the heat equation with respect to the spatial variables, we prove the convergence result given as $I_{NRT}(G)<\infty$ if and only if $\overline{D}\subset \overline{G}$, provided that the solution to the heat equation cannot be analytically extended across the boundary of the cavity. Thus, we complete the theoretical study of both methods. Here the analytic extension of solutions does not require the property that the solutions are real analytic with respect to the space variables. However, for the proof of the mentioned convergence result, we fully use this property.

math.AP↗

arXiv2025

Direct inversion scheme of time-domain fluorescence diffuse optical tomography by asymptotic analysis of peak time

This paper proposes a direct inversion scheme for fluorescence diffuse optical tomography (FDOT) to reconstruct the location of a point target using the measured peak time of the temporal response functions. A sphere is defined for the target, with its radius determined by the peak time, indicating that the target lies on the sphere. By constructing a tetrahedron with edges determined by the radii, we identify the location of the target as the vertex of the tetrahedron. Asymptotically, we derive the relationship between the radius of the sphere and the peak time. Several numerical tests are implemented to demonstrate the accuracy and performance of the asymptotic relationship and the inversion scheme.

math.AP↗

arXiv2025

Convergence Analysis of Levenberg-Marquardt Method for Inverse Problem with H\"{o}lder Stability Estimate

We analyze convergence of the Levenberg-Marquardt method for solving nonlinear inverse problems in Hilbert spaces. Specifically, we establish local convergence and convergence rates for a class of inverse problems that satisfy H\"{o}lder stability estimate. Furthermore, based on what we found in the mentioned analysis, we develop global reconstruction algorithms for solving inverse problems with finite measurements for exact and noisy data, respectively.

math.FA↗

arXiv2024

Approximate peak time to time-domain fluorescence diffuse optical tomography for nonzero fluorescence lifetime

This paper concerns an inverse problem for fluorescence diffuse optical tomography (FDOT) reconstructing locations of multiple point targets from the measured temporal response functions. The targets are multiple fluorescent point objects with a nonzero fluorescence lifetime at unknown locations. Peak time, when the temporal response function of the fluorescence reaches its maximum, is a robust parameter of the temporal response function in FDOT because it is most less suffered by the artifacts, such as noise, and is easily determined by experiments. We derive an approximate peak time equation based on asymptotic analysis in an explicit way in the case of nonzero fluorescence lifetime when there are single and multiple point targets. The performance of the approximation is numerically verified. Then, we develop a bisection algorithm to reconstruct the location of a single point target from the algorithm proposed in [4] for the case of zero fluorescence lifetime. Moreover, we propose a boundary-scan algorithm for the reconstruction of locations of multiple point targets. Finally, several numerical experiments are implemented to show the efficiency and robustness of the addressed algorithms.

math.NA↗

arXiv2024

The Calder\'on problem for the Schr\"odinger equation in transversally anisotropic geometries with partial data

We study the partial data Calder\'on problem for the anisotropic Schr\"{o}dinger equation \begin{equation} \label{eq: a1} (-\Delta_{\widetilde{g}}+V)u=0\text{ in }\Omega\times (0,\infty), \end{equation} where $\Omega\subset\mathbb{R}^n$ is a bounded smooth domain, $\widetilde{g}=g_{ij}(x)dx^{i}\otimes dx^j+dy\otimes dy$ and $V$ is translationally invariant in the $y$ direction. Our goal is to recover both the metric $g$ and the potential $V$ from the (partial) Neumann-to-Dirichlet (ND) map on $\Gamma\times \{0\}$ with $\Gamma\Subset \Omega$. Our approach can be divided into three steps: Step 1. Boundary determination. We establish a novel boundary determination to identify $(g,V)$ on $\Gamma$ with help of suitable approximate solutions for the Schr\"odinger equation with inhomogeneous Neumann boundary condition. Step 2. Relation to a nonlocal elliptic inverse problem. We relate inverse problems for the Schr\"odinger equation with the nonlocal elliptic equation \begin{equation} \label{eq: a2} (-\Delta_g+V)^{1/2}v=f\text{ in }\Omega, \end{equation} via the Caffarelli--Silvestre type extension, where the measurements are encoded in the source-to-solution map. The nonlocality of this inverse problem allows us to recover the associated heat kernel. Step 3. Reduction to an inverse problem for a wave equation. Combining the knowledge of the heat kernel with the Kannai type transmutation formula, we transfer the inverse problem for the nonlocal equation to an inverse problem for the wave equation \begin{equation} \label{eq: a3} (\partial_t^2-\Delta_g+V)w=F\text{ in }\Omega\times (0,\infty), \end{equation} where the measurement operator is also the source-to-solution map. We can finally determine $(g,V)$ on $\Omega\setminus\Gamma$ by solving the inverse problem for the wave equation.

