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Gen Nakamura

Publications and source records attributed to Gen Nakamura.

At least 37 records · Page 2Linked to original sources
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↗
arXiv2021

Semiclassical analysis of elastic surface waves

In this paper, we present a semiclassical description of surface waves or modes in an elastic medium near a boundary, in spatial dimension three. The medium is assumed to be essentially stratified near the boundary at some scale comparable to the wave length. Such a medium can also be thought of as a surficial layer (which can be thick) overlying a half space. The analysis is based on the work of Colin de Verdière on acoustic surface waves. The description is geometric in the boundary and locally spectral "beneath" it. Effective Hamiltonians of surface waves correspond with eigenvalues of ordinary differential operators, which, to leading order, define their phase velocities. Using these Hamiltonians, we obtain pseudodifferential surface wave equations. We then construct a parametrix. Finally, we discuss Weyl's formulas for counting surface modes, and the decoupling into two classes of surface waves, that is, Rayleigh and Love waves, under appropriate symmetry conditions.

math.AP↗
arXiv2021

Classical unique continuation property for multi-terms time fractional diffusion equations

As for the unique continuation property (UCP) of solutions in $(0,T)\timesΩ$ with a domain $Ω\subset{\mathbb R}^n,\,n\in{\mathbb N}$ for a multi-terms time fractional diffusion equation, we have already shown it by assuming that the solutions are zero for $t\le0$ (see \cite{LN2019}). Here the strongly elliptic operator for this diffusion equation can depend on time and the orders of its time fractional derivatives are in $(0,2)$. This paper is a continuation of the previous study. The aim of this paper is to drop the assumption that the solutions are zero for $t\le0$. We have achieved this aim by first using the usual Holmgren transformation together with the argument in \cite{LN2019} to derive the UCP in $(T_1,T_2)\times B_1$ for some $0<T_1<T_2<T$ and a ball $B_1\subsetΩ$. Then if $u$ is the solution of the equation with $u=0$ in $(T_1,T_2)\times B_1$, we show $u=0$ also in $((0,T_1]\cup[T_2,T))\times B_r$ for some $r<1$ by using the argument in \cite{LN2019} which uses two Holmgren type transformations different from the usual one. This together with spatial coordinates transformation, we can obtain the usual UCP which we call it the classical UCP given in the title of this paper for our time fractional diffusion equation.

math.AP↗
arXiv2020

Inverse Initial Boundary Value Problem for a Non-linear Hyperbolic Partial Differential Equation

In this article we are concerned with an inverse initial boundary value problem for a non-linear wave equation in space dimension $n\geq 2$. In particular we consider the so called interior determination problem. This non-linear wave equation has a trivial solution, i.e. zero solution. By linearizing this equation at the trivial solution, we have the usual linear wave equation with a time independent potential. For any small solution $u=u(t,x)$ of this non-linear equation, it is the perturbation of linear wave equation with time-independent potential perturbed by a divergence with respect to $(t,x)$ of a vector whose components are quadratics with respect to $\nabla_{t,x} u(t,x)$. By ignoring the terms with smallness $O(|\nabla_{t,x} u(t,x)|^3)$, we will show that we can uniquely determine the potential and the coefficients of these quadratics by many boundary measurements at the boundary of the spacial domain over finite time interval and the final overdetermination at $t=T$. In other words, the measurement is given by the so-called the input-output map (see (1.5)).

math.AP↗
arXiv2020

Duality between range and no-response tests and its application for inverse problems

In this paper we will show the duality between the range test (RT) and no-response test (NRT) for the inverse boundary value problem for the Laplace equation in $Ω\setminus\overline D$ with an obstacle $D\SubsetΩ$ whose boundary $\partial D$ is visible from the boundary $\partialΩ$ of $Ω$ and a measurement is given as a set of Cauchy data on $\partialΩ$. Here the Cauchy data is given by a unique solution $u$ of the boundary value problem for the Laplace equation in $Ω\setminus\overline D$ with homogeneous and inhomogeneous Dirichlet boundary condition on $\partial D$ and $\partialΩ$, respectively. These testing methods are domain sampling method to estimate the location of the obstacle using test domains and the associated indicator functions. Also both of these testing methods can test the analytic extension of $u$ to the exterior of a test domain. Since these methods are defined via some operators which are dual to each other, we could expect that there is a duality between the two methods. We will give this duality in terms of the equivalence of the pre-indicator functions associated to their indicator functions. As an application of the duality, the reconstruction of $D$ using the RT gives the reconstruction of $D$ using the NRT and vice versa. We will also give each of these reconstructions without using the duality if the Dirichlet data of the Cauchy data on $\partialΩ$ is not identically zero and the solution to the associated forward problem does not have any analytic extension across $\partial D$. Moreover, we will show that these methods can still give the reconstruction of $D$ if $D$ is a convex polygon and it satisfies one of the following two properties: all of its corner angles are irrational and its diameter is less than its distance to $\partialΩ$.

