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  • Author: H. T. Banks x
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Abstract

We present a mathematical and statistical computational framework for inverse problems involving delay or hysteretic differential equations. We demonstrate efficacy of the methodology in the context of models for insect maturation and mortality due to insecticide exposure.

We consider a class of probability measure dependent dynamical systems which arise in the study of multiscale phenomena in diverse fields such as immunological population dynamics, viscoelasticity of polymers and rubber, and polarization in dielectric materials. We develop an inverse problem framework for studying systems with distributed temporal delays. In particular, we establish conditions for existence and uniqueness of the forward problem and well-posedness (including method stability under numerical approximations) for the inverse problem of estimating the probability measures. We show that a motivating class of models of HIV infection dynamics satisfies all the conditions of our framework, thereby providing a theoretical foundation for inverse problem computations with these models.

We consider the interrogation by means of a pulsed planar electromagnetic wave of a dielectric slab with a supraconductive backing. Previous work using a weak formulation with finite elements (FE) demonstrated the ability to determine material parameters and the slab thickness in the inverse problem. In this work we report on results using Proper Orthogonal Decomposition (POD) to create a more efficient set of basis functions than the standard FE basis functions. We first demonstrate the ability of the reduced basis POD formulation to capture the electromagnetic behavior in the case of the forward problem. We then apply the POD formulation to the inverse scattering problem with unknown parameters and show that the POD formulation provides a considerable reduction in computational time over standard FE methods with comparable ability to recover the unknown parameter values.

Abstract

We consider the problem of estimating amodeling parameter θ using a weighted least squares criterion for given data y by introducing an abstract framework involving generalized measurement procedures characterized by probability measures. We take an optimal design perspective, the general premise (illustrated via examples) being that in any data collected, the information content with respect to estimating θ may vary considerably from one time measurement to another, and in this regard some measurements may be much more informative than others. We propose mathematical tools which can be used to collect data in an almost optimal way, by specifying the duration and distribution of time sampling in the measurements to be taken, consequently improving the accuracy (i.e., reducing the uncertainty in estimates) of the parameters to be estimated.

We recall the concepts of traditional and generalized sensitivity functions and use these to develop a strategy to determine the “optimal” final time T for an experiment; this is based on the time evolution of the sensitivity functions and of the condition number of the Fisher information matrix. We illustrate the role of the sensitivity functions as tools in optimal design of experiments, in particular in finding “best” sampling distributions. Numerical examples are presented throughout to motivate and illustrate the ideas.

We compare parametric and nonparametric estimation methods in the context of PBPK modeling using simulation studies. We implement a Monte Carlo Markov Chain simulation technique in the parametric method, and a functional analytical approach to estimate the probability distribution function directly in the non-parametric method. The simulation results suggest an advantage for the parametric method when the underlying model can capture the true population distribution. On the other hand, our calculations demonstrate some advantages for a nonparametric approach since it is a more cautious (and hence safer) way to assess the distribution when one does not have sufficient knowledge to assume a population distribution form or parametrization. The parametric approach has obvious advantages when one has significant a priori information on the distributions sought, although when used in the nonparametric method, prior information can also significantly facilitate estimation.

We review the asymptotic theory for standard errors in classical ordinary least squares (OLS) inverse or parameter estimation problems involving general nonlinear dynamical systems where sensitivity matrices can be used to compute the asymptotic covariance matrices. We discuss possible pitfalls in computing standard errors in regions of low parameter sensitivity and/or near a steady state solution of the underlying dynamical system.

Abstract.

Optimal design methods (designed to choose optimal sampling distributions through minimization of a specific cost function related to the resulting error in parameter estimates) for inverse or parameter estimation problems are considered. We compare a recent design criteria, SE-optimal design (standard error optimal design) with the more traditional D-optimal and E-optimal designs. The optimal sampling distributions from each design are used to compute and compare standard errors; here the standard errors for parameters are computed using the optimal mesh along with Monte Carlo simulations as compared to asymptotic theory based standard errors. We illustrate ideas with two examples: the Verhulst–Pearl logistic population model and the standard harmonic oscillator model.

We compare an inverse problem approach to parameter estimation with homogenization techniques for characterizing the electrical response of composite dielectric materials in the time domain. We first consider an homogenization method, based on the periodic unfolding method, to identify the dielectric response of a complex material with heterogeneous micro-structures which are described by spatially periodic parameters. We also consider electromagnetic interrogation problems for complex materials assuming multiple polarization mechanisms with distributions of parameters. An inverse problem formulation is devised to determine effective polarization parameters specific to the interrogation problem. We compare the results of these two approaches with the classical Maxwell-Garnett mixing model and a simplified model with a weighted average of parameters. Numerical results are presented for a specific example involving a mixture of ethanol and water (modeled with multiple Debye mechanisms). A comparison between each approach is made in the frequency domain (e.g., Cole-Cole diagrams), as well as in the time domain (e.g., plots of susceptibility kernels).