The tangential condition was introduced in [Hanke et al., 95] as a sufficient condition for convergence of the Landweber iteration for solving ill-posed problems. In this paper we present a series of time dependent benchmark inverse problems for which we can verify this condition.
In this paper we introduce a new approach to compute rigorously solutions of Cauchy problems for a class of semi-linear parabolic partial differential equations. Expanding solutions with Chebyshev series in time and Fourier series in space, we introd
uce a zero finding problem $F(a)=0$ on a Banach algebra $X$ of Fourier-Chebyshev sequences, whose solution solves the Cauchy problem. The challenge lies in the fact that the linear part $mathcal{L} := DF(0)$ has an infinite block diagonal structure with blocks becoming less and less diagonal dominant at infinity. We introduce analytic estimates to show that $mathcal{L}$ is a boundedly invertible linear operator on $X$, and we obtain explicit, rigorous and computable bounds for the operator norm $| mathcal{L}^{-1}|_{B(X)}$. These bounds are then used to verify the hypotheses of a Newton-Kantorovich type argument which shows that the (Newton-like) operator $mathcal{T}(a) := a - mathcal{L}^{-1} F(a)$ is a contraction on a small ball centered at a numerical approximation of the Cauchy problem. The contraction mapping theorem yields a fixed point which corresponds to a classical (strong) solution of the Cauchy problem. The approach is simple to implement, numerically stable and is applicable to a class of PDE models, which include for instance Fishers equation, the Kuramoto-Sivashinsky equation, the Swift-Hohenberg equation and the phase-field crystal (PFC) equation. We apply our approach to each of these models and report plausible experimental results, which motivate further research on the method.
We show that the Strang splitting method applied to a diffusion-reaction equation with inhomogeneous general oblique boundary conditions is of order two when the diffusion equation is solved with the Crank-Nicolson method, while order reduction occur
s in general if using other Runge-Kutta schemes or even the exact flow itself for the diffusion part. We prove these results when the source term only depends on the space variable, an assumption which makes the splitting scheme equivalent to the Crank-Nicolson method itself applied to the whole problem. Numerical experiments suggest that the second order convergence persists with general nonlinearities.
We are concerned with the problem of recovering the radial kernel $k$, depending also on time, in a parabolic integro-differential equation $$D_{t}u(t,x)={cal A}u(t,x)+int_0^t k(t-s,|x|){cal B}u(s,x)ds +int_0^t D_{|x|}k(t-s,|x|){cal C}u(s,x)ds+f(t,x)
,$$ ${cal A}$ being a uniformly elliptic second-order linear operator in divergence form. We single out a special class of operators ${cal A}$ and two pieces of suitable additional information for which the problem of identifying $k$ can be uniquely solved locally in time when the domain under consideration is a spherical corona or an annulus.
We are concerned with the problem of recovering the radial kernel $k$, depending also on time, in the parabolic integro-differential equation $$D_{t}u(t,x)={cal A}u(t,x)+int_0^t k(t-s,|x|){cal B}u(s,x)ds +int_0^t D_{|x|}k(t-s,|x|){cal C}u(s,x)ds+f(t,
x),$$ ${cal A}$ being a uniformly elliptic second-order linear operator in divergence form. We single out a special class of operators ${cal A}$ and two pieces of suitable additional information for which the problem of identifying $k$ can be uniquely solved locally in time when the domain under consideration is a ball or a disk.
We consider a parabolic sine-Gordon model with periodic boundary conditions. We prove a fundamental maximum principle which gives a priori uniform control of the solution. In the one-dimensional case we classify all bounded steady states and exhibit
some explicit solutions. For the numerical discretization we employ first order IMEX, and second order BDF2 discretization without any additional stabilization term. We rigorously prove the energy stability of the numerical schemes under nearly sharp and quite mild time step constraints. We demonstrate the striking similarity of the parabolic sine-Gordon model with the standard Allen-Cahn equations with double well potentials.
Barbara Kaltenbacher
,Tram Thi Ngoc Nguyen
,Otmar Scherzer
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(2019)
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"The tangential cone condition for some coefficient identification model problems in parabolic PDEs"
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Tram Thi Ngoc Nguyen
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