204 research outputs found
Reconstruction of multiplicative space- and time-dependent sources
This paper presents a numerical regularization approach to the simultaneous determination of multiplicative space- and time-dependent source functions in a nonlinear inverse heat conduction problem with homogeneous Neumann boundary conditions together with specified interior and final time temperature measurements. Under these conditions a unique solution is known to exist. However, the inverse prob- lem is still ill-posed since small errors in the input interior temperature data cause large errors in the output heat source solution. For the numerical discretisation, the boundary element method combined with a regularized nonlinear optimization are utilized. Results obtained from several numerical tests are provided in order to illustrate the efficiency of the adopted computational methodology
On the Calibration of a Size-Structured Population Model from Experimental Data
The aim of this work is twofold. First, we survey the techniques developed in
(Perthame, Zubelli, 2007) and (Doumic, Perthame, Zubelli, 2008) to reconstruct
the division (birth) rate from the cell volume distribution data in certain
structured population models. Secondly, we implement such techniques on
experimental cell volume distributions available in the literature so as to
validate the theoretical and numerical results. As a proof of concept, we use
the data reported in the classical work of Kubitschek [3] concerning
Escherichia coli in vitro experiments measured by means of a Coulter
transducer-multichannel analyzer system (Coulter Electronics, Inc., Hialeah,
Fla, USA.) Despite the rather old measurement technology, the reconstructed
division rates still display potentially useful biological features
Inverse bifurcation analysis: application to simple gene systems
BACKGROUND: Bifurcation analysis has proven to be a powerful method for understanding the qualitative behavior of gene regulatory networks. In addition to the more traditional forward problem of determining the mapping from parameter space to the space of model behavior, the inverse problem of determining model parameters to result in certain desired properties of the bifurcation diagram provides an attractive methodology for addressing important biological problems. These include understanding how the robustness of qualitative behavior arises from system design as well as providing a way to engineer biological networks with qualitative properties. RESULTS: We demonstrate that certain inverse bifurcation problems of biological interest may be cast as optimization problems involving minimal distances of reference parameter sets to bifurcation manifolds. This formulation allows for an iterative solution procedure based on performing a sequence of eigen-system computations and one-parameter continuations of solutions, the latter being a standard capability in existing numerical bifurcation software. As applications of the proposed method, we show that the problem of maximizing regions of a given qualitative behavior as well as the reverse engineering of bistable gene switches can be modelled and efficiently solved
Determination of the characteristic directions of lossless linear optical elements
We show that the problem of finding the primary and secondary characteristic
directions of a linear lossless optical element can be reformulated in terms of
an eigenvalue problem related to the unimodular factor of the transfer matrix
of the optical device. This formulation makes any actual computation of the
characteristic directions amenable to pre-implemented numerical routines,
thereby facilitating the decomposition of the transfer matrix into equivalent
linear retarders and rotators according to the related Poincare equivalence
theorem. The method is expected to be useful whenever the inverse problem of
reconstruction of the internal state of a transparent medium from optical data
obtained by tomographical methods is an issue.Comment: Replaced with extended version as published in JM
A General Inverse Problem for the Growth-Fragmentation Equation
The growth-fragmentation equation arises in many different contexts, ranging
from cell division, protein polymerization, biopolymers, neurosciences etc.
Direct observation of temporal dynamics being often difficult, it is of main
interest to develop theoretical and numerical methods to recover reaction rates
and parameters of the equation from indirect observation of the solution.
Following the work done in (Perthame, Zubelli, 2006) and (Doumic, Perthame,
Zubelli, 2009) for the specific case of the cell division equation, we address
here the general question of recovering the fragmentation rate of the equation
from the observation of the time-asymptotic solution, when the fragmentation
kernel and the growth rates are fully general. We give both theoretical results
and numerical methods, and discuss the remaining issues
Singular Value Homogenization: a simple preconditioning technique for linearly constrained optimization and its potential applications in medical therapy
A modified regularization method for an inverse heat conduction problem with only boundary value
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Kernel reconstruction for delayed neural field equations
Understanding the neural field activity for realistic living systems is a challenging task in contemporary neuroscience. Neural fields have been studied and developed theoretically and numerically with considerable success over the past four decades. However, to make effective use of such models, we need to identify their constituents in practical systems. This includes the determination of model parameters and in particular the reconstruction of the underlying effective connectivity in biological tissues. In this work, we provide an integral equation approach to the reconstruction of the neural connectivity in the case where the neural activity is governed by a delay neural field equation. As preparation, we study the solution of the direct problem based on the Banach fixed point theorem. Then we reformulate the inverse problem into a family of integral equations of the first kind. This equation will be vector valued when several neural activity trajectories are taken as input for the inverse problem. We employ spectral regularization techniques for its stable solution. A sensitivity analysis of the regularized kernel reconstruction with respect to the input signal u is carried out, investigating the Frechet differentiability of the kernel with respect to the signal. Finally, we use numerical examples to show the feasibility of the approach for kernel reconstruction, including numerical sensitivity tests, which show that the integral equation approach is a very stable and promising approach for practical computational neuroscience
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