264 citations · 659 across the 23 of their papers we have counts for
26 papers
Identification of space-dependent coefficients in two competing terms of a nonlinear subdiffusion equation
Barbara Kaltenbacher, William Rundell
We consider a (sub)diffusion equation with a nonlinearity of the form , where and are space dependent functions. Prominent examples are the Fisher-KPP, the Frank-…
Reconstruction of space-dependence and nonlinearity of a reaction term in a subdiffusion equation
Barbara Kaltenbacher, William Rundell
In this paper we study the simultaneous reconstruction of two coefficients in a reaction-subdiffusion equation, namely a nonlinearity and a space dependent factor. The fact that th…
Regularising the Cauchy problem for Laplace's equation by fractional operators
Barbara Kaltenbacher an William Rundell
In this paper we revisit the classical Cauchy problem for Laplace's equation as well as two further related problems in the light of regularisation of this highly ill-conditioned p…
Nonlinearity parameter imaging in the frequency domain
Barbara Kaltenbacher, William Rundell
Nonlinearity parameter tomography leads to the problem of identifying a coefficient in a nonlinear wave equation (such as the Westervelt equation) modeling ultrasound propagation.…
On the simultanenous identification of the nonlinearity coefficient and the sound speed in the Westervelt equation
Barbara Kaltenbacher, William Rundell
This paper considers the Westervelt equation, one of the most widely used models in nonlinear acoustics, and seeks to recover two spatially-dependent parameters of physical importa…
Determining damping terms in fractional wave equations
Barbara Kaltenbacher, William Rundell
This paper deals with the inverse problem of recovering an arbitrary number of fractional damping terms in a wave equation. We develop several approaches on uniqueness and reconstr…
Determining the nonlinearity in an acoustic wave equation
Barbara Kaltenbacher, William Rundell
We consider an undetermined coefficient inverse problem for a nonlinear partial differential equation describing high intensity ultrasound propagation as widely used in medical ima…
On an inverse problem of nonlinear imaging with fractional damping
Barbara Kaltenbacher, William Rundell
This paper considers the attenuated Westervelt equation in pressure formulation. The attenuation is by various models proposed in the literature and characterised by the inclusion…
On the identification of the nonlinearity parameter in the Westervelt equation from boundary measurements
Barbara Kaltenbacher, William Rundell
We consider an undetermined coefficient inverse problem for a non-\\linear partial differential equation occurring in high intensity ultrasound propagation as used in acoustic tomo…
On uniqueness and reconstruction of a nonlinear diffusion term in a parabolic equation
Barbara Kaltenbacher, William Rundell
The problem of recovering coefficients in a diffusion equation is one of the basic inverse problems. Perhaps the most important term is the one that couples the length and time sca…
On the simultaneous recovery of the conductivity and the nonlinear reaction term in a parabolic equation
Barbara Kaltenbacher, William Rundell
This paper considers the inverse problem of recovering both the unknown, spatially-dependent conductivity and the nonlinear reaction term in a reaction-diffusion equa…
Uniqueness for an inverse coefficient problem for a one-dimensional time-fractional diffusion equation with non-zero boundary conditions
W. Rundell, M. Yamamoto
We consider initial boundary value problems for one-dimensional diffusion equation with time-fractional derivative of order which are subject to non-zero Neumann bound…
The Inverse Problem of Reconstructing Reaction-Diffusion Systems
Barbara Kaltenbacher, William Rundell
This paper considers the inverse problem of recovering state-dependent source terms in a reaction-diffusion system from overposed data consisting of the values of the state variabl…
Regularization of a backwards parabolic equation by fractional operators
Barbara Kaltenbacher, William Rundell
The backwards diffusion equation is one of the classical ill-posed inverse problems, related to a wide range of applications, and has been extensively studied over the last 50 year…
On the identification of source term in the heat equation from sparse data
William Rundell, Zhidong Zhang
We consider the recovery of a source term for the nonhomogeneous heat equation in where is a bounded domain in with smooth…
Recovery of multiple coefficients in a reaction-diffusion equation
Barbara Kaltenbacher, William Rundell
This paper considers the inverse problem of recovering both the unknown, spatially-dependent conductivity and the potential in a parabolic equation from overposed dat…
On the identification of a nonlinear term in a reaction-diffusion equation
Barbara Kaltenbacher, William Rundell
Reaction-diffusion equations are one of the most common partial differential equations used to model physical phenomenon. They arise as the combination of two physical processes: a…
Recovery of a potential in a fractional diffusion equation
William Rundell, Masahiro Yamamoto
We consider the determination of an unknown potential form a fractional diffusion equation subject to overposed lateral boundary data. We show that this data allows recovery…
Recovering an Unknown Source in a Fractional Diffusion Problem
William Rundell, Zhidong Zhang
A standard inverse problem is to determine a source which is supported in an unknown domain from external boundary measurements. Here we consider the case of a time-dependent s…
A direct method for reconstructing inclusions and boundary conditions from electrostatic data
Isaac Harris
In this paper, we will discuss the use of a Sampling Method to reconstruct impenetrable inclusions from Electrostatic Cauchy data. We consider the case of a perfectly conducting an…
Fraction diffusion: Recovering the distributed fractional derivative from overposed data
William Rundell, Zhidong Zhang
There has been considerable recent study in "sub-diffusion" models that replace the standard parabolic equation model by a one with a fractional derivative in the time variable. Th…
Strong maximum principle for fractional diffusion equations and an application to an inverse source problem
Yikan Liu, William Rundell, Masahiro Yamamoto
The strong maximum principle is a remarkable characterization of parabolic equations, which is expected to be partly inherited by fractional diffusion equations. Based on the corre…
A Tutorial on Inverse Problems for Anomalous Diffusion Processes
Bangti Jin, William Rundell
Over the last two decades, anomalous diffusion processes in which the mean squares variance grows slower or faster than that in a Gaussian process have found many applications. At…
A Finite Element Method for the Fractional Sturm-Liouville Problem
Bangti Jin, Raytcho Lazarov, Joseph Pasciak +1
In this work, we propose an efficient finite element method for solving fractional Sturm-Liouville problems involving either the Caputo or Riemann-Liouville derivative of order $α\…