math.AP↗
arXiv2020

Born series for the photon diffusion equation perturbing the Robin boundary condition

The photon diffusion equation is solved making use of the Born series for the Robin boundary condition. We develop a general theory for arbitrary domains with smooth enough boundaries and explore the convergence. The proposed Born series is validated by numerical calculation in the three-dimensional half space. It is shown that in this case the Born series converges regardless the value of the impedance term in the Robin boundary condition.

math-ph↗
arXiv2020

Short-time approximate solutions of an equation modeling a camphor motion

As a profound example of spontaneous motion, we analyze the motion of a camphor particle on a water surface. The motion is modeled as an initial-boundary value problem for a coupled nonlinear system of a diffusion equation and an ordinary differential equation in a two-dimensional domain. Since it seems that the well-posedness of this initial boundary value problem is missing, we provided its proof. Then, by constructing an approximate solution to this initial boundary value problem, we gave a mathematically rigorous interpretation of a camphor motion. That is we showed that the motion of camphor locally in time has a self-avoiding orbit. We also gave the numerical performance of the approximate solution.

math.AP↗
arXiv2019

The factorization method for recovering cavities in a heat conductor

In this paper, we develop a factorization method to reconstruct cavities in a heat conductor by knowing the Neumann-to-Dirichlet map at the boundary of this conductor. The factorization method is a very well known reconstruction method for inverse problems described by self-adjoint elliptic equations. It enables us to reconstruct the boundaries of unknown targets such as cavities and inclusions. This method was successfully applied for examples to the inverse scattering problem by obstacles for the acoustic wave and the electrical impedance tomography problem. The forward model equations for these problems are the Helmholtz equation and the conductivity equation which are both self-adjoint equations. On the other hand, the heat equation is a typical non-selfadjoint equation. This paper is giving the first attempt to study the factorization method for inverse problems whose forward model equations are non-selfadjoint. We emphasize that an auxiliary operator introduced in this paper is the key which makes this attempt successful. This operator connects the back projection operators for the forward and backward heat equations.

math-ph↗
arXiv2019

A hybrid inversion scheme combining Markov chain Monte Carlo and iterative methods for determining optical properties of random media

Near-infrared spectroscopy (NIRS) including diffuse optical tomography is an imaging modality which makes use of diffuse light propagation in random media. When optical properties of a random medium is investigated from boundary measurements of reflected or transmitted light, iterative inversion schemes such as the Levenberg-Marquardt algorithm are known to fail when initial guesses are not close to the true value of the coefficient to be reconstructed. In this paper, we investigate how this weakness of iterative schemes is overcome by the use of Markov chain Monte Carlo. Using time-resolved measurements performed against a polyurethane-based phantom, we present a case that the Levenberg-Marquardt algorithm fails to work but the proposed hybrid method works well. Then with a toy model of diffuse optical tomography we illustrate that the evenberg-Marquardt method fails when it is trapped by a local minimum but the hybrid method can escape from local minima by using the Metropolis-Hastings Markov chain Monte Carlo algorithm until it reaches the valley of the global minimum. The proposed hybrid scheme can be applied to different inverse problems in NIRS which are solved iteratively. We find that for both numerical and phantom experiments optical properties such as the absorption and reduced scattering coefficients can be retrieved without being trapped by a local minimum when Monte Carlo simulation is run only about $100$ steps before switching to an iterative method. The hybrid method is compared with simulated annealing. Although the Metropolis-Hastings MCMC arrives at the steady state at about $10000$ Monte Carlo steps, in the hybrid method the Monte Carlo simulation can be stopped way before the burn-in time.

physics.comp-ph↗
arXiv2019

Reconstruction for the coefficients of a quasilinear elliptic partial differential equation

In this paper we consider an inverse coefficients problem for a quasilinear elliptic equation of divergence form $\nabla\cdot\vec{C}(x,\nabla u(x))=0$, in a bounded smooth domain $Ω$. We assume that $\overrightarrow{C}(x,\vec{p})=γ(x)\vec{p}+\vec{b}(x)|\vec{p}|^2+\mathcal{O}(|\vec{p}|^3)$, by expanding $\overrightarrow{C}(x,\vec{p})$ around $\vec{p}=0$. We give a reconstruction method for $γ$ and $\vec{b}$ from the Dirichlet to Neumann map defined on $\partialΩ$.

math.AP↗
arXiv2019

Reconstruction of unknown cavity by single measurement

In this paper we propose a domain sampling type reconstruction scheme for an inverse boundary value problem to identify an unknown cavity by single measurement on the accessible boundary of a known electric or heat conductive medium. Here the single measurement is to give single current or heat flux which can have a small support over the boundary, and we measure the corresponding voltage or temperature over the whole boundary. For this inverse boundary value problem, we adapted the single NRT (no response test) introduced by Luke and Potthast for inverse scattering problem and show that it can provide such a domain sampling type reconstruction scheme.

