Interval-based projection method for under-constrained numerical systems

Ishii, Daido; Goldsztejn, Alexandre; Jermann, Christophe

doi:10.1007/s10601-012-9126-y

Cited by 18 publications

(16 citation statements)

References 14 publications

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“…push (L, x (2) ) 15: cycle loop 16: else if (wid x < ε) then 17: {The box x is too small for bisection} 18: push (L pos , x) 19: end if 20: if (x was discarded or x was stored) then 21: x = pop (L) 22: if (L was empty) then 23: {all boxes have been considered} 24: return L ver , L pos 25: end if 26: else 27: bisect (x), obtaining x (1) and x (2) 28:…”

Section: Algorithm 1 Ibpmentioning

confidence: 99%

“…Many of them are not well-determined, but underdetermined, i.e., having fewer equations than unknowns (m < n), which means they have uncountably many solutions and their solution sets do not consist of isolated points, but are manifolds. In particular, we encounter such systems in robotics [19], stability theory of dynamical systems [35], differential equations solving [31] and multicriteria analysis [30].…”

Section: Introductionmentioning

confidence: 99%

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Presentation of a highly tuned multithreaded interval solver for underdetermined and well-determined nonlinear systems

Kubica

2015

Numer Algor

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The paper summarizes author's investigations in tuning a multithreaded interval branch-and-prune algorithm for nonlinear systems and presents the developed solver. New results for using the box-consistency enforcing operator and a new variant of the initial exclusion phase are presented. Also, a new heuristic to choose the coordinate for bisection is considered. Extensive numerical experiments are analyzed to provide the satisfying version of the algorithm.

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Section: Algorithm 1 Ibpmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Presentation of a highly tuned multithreaded interval solver for underdetermined and well-determined nonlinear systems

Kubica

2015

Numer Algor

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“…In the general case, the guard condition will be expressed as a set of inequalities involving several noise symbols, for which we will have to compute inner boxes, or joint interval inner approximations, for instance using the work of Isshii et al [22] on an interval-based projection method for under-constrained systems, that relies on similar ideas as described in Section 4.2 for the computation of boxes guaranteed to be in the image of a vector-valued function.…”

Section: Guard Conditions In Hybrid Automatamentioning

confidence: 99%

Inner approximated reachability analysis

Goubault

Mullier

Putot

et al. 2014

Proceedings of the 17th International Conference on Hybrid Systems: Computation and Control

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Computing a tight inner approximation of the range of a function over some set is notoriously difficult, way beyond obtaining outer approximations. We propose here a new method to compute a tight inner approximation of the set of reachable states of non-linear dynamical systems on a bounded time interval. This approach involves affine forms and Kaucher arithmetic, plus a number of extra ingredients from set-based methods. An implementation of the method is discussed, and illustrated on representative numerical schemes, discrete-time and continuous-time dynamical systems.

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“…In the context of certification of manifolds using parallelotopes, inflating a parallelotope consists in inflating its characteristic box u corresponding to the nonparametric dimensions. To this end, Algorithm 2 implements a two-stage inflation process adapted from a box-inflation process proposed in [12,19]: The main inflation is performed by the interval Newton operator itself at line 2.4, applying it without intersecting the previous domain so as to allow shifting and inflating the box. When this inflation process succeeds (informally the initial parallelotope has to be small enough and close enough to a regular manifold), it often converges to a limit parallelotope from the inside, which does not allow observing the strict contraction necessary to certify the manifold.…”

Section: Manifold Certification Via Parallelotope Computationmentioning

confidence: 99%

“…Note that each iteration of Algorithm 2 involves the interval evaluation of a function, that of a Jacobian matrix and O(n 3 ) interval operations for the right preconditioning of the interval Jacobian. The properties of Algorithm 2 are summarized in the following theorem (whose proof is derived straightforwardly from [10,11,19] …”

Section: Manifold Certification Via Parallelotope Computationmentioning

confidence: 99%

Certified Parallelotope Continuation for One-Manifolds

Martin¹,

Goldsztejn²,

Granvilliers³

et al. 2013

SIAM J. Numer. Anal.

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Starting from an initial solution, continuation methods efficiently produce a sequence of points on a manifold typically defined as the solution set of an underconstrained system of equations. They have a wide range of applications ranging from curve plotting to polynomial root-finding by homotopy. However, classical methods cannot guarantee that the returned points all belong to the same connected component of the manifold, i.e., they may jump from one component to another. Trying to overcome this issue has given birth to several sophisticated heuristics on the one hand and to guaranteed methods based on rigorous computations on the other hand. In this paper we introduce a new rigorous predictor corrector continuation method based on interval computations. Its novelty lies in the fact that it uses parallelotopes as defined in A. Goldsztejn and L. Granvilliers, A new framework for sharp and efficient resolution of NCSP with manifolds of solutions, Constraints, 15 (2010), pp. 190-212, to enclose consecutive portions of the followed manifold. Though computationally more expensive than regular interval computations, the fact that their orientation can be automatically adapted to the local topology of the followed curve makes parallelotopes a very suitable tool for a certified continuation method, as shown by reported experimental results. Introduction.Continuation methods [1] allow exploring step by step a solution manifold, usually defined as the solution set of an underconstrained system of equations F (x) = 0 with F : R n → R m and m < n. This paper focuses on systems where n − m = 1, whose solution manifolds are curves, and proposes a robust singleparameter continuation method. It has a wide range of applications, e.g., polynomial root finding via homotopy [3], nonlinear eigenvalue problems [4], biobjective optimization [18,33], and robot path planning [24]. The most simple and effective continuation method is the predictor corrector algorithm. It includes the simple embedding method and several of its improvements such as the parameter switching [28,29] and the pseudo-arclength methods that have the definite advantage over the simple embedding method that they naturally track folds [6]. They are all, however, subject to jumping between disconnected components without any prompt. For a given system, there exist some theoretical upper bounds on the step size that enforce the connectivity of the continuation, e.g., Theorem 5.2.1 in [1], but they are impractical. While this may be acceptable in some contexts since solutions are eventually computed, such jumps contradict the essence of continuation. Furthermore, connectivity is mandatory in some applications, e.g., robot command synthesis, where such jumps would result in a nonfeasible command path, or in homotopy methods, where such jumps would prevent computing all the solutions of the original system.

show abstract

Interval-based projection method for under-constrained numerical systems

Cited by 18 publications

References 14 publications

Presentation of a highly tuned multithreaded interval solver for underdetermined and well-determined nonlinear systems

Presentation of a highly tuned multithreaded interval solver for underdetermined and well-determined nonlinear systems

Inner approximated reachability analysis

Certified Parallelotope Continuation for One-Manifolds

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