A robust approach for finding all well-separated solutions of sparse systems of nonlinear equations

Abstract

Tearing is a long-established decomposition technique, widely adapted across many engineering fields. It reduces the task of solving a large and sparse nonlinear system of equations to that of solving a sequence of low-dimensional ones. The most serious weakness of this approach is well-known: It may suffer from severe numerical instability. The present paper resolves this flaw for the first time. The new approach requires reasonable bound constraints on the variables. The worst-case time complexity of the algorithm is exponential in the size of the largest subproblem of the decomposed system. Although there is no theoretical guarantee that all solutions will be found in the general case, increasing the so-called sample size parameter of the method improves robustness. This is demonstrated on two particularly challenging problems. Our first example is the steady-state simulation a challenging distillation column, belonging to an infamous class of problems where tearing often fails due to numerical instability. This column has three solutions, one of which is missed using tearing, but even with problem-specific methods that are not based on tearing. The other example is the Stewart-Gough platform with 40 real solutions, an extensively studied benchmark in the field of numerical algebraic geometry. For both examples, all solutions are found with a fairly small amount of sampling.

Improving Convergence Behavior of Nonlinear Equation Systems in Intensified Process Models by Decomposition Methods

“…Further test cases will be performed to identify a limit for the number of tearing variables where the numerical costs are still reasonable low. Beside this, methods as the one of Baharev et al (2017) that systematically generate initial values close to the solution will be examined in order to provide a sufficient set of initial values for the tearing variables and achieve a certain level of efficiency during the iteration of a BBTF decomposed system.…”

A manifold-based approach to sparse global constraint satisfaction problems

“…4. The algorithm of the present paper is a significant improvement over older algorithms discussed in [3, 4], both algorithmically and on the implementation level. The entire algorithm has been redesigned and rewritten from scratch, and in particular, the backsolve step is radically different.…”

“…2 is written, we try all the possible matches in a brute-force manner. The previous implementation of the algorithm also worked [3] this way. Numerical evidence shows that it can be very wasteful: If two distinct points in the point cloud are close in their components, it is very likely that the matches will have very similar objective value in (9) too; there is little to no benefit in trying both of them.…”

Failure Modes of Tearing and a Novel Robust Approach

“…8.9.2, p. 442); the emphasis is ours. If we want to find all well-separated solutions of a nonlinear system of equations, the kind of applications discussed in (Baharev et al, 2016), this is unacceptable.…”

“…Staircase sampling was inspired by (Baharev and Neumaier, 2014), and proposed in (Baharev et al, 2016) to mitigate all of the issues listed in Sections 3 and 4. A detailed presentation of this method is outside the scope of this paper; here we only sketch the basic idea.…”