Adaptive Newton-Type Schemes Based on Projections

Abstract

In this work we present and discuss a possible globalization concept for Newton-type methods. We consider nonlinear problems f (x) = 0 in R n using the concepts from ordinary differential equations as a basis for the proposed numerical solution procedure. Thus, the starting point of our approach is within the framework of solving ordinary differential equations numerically. Accordingly, we are able to reformulate general Newton-type iteration schemes using an adaptive step size control procedure. In doing so, we derive and discuss a discrete adaptive solution scheme, thereby trying to mimic the underlying continuous problem numerically without losing the famous quadratic convergence regime of the classical Newton method in a vicinity of a regular solution. The derivation of the proposed adaptive iteration scheme relies on a simple orthogonal projection argument taking into account that, sufficiently close to regular solutions, the vector field corresponding to the Newton scheme is approximately linear. We test and exemplify our adaptive root-finding scheme using a few low-dimensional examples. Based on the presented examples, we finally show some performance data.

A global Newton-type scheme based on a simplified Newton-type approach

“…Here, t j signifies some adaptively chosen step size (see, e.g., [1,3,18,19] for some highly efficient step size methodologies). Again, we notice that for M(x) = J f (x) and t j ≡ 1, the iteration (5) is simply the standard Newton method.…”

“…The computation of t ∈ (0, 1] typically relies on a computational upper bound with respect to the distance x(t n ) − x n . There exists different suggested approaches towards an effective computation of the step size t (see, e.g., [1,3,4,8,12,18,19]). Here we use the adaptive step size control given in [1].…”

“…There exists different suggested approaches towards an effective computation of the step size t (see, e.g., [1,3,4,8,12,18,19]). Here we use the adaptive step size control given in [1]. This adaptive choice of the step size t consists mainly of two parts: A prediction for the step size t and a correction of the step size whenever…”

“…Algorithm 1 reproduces the classical Newton scheme-apart from the simplified Newton scheme given in step 13. Furthermore, the adaptive scheme from [1] needs a lower bound t lower for the step size t n in (3). Indeed, if t n degenerates to 0, the iterative scheme is not well defined in the sense that it must be classified as not convergent.…”

“…• initial value x 0 ∈ U , • error tolerance ε > 0 respectively. 2: δ 0 ← −M(x 0 ) −1 f (x 0 ) compute a first correction 3: t ← min (1, t) compute an initial step size based on an adaptive procedure; see, e.g., [1,18,19] 4: x s ← x 0 5: for k = 1, 2, . .…”