math.AP↗
arXiv2019

Uniqueness for the inverse boundary value problem of piecewise homogeneous anisotropic elasticity in the time domain

We consider the inverse boundary value problem of recovering piecewise homogeneous elastic tensor and piecewise homogeneous mass density from a localized lateral Dirichlet-to-Neumann or Neumann-to-Dirichlet map for the elasticity equation in the space-time domain. We derive uniqueness for identifying these tensor and density on all domains of homogeneity that may be reached from the part of the boundary where the measurements are taken by a chain of subdomains whose successive interfaces contain a curved portion.

math.AP↗
arXiv2019

Inverse Boundary Value Problem for Non-linear Hyperbolic Partial Differential Equations

In this article we are concerned with an inverse boundary value problem for a non-linear wave equation of divergence form with space dimension $n\geq 3$. In particular the so called the interior determination problem. This non-linear wave equation has a trivial solution, i.e. zero solution. By linearizing this equation at the trivial solution, we have the usual linear isotropic wave equation with the speed $\sqrt{γ(x)}$ at each point $x$ in a given spacial domain. For any small solution $u=u(t,x)$ of this non-linear equation, we have the linear isotropic wave equation perturbed by a divergence with respect to $x$ of a vector whose components are quadratics with respect to $\nabla_x u(t,x)$ by ignoring the terms with smallness $O(|\nabla_x u(t,x)|^3)$. We will show that we can uniquely determine $γ(x)$ and the coefficients of these quadratics by many boundary measurements at the boundary of the spacial domain over finite time interval. More precisely the boundary measurements are given as the so-called the hyperbolic Dirichlet to Neumann map.

math.AP↗
arXiv2018

Unique recovery of piecewise analytic density and stiffness tensor from the elastic-wave Dirichlet-to-Neumann map

We study the recovery of piecewise analytic density and stiffness tensor of a three-dimensional domain from the local dynamical Dirichlet-to-Neumann map. We give global uniqueness results if the medium is transversely isotropic with known axis of symmetry or orthorhombic with known symmetry planes on each subdomain. We also obtain uniqueness of a fully anisotropic stiffness tensor, assuming that it is piecewise constant and that the interfaces which separate the subdomains have curved portions. The domain partition need not to be known. Precisely, we show that a domain partition consisting of subanalytic sets is simultaneously uniquely determined.

math.AP↗
arXiv2018

Solvability of interior transmission problem for the diffusion equation by constructing its Green function

Consider the interior transmission problem arising in inverse boundary value problems for the diffusion equation with discontinuous diffusion coefficients. We prove the unique solvability of the interior transmission problem by constructing its Green function. First, we construct a local parametrix for the interior transmission problem near the boundary in the Laplace domain, by using the theory of pseudo-differential operators with a large parameter. Second, by carefully analyzing the analyticity of the local parametrix in the Laplace domain and estimating it there, a local parametrix for the original parabolic interior transmission problem is obtained via the inverse Laplace transform. Finally, using a partition of unity, we patch all the local parametrices and the fundamental solution of the diffusion equation to generate a global parametrix for the parabolic interior transmission problem, and then compensate it to get the Green function by the Levi method. The uniqueness of the Green function is justified by using the duality argument, and then the unique solvability of the interior transmission problem is concluded. We would like to emphasize that the Green function for the parabolic interior transmission problem is constructed for the first time in this paper. It can be applied for active thermography and diffuse optical tomography modeled by diffusion equations to identify an unknown inclusion and its physical property.

math.AP↗
arXiv2017

Unique continuation property for multi-terms time fractional diffusion equations

A Carleman estimate and the unique continuation property of solutions for a multi-terms time fractional diffusion equation up to order $α\,\,(0<α<2)$ and general time dependent second order strongly elliptic time elliptic operator for the diffusion. The estimate is derived via some subelliptic estimate for an operator associated to this equation using calculus of pseudo-differential operators. A special Holmgren type transformation which is linear with respect to time is used to show the local unique continuation of solutions. We developed a new argument to derive the global unique continuation of solutions. Here the global unique continuation means as follows. If u is a solution of the multi-terms time fractional diffusion equation in a domain over the time interval $(0,T)$, then a zero set of solution over a subdomain of $Ω$ can be continued to $(0,T)\timesΩ$.

math.AP↗
arXiv2017

Strong unique continuation for two-dimensional anisotropic elliptic systems

In this paper, we give the strong unique continuation property for a general two dimensional anisotropic elliptic system with real coefficients in a Gevrey class under the assumption that the principal symbol of the system has simple characteristics. The strong unique continuation property is derived by obtaining some Carleman estimate. The derivation of the Carleman estimate is based on transforming the system to a larger second order elliptic system with diagonal principal part which has complex coefficients.

math.AP↗