Interval analysis: theory and applications
Journal of Computational and Applied Mathematics - Special issue on numerical analysis in the 20th century vol. 1: approximation theory
A Rigorous Global Optimization Algorithm for Problems with Ordinary Differential Equations
Journal of Global Optimization
Some recent advances in validated methods for IVPs for ODEs
Applied Numerical Mathematics
Global Optimization with Nonlinear Ordinary Differential Equations
Journal of Global Optimization
Bounding the Solutions of Parameter Dependent Nonlinear Ordinary Differential Equations
SIAM Journal on Scientific Computing
Validated solutions of initial value problems for parametric ODEs
Applied Numerical Mathematics
On Taylor Model Based Integration of ODEs
SIAM Journal on Numerical Analysis
Interval Tools for ODEs and DAEs
SCAN '06 Proceedings of the 12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics
Global Optimization Of Linear Hybrid Systems With Varying Transition Times
SIAM Journal on Control and Optimization
Introduction to Interval Analysis
Introduction to Interval Analysis
Towards global bilevel dynamic optimization
Journal of Global Optimization
McCormick-Based Relaxations of Algorithms
SIAM Journal on Optimization
Generalized McCormick relaxations
Journal of Global Optimization
Improved relaxations for the parametric solutions of ODEs using differential inequalities
Journal of Global Optimization
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This paper presents a discretize-then-relax methodology to compute convex/concave bounds for the solutions of a wide class of parametric nonlinear ODEs. The procedure builds upon interval methods for ODEs and uses the McCormick relaxation technique to propagate convex/concave bounds. At each integration step, a two-phase procedure is applied: a priori convex/concave bounds that are valid over the entire step are calculated in the first phase; then, pointwise-in-time convex/concave bounds at the end of the step are obtained in the second phase. An approach that refines the interval state bounds by considering subgradients and affine relaxations at a number of reference parameter values is also presented. The discretize-then-relax method is implemented in an object-oriented manner and is demonstrated using several numerical examples.