Numerical methods in inverse problems

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• MASAHİRO YAMAMOTO

We consider a one-dimensional fractional diffusion equation: ∂ α t u(x, t) = ∂ ∂x p(x) ∂u ∂x (x, t) , 0 < x < ℓ, where 0 < α < 1 and ∂ α t denotes the Caputo derivative in time of order α.We attach the homogeneous Neumann boundary condition at x = 0, ℓ and the initial value given by the Dirac delta function.We prove t…

• MASAHİRO YAMAMOTO

In this review, concerning parabolic equations, we give self-contained descriptions on (1) derivations of Carleman estimates;(2) methods for applications of the Carleman estimates to estimates of solutions and to inverse problems.Moreover limited to parabolic equations, we survey the previous and recent results in vie…

• MASAHİRO YAMAMOTO

We consider a backward problem in time for a time-fractional partial differential equation in one-dimensional case, which describes the diffusion process in porous media related with the continuous time random walk problem. Such a backward problem is of practically great importance because we often do not know the ini…

• MASAHİRO YAMAMOTO

We consider a system with a suitable boundary condition, where is a bounded domain, is a uniformly elliptic operator of the second order whose coefficients are suitably regular for , is fixed, and a function satisfies Our inverse problems are determinations of g using overdetermining data or , where and . Our main res…

For the solution u ( p ) = u ( p )( x , t ) to ∂ t 2 u ( x , t )-Δ u ( x , t )- p ( x ) u ( x , t ) = 0 in Ω × (0, T ) and (∂ u /∂ν)| ∂Ω × (0, T ) = 0 with given u (·,0) and ∂ t u (·,0), we consider an inverse problem of determining p ( x ), x ∊Ω, from data u |ω × (0, T ) . Here Ω⊂ n , n = 1,2,3, is a bounded domain,…

This paper deals with an inverse problem of simultaneously identifying the space-dependent diffusion coefficient and the fractional order in the 1D timefractional diffusion equation with smooth initial functions by using boundary measurements. The uniqueness results for the inverse problem are proved on the basis of t…

• MASAHİRO YAMAMOTO

We consider an inverse problem of determining p(x), x ∈ Ω in (∂2 u/∂t 2)(x, t) − Δu(x, t) − p(x)u(x, t) = 0 in Ω × (0, T) and (∂u/∂ν)|∂Ω×(0, T) = 0 with given u(·, 0) and (∂u/∂t)(·, 0). Here Ω ⊂ R n , n = 1, 2, 3, is a bounded domain. We prove the Lipschitz stability in determining p from u|∂Ω×(0,T), under the assumpt…

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Let u(f) be the solution to a hyperbolic equation in a bounded domain Omega subset R ': u"(x, t) = Delta u(x,t) + sigma (t)f(x) (x in Omega, 0 < t < T) u(x,0) = 0 u'(x,0) = 0 (x in Omega) u(x,t) = 0 (x in partial Omega, 0 < t < T). We assume that sigma in C 1 [0,T] is a known function, sigma(0) not= 0, and f in L 2 (O…

In this paper, based on the conditional stability estimate for ill-posed inverse problems, we propose a new strategy for a priori choice of regularizing parameters in Tikhonov's regularization and we show that it can be applied to a wide class of inverse problems. The convergence rate of the regularized solutions is a…

We prove for a two-dimensional bounded domain that the Cauchy data for the Schrödinger equation measured on an arbitrary open subset of the boundary uniquely determines the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an arbitrary open subset of the b…

• MASAHİRO YAMAMOTO