Abstract
We consider square, sparse nonlinear systems of equations whose Jacobian is structurally nonsingular, with reasonable bound constraints on all variables. We propose an algorithm for finding good approximations to all well-separated solutions of such systems. We assume that the input system is ordered such that its Jacobian is in bordered block lower triangular form with small diagonal blocks and with small border width; this can be performed fully automatically with off-the-shelf decomposition methods. Five decades of numerical experience show that models of technical systems tend to decompose favorably in practice. Once the block decomposition is available, we reduce the task of solving the large 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 proposed method resolves this issue with the novel backsolve step. We study the effect of the decomposition on a sequence of challenging problems. Beyond a certain problem size, the computational effort of multistart (no decomposition) grows exponentially. In contrast, thanks to the decomposition, for the proposed method the computational effort grows only linearly with the problem size. It depends on the problem size and on the hyperparameter settings whether the decomposition and the more sophisticated algorithm pay off. Although there is no theoretical guarantee that all solutions will be found in the general case, increasing the so-called sample size hyperparameter improves the robustness of the proposed method.
Highlights
1.1 AimsWe consider square nonlinear systemsF(x) = 0, x ≤ x ≤ x, (1)where F : Rn → Rn is a continuously differentiable vector-valued function, and whose Jacobian is structurally nonsingular; x and x denote the vector of lower and upper bounds, respectively on the components of x
The so-called bordered block lower triangular form is illustrated in Fig. 1, and formally defined as follows
Decomposing to bordered block lower triangular form has a long tradition in engineering applications: It is usually referred to as tearing, diakoptics, or sequential modular approach, depending on the engineering discipline
Summary
Where F : Rn → Rn is a continuously differentiable vector-valued function, and whose Jacobian is structurally nonsingular; x and x denote the vector of lower and upper bounds, respectively on the components of x. The task we pose is to find a reasonably small set of points such that every solution of (1) is close to one of the points in this set. An algorithm solving this task finds in particular good approximations to all well-separated solutions. We assume that (1) has already been ordered such that its Jacobian is in bordered block lower triangular form with small blocks and with small border width; the formal definition of bordered block lower triangular forms is given in Sect. Further (less limiting) assumptions are given in Sect.
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