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Masahiro Yamamoto

Publications and source records attributed to Masahiro Yamamoto.

At least 37 records · Page 2Linked to original sources
arXiv2024

One-dimensional coefficient inverse problems by transformation operators

We prove the uniqueness for an inverse problem of determining a matrix coefficient $P(x)$ of a system of evolution equations $σ\ppp_t u = \ppp_x^2 u(t,x) - P(x) u(t,x)$ for $0 0$ and $T>0$ are arbitrarily given. The uniqueness results assert that two solutions have the same Cauchy data at $x=0$ over $(0,T)$ and the same initial value or the final value which is positive on $[0,\ell]$, then the zeroth-order coefficient is uniquely determined on $[0,\ell]$. The uniqueness for inverse coefficient problem for a system of evolution equations without boundary conditions over the whole boundary is an open problem even in the one-dimension in the case where only initial value is given as spatial data. Moreover, in the case of the zero initial condition, we prove the uniqueness in the half of the spatial interval.

math.AP↗
arXiv2024

Global and local existence of solutions for nonlinear systems of time-fractional diffusion equations

In this paper, we consider initial-boundary value problems for two-component nonlinear systems of time-fractional diffusion equations with the homogeneous Neumann boundary condition and non-negative initial values. The main results are the existence of solutions global in time and the blow-up. Our approach involves the truncation of the nonlinear terms, which enables us to handle all local Lipschitz continuous nonlinear terms, provided their sum is less than or equal to zero. By employing a comparison principle for the corresponding linear system, we establish also the non-negativity of the nonlinear system.

math.AP↗
arXiv2024

Beam dynamics study of the high-power electron beam irradiator using niobium-tin superconducting cavity

A compact accelerator design for irradiation purposes is being proposed at KEK. This design targets an energy of 10 MeV and a current of 50 mA. Current design includes a 100 kV thermionic DC electron gun with an RF grid, 1-cell normal-conducting buncher cavity, and Nb$_{3}$Sn superconducting cavities to accelerate the beam to the final energy of 10 MeV. The goal of the present beam dynamics study is the beam loss suppression (to the ppm level), since it results in a thermal load on the cavity. Then the beam performance at the accelerator exit should be confirmed. The main issue was to transport the beam without loss, since the initial electron energy (100 keV) is low, and the beam parameters are intricately correlated. In addition, the space charge effect is considerable. For this reason, simultaneous optimization of multiple parameters was necessary. Here we report optimization results and their effect on the design of the machine.

physics.acc-ph↗
arXiv2024

Blowup in $L^1(Ω)$-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms

This article is concerned with semilinear time-fractional diffusion equations with polynomial nonlinearity $u^p$ in a bounded domain $Ω$ with the homogeneous Neumann boundary condition and positive initial values. In the case of $p>1$, we prove the blowup of solutions $u(x,t)$ in the sense that $\|u(\,\cdot\,,t)\|_{L^1(Ω)}$ tends to $\infty$ as $t$ approaches some value, by using a comparison principle for the corresponding ordinary differential equations and constructing special lower solutions. Moreover, we provide an upper bound for the blowup time. In the case of $0<p<1$, we establish the global existence of solutions in time based on the Schauder fixed-point theorem.

math.AP↗
arXiv2023

Reconstruction of degenerate conductivity region for parabolic equations

We consider an inverse problem of reconstructing a degeneracy point in the diffusion coefficient in a one-dimensional parabolic equation by measuring the normal derivative on one side of the domain boundary. We analyze the sensitivity of the inverse problem to the initial data. We give sufficient conditions on the initial data for uniqueness and stability for the one-point measurement and show some examples of positive and negative results. On the other hand, we present more general uniqueness results, also for the identification of an initial data by measurements distributed over time. The proofs are based on an explicit form of the solution by means of Bessel functions of the first type. Finally, the theoretical results are supported by numerical experiments.

math.AP↗
arXiv2023

Determination of source terms and coefficients dependent on $n-1$ spatial variable of parabolic and Schrödinger equations

