Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
A-BDF: A Generalization of the Backward Differentiation Formulae
SIAM Journal on Numerical Analysis
Numerical Initial Value Problems in Ordinary Differential Equations
Numerical Initial Value Problems in Ordinary Differential Equations
GALAHAD, a library of thread-safe Fortran 90 packages for large-scale nonlinear optimization
ACM Transactions on Mathematical Software (TOMS)
A-EBDF: an adaptive method for numerical solution of stiff systems of ODEs
Mathematics and Computers in Simulation
A Multidimensional Filter Algorithm for Nonlinear Equations and Nonlinear Least-Squares
SIAM Journal on Optimization
Mathematical Programming: Series A and B
A Globally Convergent Linearly Constrained Lagrangian Method for Nonlinear Optimization
SIAM Journal on Optimization
On the stability of exponential fitting BDF algorithms
Journal of Computational and Applied Mathematics - Special issue: Selected papers of the international conference on computational methods in sciences and engineering (ICCMSE-2003)
Brief paper: Rigorous parameter reconstruction for differential equations with noisy data
Automatica (Journal of IFAC)
SIAM Journal on Scientific Computing
Parameter range reduction for ODE models using monotonic discretizations
Journal of Computational and Applied Mathematics
Telescoping strategies for improved parameter estimation of environmental simulation models
Computers & Geosciences
Hi-index | 7.30 |
We consider fitting an ODE model to time series data of the system variables. We assume that the parameters of the model have some initial range of possible values and the goal is to reduce these ranges to produce a smaller parameter region from which to start a global nonlinear optimization algorithm. We introduce the class of cumulative backward differentiation formulas (CBDFs) and show that they inherit the accuracy and stability properties of their generating backward differentiation formulas (BDFs). Discretizing the system with these CBDFs and applying consistency conditions results in reductions of the parameter ranges. We show that these reductions are better than can be obtained simply using BDFs. In addition CBDFs inherit any monotonicity properties with respect to the parameters that the vector field possesses, and we exploit these properties to make the consistency checking more efficient. We illustrate with several examples, analyze some of the behavior of our range reduction method, and discuss how the method could be extended and improved.