Hull of a system of linear simultaneous algebraic interval equations.

摘要

A general method for calculating the hull of a system of simultaneous algebraic interval equations derived from linear partial differential equations has been developed. Entries in both the coefficient matrix and the right hand side can be interval quantities. A direct application of interval arithmetic to solve such a system is infeasible because of dependencies in the coefficient matrix. The proposed solution scheme therefore utilizes a combination of interval and point arithmetic.; Since the coefficient matrix of the underlying point system is inverse positive, it is shown that the solution vector varies montonically with respect to the right hand side vector. Therefore, the right hand side interval vector is replaced by two point vectors, each representing a separate system: a minimal system which provides the lower bound of the hull, and a maximal system which provides the upper bound of the hull. With respect to parameters in the coefficient matrix, the solution is not only nonlinear, but can also be nonconvex. Hence, available gradient based iterative methods for solving nonlinear systems cannot yield the hull with guarantee. A global interval-based approach is therefore implemented to isolate stationary points in the parameter space. The hull is then determined by evaluating bounds at the stationary points and at the comers of the parameter space.; The solution scheme relies on monotonicity of terms in the inverse coefficient matrix with respect to parameters in the untied parameter space. While this approach enables the class of problem under consideration to be solved, it is computationally intensive. It is therefore crucial that conditions for extended monotonicity be researched and established in the tied parameter space. This will reduce the problem dimensionality and thereby enable solution of large problems in an efficient manner.; The proposed solution scheme is validated using an example problem for which the solution space was constructed by conducting a set of point evaluations. The applicability and feasibility of applying interval-based techniques for solving some intractable engineering problems is also discussed. In particular, the problems of global parameter estimation (which can also be nonconvex) and probabilistic analysis of systems are studied and found to be worthy of further investigations for application of interval-based methods.