We establish the uniqueness in the determination of a source term or a coefficient of the zeroth order term of a second-order parabolic equation. Moreover we consider the determination of a potential of the Schrödinger equation. For a parabolic equation, an unknown source term and coefficient depend on the time and $n-1$ spatial variables. For the Schrödinger equation, we assume that a potential depends on $n-1$ spatial variables. The data are taken on a part of the boundary satisfying some geometric condition.

math.AP↗
arXiv2023

Blow-up for time-fractional diffusion equations with superlinear convex semilinear terms

This article is concerned with a semilinear time-fractional diffusion equation with a superlinear convex semilinear term in a bounded domain $Ω$ with the homogeneous Dirichlet, Neumann, Robin boundary conditions and non-negative and not identically vanishing initial value. The order of the fractional derivative in time is between $1$ and $0$, and the elliptic part is with time-independent coefficients. We prove (i) The solution with any initial value blow-up if the eigenvalue $λ_1$ of the elliptic operator with the minimum real part is non-positive. (ii) Otherwise, the solution blows up if a weighted $L^1$-norm of initial value is greater than some critical value give by $λ_1$. We provide upper estimates of the blow-up times. The key is a comparison principle for time-fractional ordinary differential equations.

math.AP↗
arXiv2023

Evolution of Natural Language Processing Technology: Not Just Language Processing Towards General Purpose AI

Since the invention of computers, communication through natural language (actual human language) has been a dream technology. However, natural language is extremely difficult to mathematically formulate, making it difficult to realize as an algorithm without considering programming. While there have been numerous technological developments, one cannot say that any results allowing free utilization have been achieved thus far. In the case of language learning in humans, for instance when learning one's mother tongue or foreign language, one must admit that this process is similar to the adage "practice makes perfect" in principle, even though the learning method is significant up to a point. Deep learning has played a central role in contemporary AI technology in recent years. When applied to natural language processing (NLP), this produced unprecedented results. Achievements exceeding the initial predictions have been reported from the results of learning vast amounts of textual data using deep learning. For instance, four arithmetic operations could be performed without explicit learning, thereby enabling the explanation of complex images and the generation of images from corresponding explanatory texts. It is an accurate example of the learner embodying the concept of "practice makes perfect" by using vast amounts of textual data. This report provides a technological explanation of how cutting-edge NLP has made it possible to realize the "practice makes perfect" principle. Additionally, examples of how this can be applied to business are provided. We reported in June 2022 in Japanese on the NLP movement from late 2021 to early 2022. We would like to summarize this as a memorandum since this is just the initial movement leading to the current large language models (LLMs).

cs.CL↗
arXiv2023

Simultaneous determination of initial value and source term for time-fractional wave-diffusion equations

We consider initial boundary value problems for time fractional diffusion-wave equations: $$ d_t^α u = -Au + μ(t)f(x) $$ in a bounded domain where $μ(t)f(x)$ describes a source and $α\in (0,1) \cup (1,2)$, and $-A$ is a symmetric ellitpic operator with repect to the spatial variable $x$. We assume that $μ(t) = 0$ for $t > T$:some time and choose $T_2>T_1>T$. We prove the uniqueness in simultaneously determining $f$ in $Ω$, $μ$ in $(0,T)$, and initial values of $u$ by data $u\vert_{ω\times (T_1,T_2)}$, provided that the order $α$ does not belong to a countably infinite set in $(0,1) \cup (1,2)$ which is characterized by $μ$. The proof is based on the asymptotic behavior of the Mittag-Leffler functions.

math.AP↗
arXiv2023

Lipschitz stability for determination of states and inverse source problem for the mean field game equations

In a bounded domain $Ω\subset \mathbb{R}^d$ over time interval $(0,T)$, we consider mean field game equations whose principal coefficients depend on the time and state variables with a general Hamiltonian. We attach the non-zero Robin boundary condition. We first prove the Lipschitz stability in $Ω\times (\varepsilon, T-\varepsilon)$ with given $\varepsilon>0$ for the determination of the solutions by Dirichlet data on arbitrarily chosen subboundary of $\partialΩ$. Next we prove the Lipschitz stability for an inverse problem of determining spatially varying factors of source terms and a coefficient by extra boundary data and spatial data at an intermediate time.

