Alexandre Goldsztejn

A Right-Preconditioning Process for the Formal-Algebraic Approach to Inner and Outer Estimation of AE-Solution Sets

Aright-preconditioning process for linear interval systems has been presented by Neumaier in 1987. It allows the construction of an outer estimate of the united solution set of a square linear interval system in the form of a parallelepiped. The denomination “right-preconditioning” is used to describe the preconditioning processes which involve the matrix product AC in contrast to the (usual) left-preconditioning processes which involve the matrix product AC, where A and C are respectively the interval matrix of the studied linear interval system and the preconditioning matrix.The present paper presents a new right-preconditioning process similar to the one presented by Neumaier in 1987 but in the more general context of the inner and outer estimations of linear AEsolution sets. Following the spirit of the formal-algebraic approach to AE-solution sets estimation, summarized by Shary in 2002, the new right-preconditioning process is presented in the form of two new auxiliary interval equations. Then, the resolution of these auxiliary interval equations leads to inner and outer estimates of AE-solution sets in the form of parallelepipeds. This right-preconditioning process has two advantages: on one hand, the parallelepipeds estimates are often more precise than the interval vectors estimates computed by Shary. On the other hand, in many situations, it simplifies the formal algebraic approach to inner estimation of AE-solution sets. Therefore, some AE-solution sets which were almost impossible to inner estimate with interval vectors, become simple to inner estimate using parallelepipeds. These benefits are supported by theoretical results and by some experimentations on academic examples of linear interval systems.

A branch and prune algorithm for the approximation of non-linear AE-solution sets

Non-linear AE-solution sets are a special case of parametric systems of equations where universally quantified parameters appear first. They allow to model many practical situations. A new branch and prune algorithm dedicated to the approximation of non-linear AE-solution sets is proposed. It is based on a new generalized interval (intervals whose bounds are not constrained to be ordered) parametric Hansen-Sengupta operator. In spite of some restrictions on the form of the AE-solution set which can be approximated, it allows to solve problems which were before out of reach of previous numerical methods. Some promising experimentations are presented.

Interval-based projection method for under-constrained numerical systems

This paper presents an interval-based method that follows the branch-and-prune scheme to compute a verified paving of a projection of the solution set of an under-constrained system. Benefits of this algorithm include anytime solving process, homogeneous verification of inner boxes, and applicability to generic problems, allowing any number of (possibly nonlinear) equality and inequality constraints. We present three key improvements of the algorithm dedicated to projection problems: (i) The verification process is enhanced in order to prove faster larger boxes in the projection space. (ii) Computational effort is saved by pruning redundant portions of the solution set that would project identically. (iii) A dedicated branching strategy allows reducing the number of treated boxes. Experimental results indicate that various applications can be modeled as projection problems and can be solved efficiently by the proposed method.

A branch and prune algorithm for the computation of generalized aspects of parallel robots

Parallel robots enjoy enhanced mechanical characteristics that have to be contrasted with a more complicated design. In particular, they often have parallel singularities at some poses, and the robots may become uncontrollable, and could even be damaged, in such configurations. The computation of the connected components in the set of nonsingular reachable configurations, called generalized aspects, is therefore a key issue in their design. This paper introduces a new method, based on numerical constraint programming, to compute a certified enclosure of the generalized aspects. Though this method does not allow counting their number rigorously, it constructs inner approximations of the nonsingular workspace that allow commanding parallel robots safely. It also provides a lower-bound on the exact number of generalized aspects. It is moreover the first general method able to handle any parallel robot in theory, though its computational complexity currently restricts its usage to robots with three degrees of freedom. Finally, the constraint programming paradigm it relies on makes it possible to consider various additional constraints (e.g., collision avoidance), making it suitable for practical considerations.

Certified Parallelotope Continuation for One-Manifolds

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 consecuti...

Tinkerbell Is Chaotic

Shadowing is a method of backward error analysis that plays a important role in hyperbolic dynamics. In this paper, the shadowing by containment framework is revisited, including a new shadowing theorem. This new theorem has several advantages with respect to existing shadowing theorems: It does not require injectivity or differentiability, and its hypothesis can be easily verified using interval arithmetic. As an application of this new theorem, shadowing by containment is shown to be applicable to infinite length orbits and is used to provide a computer assisted proof of the presence of chaos in the well-known noninjective Tinkerbell map.

A New Framework for Sharp and Efficient Resolution of NCSP with Manifolds of Solutions

When numerical CSPs are used to solve systems of nequations with nvariables, the interval Newton operator plays a key role: It acts like a global constraint, hence achieving a powerful contraction, and proves rigorously the existence of solutions. However, both advantages cannot be used for under-constrained systems of equations, which have manifolds of solutions. A new framework is proposed in this paper to extend the advantages of the interval Newton to under-constrained systems of equations. This is done simply by permitting domains of CSPs to be parallelepipeds instead of the usual boxes.

Including ordinary differential equations based constraints in the standard CP framework

Coupling constraints and ordinary differential equations has numerous applications. This paper shows how to introduce constraints involving ordinary differential equations into the numerical constraint satisfaction problem framework in a natural and efficient way. Slightly adapted standard filtering algorithms proposed in the numerical constraint satisfaction problem framework are applied to these constraints leading to a branch and prune algorithm that handles ordinary differential equations based constraints. Preliminary experiments are presented.

On continuation methods for non-linear bi-objective optimization: towards a certified interval-based approach

The global resolution of constrained non-linear bi-objective optimization problems (NLBOO) aims at covering their Pareto-optimal front which is in general a one-manifold in