Bernard Mourrain

We present an efficient, globally G1--continuous scheme, for extracting Bézier patches from any mesh having vertices of valence four and polygonal faces, that is, the mesh topology that arises in Doo-Sabin subdivision. The Bézier points are given explicitly as local, weighted averages in the vicinity of each vertex of the mesh, yielding bi-quadratic patches in regular regions and bi-quartic or bi-quintic patches in the vicinity of irregular regions. In particular, we first derive simple bi-quadratic averaging masks that produce quadratic patches that join with C1 continuity in regular regions. In the vicinity of irregular faces, we elevate the degree, then impose certain symmetric gluing data of degree two on the patch interfaces, and compute explicitly masks as solution of the G1 constraints. The resulting scheme, named G1ADS, ensures machine precision adherence to the G1 conditions, reproduces quadratic C1 B-splines at the regular regions, and enjoys a minimal number of degree elevated patches in the vicinity of irregular regions. We evaluate the performance of G1ADS quantitatively and qualitatively on several challenging benchmarks, in terms of accuracy, curvature and isophote analysis, and we compare with the state-of-the-art method. Our results demonstrate that G1ADS is efficient, robust and more accurate, producing high quality surfaces that converge to the respective Doo-Sabin limit surface.

We study the problem of representing multivariate polynomials with rational coefficients, which are nonnegative and strictly positive on finite semialgebraic sets, using rational sums of squares. We focus on the case of finite semialgebraic sets S S defined by equality constraints, generating a zero-dimensional ideal I I , and by nonnegative sign constraints. First, we obtain existential results. We prove that a strictly positive polynomial f f with coefficients in a subfield K K of R \mathbb {R} has a representation in terms of weighted Sums-of-Squares with coefficients in this field, even if the ideal I I is not radical. We generalize this result to the case where f f is nonnegative on S S and ( f ) + ( I : f ) = 1 (f) + (I:f)=1 . We deduce that nonnegative polynomials with coefficients in K K can be represented in terms of Sum-of-Squares of polynomials with coefficients in K K , when the ideal is radical. Second, we obtain degree bounds for such Sums-of-Squares representations, which depend linearly on the regularity of the ideal and the degree of the defining equations, when they form a graded basis. Finally, we analyze the bit complexity of the Sums-of-Squares representations for polynomials with coefficients in Q \mathbb {Q} , in the case the ideal is radical. The bitsize bounds are linear in the bitsize of the input polynomials, quadratic or cubic in the Bezout bound, and linear in the regularity, generalizing and improving previous results obtained for special zero dimensional ideals. As an application in the context of polynomial optimization, we retrieve and improve results on the finite convergence and exactness of the moment/Sums-of-Squares hierarchy.

We present a novel infinite-dimensional linear programming (IDLP) based method for solving Partially Observable Markov Decision Processes (POMDPs) via Moment-SoS hierarchies. Standard belief-state IDLP formulations yield rational dynamics via the Bayesian filter, obstructing direct use of Moment-SoS relaxations. To address this, we reformulate the POMDP over unnormalized beliefs, yielding polynomial system dynamics. This formulation, however, breaks contractivity of the Bellman operator, rendering conventional IDLPs ill-posed. Our key contribution is to restore contractivity by introducing an ℓ1-based Lyapunov weight and formulating the control problem in a weighted Banach space. The resulting Bellman operator becomes contractive, yielding a well-posed IDLP with strong duality. This leads to a generalized moment problem (GMP) formulation, solvable via Moment-SoS relaxations. We validate the method on the two-armed bandit and tiger problems, generating approximations for optimal policies and optimal values.

This article addresses the Generalized Additive Decomposition (GAD) ofsymmetric tensors, that is, degree-d forms f ∈ Sd. From a geometric perspective, a GADcorresponds to representing a point on a secant of osculating varieties to the Veronesevariety, providing a compact and structured description of a tensor that captures itsintrinsic algebraic properties.We provide a linear algebra method for measuring the GAD size and prove that theminimal achievable size, which we call the GAD-rank of the considered tensor, coincideswith the rank of suitable Catalecticant matrices, under certain regularity assumptions.We provide a new explicit description of the apolar scheme associated with a GAD asthe annihilator of a polynomial-exponential series. We show that if the Castelnuovo-Mumford regularity of this scheme is sufficiently small, then both the GAD and theassociated apolar scheme are minimal and unique.Leveraging these results, we develop a numerical GAD algorithm for symmetric ten-sors that effectively exploits the underlying algebraic structure, extending existing alge-braic approaches based on eigen computation to the treatment of multiple points. Weillustrate the effectiveness and numerical stability of such an algorithm through severalexamples, including Waring and tangential decompositions.

Geometrically continuous splines are piecewise polynomial functions defined on a collection of patches which are stitched together through transition maps. They are called G r G^{r} -splines if, after composition with the transition maps, they are continuously differentiable functions to order r r on each pair of patches with stitched boundaries. This type of spline has been used to represent smooth shapes with complex topology for which (parametric) spline functions on fixed partitions are not sufficient. In this article, we develop new algebraic tools to analyze G r G^r -spline spaces. We define G r G^{r} -domains and transition maps using an algebraic approach, and establish an algebraic criterion to determine whether a piecewise function is G r G^r -continuous on the given domain. In the proposed framework, we construct a chain complex whose top homology is isomorphic to the G r G^{r} -spline space. This complex generalizes Billera-Schenck-Stillman homological complex used to study parametric splines. Additionally, we show how previous constructions of G r G^r -splines fit into this new algebraic framework, and present an algorithm to construct a basis for G r G^r -spline spaces. We illustrate how our algebraic approach works with concrete examples and prove a dimension formula for the G r G^r -spline space in terms of invariants to the chain complex. In some special cases, explicit dimension formulas in terms of the degree of splines are also given.