Determination of a Potential from Cauchy Data: Uniqueness and Distinguishability

The problem of recovering a potential q(y) in the differential equation: −∆u + q(y)u = 0 (x,y) &∈ (0, 1) × (0,1) u(0, y) = u(1, y) = u(x, 0) = 0u(x, 1) = f(x), uy(x, 1) = g(x) is investigated. The method of separation of variables reduces the recovery of q(y) to a non-standard inverse Sturm-Liouville problem. Employing asymptotic techniques and integral operators of Gel\u27fand-Levitan type, it is shown that, under appropriate conditions on the Cauchy pair (f, g ), q(y) is uniquely determined, in a local sense, up to its mean. We characterize the ill-posedness of this inverse problem in terms of the distinguishability of potentials. An estimate is derived which indicates the maximum level of measurement error under which two potentials, differing only far away from y = 1, can be resolved

A Single-Parameter Model of the Immune Response to Bacterial Invasion

The human immune response to bacterial pathogens is a remarkably complex process, involving many different cell types, chemical signals, and extensive lines of communication. Mathematical models of this system have become increasingly high-dimensional and complicated, as researchers seek to capture many of the major dynamics. In this paper, we argue that, in some important instances, preference should be given to low-dimensional models of immune response, as opposed to their high-dimensional counterparts. One such model is analyzed and shown to reflect many of the key phenomenological properties of the immune response in humans. Notably, this model includes a single parameter values, may be used to quantify the overall immuno-competence of individual hosts

A Convergent Reconstruction Method for an Elliptic Operator in Potential Form

We investigate the problem of recovering a potential q(x) in the equation -∆u + q(x)u = 0 from overspecified boundary data on the unit square in R2. The potential is characterized as a fixed point of a nonlinear operator, which is shown to be a contraction on a ball in C∝. Uniqueness of q(x) follows, as does convergence of the resulting recovery scheme. Numerical examples, demonstrating the performance of the algorithm, are presented

Biology in Mathematics at the University of Richmond

In an effort to meet the needs of science students for modeling skills, three new courses have been created at the University of Richmond: Scientific Calculus I and II, and Mathematical Models in Biology and Medicine. The courses are described, and lessons learned and future directions are discussed

Reconstruction of an Unknown Boundary Portion from Cauchy Data in \u3cem\u3eN\u3c/em\u3e- Dimensions

We consider the inverse problem of determining the shape of some inaccessible portion of the boundary of a region in n dimensions from Cauchy data for the heat equation on an accessible portion of the boundary. The inverse problem is quite ill-posed, and nonlinear. We develop a Newton-like algorithm for solving the problem, with a simple and efficient means for computing the required derivatives, develop methods for regularizing the process, and provide computational examples

A unified inter-host and in-host model of antibiotic resistance and infection spread in a hospital ward

As the battle continues against hospital-acquired infections and the concurrent rise in antibiotic resistance among many of the major causative pathogens, there is a dire need to conduct controlled experiments, in order to compare proposed control strategies. However, cost, time, and ethical considerations make this evaluation strategy either impractical or impossible to implement with living patients. This paper presents a multi-scale model that offers promise as the basis for a tool to simulate these (and other) controlled experiments. This is a “unified” model in two important ways: (i) It combines inter-host and in-host dynamics into a single model, and (ii) it links two very different modeling approaches - agent-based modeling and differential equations - into a single model. The potential of this model as an instrument to combat antibiotic resistance in hospitals is demonstrated with numerical examples

Algorithm-independent Optimal Input Fluxes for Boundary Identification in Thermal Imaging

An inverse boundary determination problem for a parabolic model, arising in thermal imaging, is considered. The focus is on intelligently choosing an effective input heat flux, so as to maximize the practical effectiveness of an inversion algorithm. Three different methods, based on different interpretations of the term “effective , are presented and analyzed, then demonstrated through numerical examples. It is noteworthy that each of these flux-selection methods is independent of the particular inversion algorithm to be used