math.AP↗
arXiv2023

Harmonic Measures and Numerical Computation of Cauchy Problems for Laplace Equations

It is well known that Cauchy problem for Laplace equations is an ill-posed problem in Hadamard's sense. Small deviations in Cauchy data may lead to large errors in the solutions. It is observed that if a bound is imposed on the solution, there exists a conditional stability estimate. This gives a reasonable way to construct stable algorithms. However, it is impossible to have good results at all points in the domain. Although numerical methods for Cauchy problems for Laplace equations have been widely studied for quite a long time, there are still some unclear points, for example, how to evaluate the numerical solutions, which means whether we can approximate the Cauchy data well and keep the bound of the solution, and at which points the numerical results are reliable? In this paper, we will prove the conditional stability estimate which is quantitatively related to harmonic measures. The harmonic measure can be used as an indicate function to pointwisely evaluate the numerical result, which further enables us to find a reliable subdomain where the local convergence rate is higher than a certain order.

math.NA↗
arXiv2023

Unique continuation for a mean field game system

For a mean field game system, we prove the unique continuation which asserts that if Cauchy data are zero on arbitrarily chosen lateral subboundary, then the solution identically vanishes.

math.AP↗
arXiv2023

Stability for backward problems in time for degenerate parabolic equations

For solution $u(x,t)$ to degenearte parabolic equations in a bounded domain $Ω$ with homogenous boundary condition, we consider backward problems in time: determine $u(\cdot,t_0)$ in $Ω$ by $u(\cdot,T)$, where $t$ is the time variable and $0\le t_0 < T$. Our main results are conditional stability under boundedness assumptions on $u(\cdot,0)$. The proof is based on a weighted $L^2$-estimate of $u$ whose weight depends only on $t$, which is an inequality of Carleman's type. Moreover our method is applied to semilinear degenerate parabolic equations.

math.AP↗
arXiv2023

Comparison principles for the time-fractional diffusion equations with the Robin boundary conditions. Part I: Linear equations

The main objective of this paper is analysis of the initial-boundary value problems for the linear time-fractional diffusion equations with a uniformly elliptic spatial differential operator of the second order and the Caputo type time-fractional derivative acting in the fractional Sobolev spaces. The boundary conditions are formulated in form of the homogeneous Neumann or Robin conditions. First we discuss the uniqueness and existence of solutions to these initial-boundary value problems. Under some suitable conditions on the problem data, we then prove positivity of the solutions. Based on these results, several comparison principles for the solutions to the initial-boundary value problems for the linear time-fractional diffusion equations are derived.

math.AP↗
arXiv2023

Uniqueness of solution to boundary value problems for time-fractional wave equations

We consider an initial boundary value problem in a bounded domain $Ω$ over a time interval $(0, T)$ for a time-fractional wave equation where the order of the fractional time derivative is between $1$ and $2$ and the spatial elliptic operator has time-independent coefficients and is not necessarily symmetric. We prove that if for arbitrarily chosen subdomain $ω\subset Ω$ and $T>0$, a solution to the problem vanishes in $ω\times (0,T)$, then $u=0$ in $Ω\times (0, T)$. The uniqueness does not require any geometric condition on $ω$.

math.AP↗
arXiv2023

Lipschitz stability for determination of states and inverse source problem for the mean field game equations

We consider solutions satisfying the zero Neumann boundary condition and a linearized mean field game equation in $Ω\times (0,T)$ whose principal coefficients depend on the time and spatial variables with general Hamiltonian, where $Ω$ is a bounded domain in $\Bbb R^d$ and $(0,T)$ is the time interval. We first prove the Lipschitz stability in $Ω\times (\varepsilon, T-\varepsilon)$ with given $\varepsilon>0$ for the determination of the solutions by Dirichlet data on arbitrarily chosen subboundary of $\partialΩ$. Next we prove the Lipschitz stability for an inverse problem of determining spatially varying factors of source terms and a coefficient by extra boundary data and spatial data at intermediate time.

math.AP↗