===affil2:
National Institute of Informatics
===firstname:
Yasunori
===firstname4:
Ben
===firstname3:
Hans
===lastname2:
Hayami
===lastname:
Aoki
===firstname5:
Akihiko
===affil6:

===lastname3:
De Sterck
===email:
yaoki@uwaterloo.ca
===lastname6:

===affil5:
Tokyo Institute of Technology
===otherauths:

===lastname4:
Holder
===affil4:
Ryerson University
===lastname7:

===affil7:

===firstname7:

===postal:
Department of Applied Mathematics
University of Waterloo
200 University Ave. West
Waterloo Ontario Canada
===firstname6:

===ABSTRACT:
We describe a new method for inverse problems that proceeds by simultaneously finding multiple solutions of the parameter estimation problem.  Estimation of the parameters of a finitely parameterized ODE based mathematical model from experimental data is required in many fields of science.  This problem can be thought of as solving a system of nonlinear equations
\begin{eqnarray}
\boldsymbol{f}(\boldsymbol{\theta})&=&\boldsymbol{y}^*\,, \label{eq_1}
\end{eqnarray}
where $\boldsymbol{f}$ is a map from $\mathbb{R}^m$ to $\mathbb{R}^n$, $\boldsymbol{\theta}$ is the parameter vector, and $\boldsymbol{y}^*$ is the experimental data.  A standard approach for solving this type of system of nonlinear equations is a gradient based method such as the Newton method or the Levenberg-Marquard method.  However, in the case of finding parameters from experimental data the problem may be more complicated than just solving \eqref{eq_1} in the following ways, and standard nonlinear solvers may not be adequate:
\begin{itemize}
\item There is no guarantee that a solution to \eqref{eq_1} exists, nor that it is unique.
\item We often do not have a good initial guess for the parameter value $\boldsymbol{\theta}$.
\item The experimental data contains a lot of error.
\end{itemize}
In order to overcome these challenges we have constructed a numerical algorithm called the Cluster Approximation method that approximately solves the system of nonlinear equations.  This method differs from traditional nonlinear equation solvers in the following ways:
\begin{itemize}
\item It finds multiple possible solutions that approximately satisfy \eqref{eq_1}.
\item It is more robust against local minima of the objective function compared to other gradient based algorithms since it constructs a linear approximation from multiple points.
\item It is faster than population based stochastic methods as it is a gradient based algorithm.
\end{itemize}
Simply put, the Cluster Approximation method combines the desirable properties of a population based global optimization method (like semi-global convergence) with properties of gradient based methods (like convergence speed), as it moves a cluster of tentative solution points using gradient like information.
We demonstrate these points through solving three different types of inverse problems in mathematical biology: underdetermined inverse problems (number of observations smaller than the number of parameters, i.e., $m>n$), overdetermined inverse problem (number of observations greater than the number of parameters, i.e., $m<n$), and parameter identifiability analysis problems (approximating an inverse image of a set).
===affil3:
University of Waterloo
===lastname5:
Konagaya
===affilother:

===title:
Cluster Approximation method for Inverse Problems: application to Model Parameter Estimation
===firstname2:
Ken