math.AP↗

arXiv2024

Exact Boundary Controllability for Reduced System Associated to Extended Maxwell Systems

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. The paper concerns about the exact boundary controllability (abbreviated by EBC) for the reduced system (abbreviated by RS) associated to the extended Maxwell model (EMM). The initial boundary value problem (abbreviated by IBP) with a mixed type boundary condition (abbreviated by MBC) in the absence of the exterior force is called the augmented system (abbreviated by AD system). Here, the MBC consists of a homogeneous displacement boundary condition and inhomogeneous traction boundary condition with a boundary control. The RS is a closed subsystem inside the AD system (see Section \ref{sec1} for the details of the EMM and the RS). For the RS, we consider the IBP for the associated AD system. By using a dissipative structure of the RS in relation with the AD system, we will prove the EBC for the RS by a modified version of Russell's principle. Also, as an application of this EBC, we will show a partial boundary controllability (abbreviated by PBC) for the Boltzmann type viscoelastic system of equations (abbreviated by BVS) associated to the EMM. That is, for a large enough time $T>0$ and any pair of given speeds $(v_0, v_1)$, there is a boundary control which steers to have $v(0)=v^0,\,v(t)=v^1$, where $v(t)=\partial_t u(t)$ is the speed of the displacement vector $u(t)$ of BVS at time $t$.

math.AP↗

arXiv2024

Anisotropic extended Burgers model, its relaxation tensor and properties of the associated Boltzmann viscoelastic system

We provide a new method for constructing the anisotropic relaxation tensor and proving its exponential decay property for the extended Burgers model (abbreviated by EBM). The EBM is an important viscoelasticity model in rheology, and used in Earth and planetary sciences. Upon having this tensor, the EBM can be converted to a Boltzmann-type viscoelastic system of equations (abbreviated by BVS). Historically, the relaxation tensor for the EBM is derived by solving the constitutive equation using the Laplace transform. (We refer to this approach by the L-method.) Since inverting the inverse Laplace transform needs a partial fractions expansion, the L-method needs to assume that the EBM elasticity tensors satisfy a commutivity condition. The new method not only avoids this condition but also enables obtaining several important properties of the relaxation tensor, including its positivity, smoothness with respect to the time variable, its exponential decay property together with its derivative, and its causality. Furthermore, we show that the BVS converted from the EBM has the exponential decay property. That is, any solution for its initial boundary value problem with homogeneous boundary data and source decays exponentially as time tends to infinity.

math.AP↗

arXiv2023

Resolvent Estimates for Viscoelastic Systems of Extended Maxwell Type and their Applications

