Exclusion regions for parameter-dependent systems of equations
For researchers working on parameter-dependent nonlinear systems, this provides a rigorous algorithmic approach to certify feasible parameter regions and solution enclosures, though it is limited to low-dimensional parameter spaces.
The paper presents a new interval-based algorithm for rigorously constructing inner estimates of feasible parameter regions and enclosures of solution sets for parameter-dependent nonlinear systems in low dimensions. The method proves existence of solutions within error bounds for all parameters in a constructed box around a regular parameter value.
This paper presents a new algorithm based on interval methods for rigorously constructing inner estimates of feasible parameter regions together with enclosures of the solution set for parameter-dependent systems of nonlinear equations in low (parameter) dimensions. The proposed method allows to explicitly construct feasible parameter sets around a regular parameter value, and to rigorously enclose a particular solution curve (resp. manifold) by a union of inclusion regions, simultaneously. The method is based on the calculation of inclusion and exclusion regions for zeros of square nonlinear systems of equations. Starting from an approximate solution at a fixed set $p$ of parameters, the new method provides an algorithmic concept on how to construct a box $\mathbf{s}$ around $p$ such that for each element $s\in \mathbf{s}$ in the box the existence of a solution can be proved within certain error bounds.