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 paper 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. This leads to the augmented system that will also be referred to as the "original" system. A reduced system is then formed which encodes essentially the EMM; it is a closed system with respect to the particle velocity and the difference between the elastic and viscous strains. Based on resolvent estimates, it is shown that the original and reduced systems generate $C_0$-groups and the reduced system generates a $C_0$-semigroup of contraction. Naturally, the EMM can be written in integrodifferential form leading explicitly to relaxation and a viscoelastic integro-differential system. However, there is a difference between the original and integrodifferential systems, in general, with consequences for whether their solutions generate semigroups or not. Finally, an energy estimate is obtained for the reduced system, and it is proven that its solutions decay exponentially as time tends to infinity. The limiting amplitude principle follows readily from these two results.

math.AP↗

arXiv2023

Uniform Decaying Property of Solutions for Anisotropic Viscoelastic Systems

The paper concerns about the uniform decaying property (abbreviated by UDP) of solutions for an anisotropic viscoelastic system in the form of integrodifferential system (abbreviated by VID system) with mixed type boundary condition. The mixed type condition consists of the homogeneous displacement boundary condition and a homogeneous traction boundary condition or with a dissipation. By using a dissipative structure of this system, we will prove the UDP in a unified way for the two cases, which are, when the time derivative of relaxation tensor decays with polynomial order and it decays with exponential order.

math.AP↗

arXiv2022

Local recovery of a piecewise constant anisotropic conductivity in EIT on domains with exposed corners

We study the local recovery of an unknown piecewise constant anisotropic conductivity in EIT (electric impedance tomography) on certain bounded Lipschitz domains $\Omega$ in $\mathbb{R}^2$ with corners. The measurement is conducted on a connected open subset of the boundary $\partial\Omega$ of $\Omega$ containing corners and is given as a localized Neumann-to-Dirichlet map. The above unknown conductivity is defined via a decomposition of $\Omega$ into polygonal cells. Specifically, we consider a parallelogram-based decomposition and a trapezoid-based decomposition. We assume that the decomposition is known, but the conductivity on each cell is unknown. We prove that the local recovery is almost surely true near a known piecewise constant anisotropic conductivity $\gamma_0$. We do so by proving that the injectivity of the Fr\'echet derivative $F'(\gamma_0)$ of the forward map $F$, say, at $\gamma_0$ is almost surely true. The proof presented, here, involves defining different classes of decompositions for $\gamma_0$ and a perturbation or contrast $H$ in a proper way so that we can find in the interior of a cell for $\gamma_0$ exposed single or double corners of a cell of $\mbox{supp}H$ for the former decomposition and latter decomposition, respectively. Then, by adapting the usual proof near such corners, we establish the aforementioned injectivity.

math.AP↗

arXiv2022

Reconstruction for the time-dependent coefficients of a quasilinear dynamical Schr{\"o}dinger equation

We study an inverse problem related to the dynamical Schr{\"o}dinger equation in a bounded domain of $\Rb^n,n\geq 2$. Since the concerned non-linear Schr\"odinger equation possesses a trivial solution, we linearize the equation around the trivial solution. Demonstrating the well-posedness of the direct problem under appropriate conditions on initial and boundary data, it is observed that the solution admits $\eps$-expansion. By taking into account the fact that the terms $\Oh(|\nabla u(t,x)|^3)$ are negligible in this context, we shall reconstruct the time-dependent coefficients such as electric potential and vector-valued function associated with quadratic nonlinearity from the knowledge of input-output map using the geometric optics solution and Fourier inversion.

math.AP↗

arXiv2021

Expression of the peak time for time-domain boundary measurements in diffuse light

Light propagation through diffusive media can be described by the diffusion equation in a space-time domain. Further, fluorescence can be described by a system of coupled diffusion equations. This paper analyzes time-domain measurements, which measure the temporal point-spread function (TPSF), at a boundary of such diffusive media with a given source and detector. We focus on the temporal position of the TPSF maximum, which we refer to as the peak time. Although some unique properties of solutions of this system have been numerically studied, we give a mathematical analysis of peak time, providing proof of the existence, uniqueness, and the explicit expression of the peak time. We clearly show the relationship between the peak time and the object position in a medium.

cs.CE↗