Simplicial Domain Research Articles

Mutation-finite quivers with real weights

Abstract We classify all mutation-finite quivers with real weights. We show that every finite mutation class not originating from an integer skew-symmetrisable matrix has a geometric realisation by reflections. We also explore the structure of acyclic representatives in finite mutation classes and their relations to acute-angled simplicial domains in the corresponding reflection groups.

Adaptive Integration and Singular Boundary Transformations

We apply and compare results of transformations used to annihilate boundary singularities for multivariate integration over hyper-rectangular and simplicial domains. While classically these transformations are applied with a product trapezoidal rule, we use adaptive methods in the PARINT software package, based on rules of higher polynomial degree for the integration over subdomains. PARINT is layered over the MPI environment (Message Passing Interface) and deploys advanced parallel computation techniques such as load balancing among processes that are distributed over a network of nodes. The message passing is performed in a non-blocking and asynchronous manner, and permits overlapping of computation and communication. Comparisons of computation times using long double vs. double precision confirm that the extended format does not considerably increase the time for long doubles. We further apply the proposed methods to problems arising from self-energy Feynman loop diagrams with massless internal lines, in particular where the corresponding integrand has singularities on the boundaries of the integration domain.

HOPF-ALGEBRAIC ASPECTS OF ITERATED STOCHASTIC INTEGRALS

An explicit formula is found for the antipode in Itô–Hopf algebras. The Hopf algebra structure is extended to iterated classical and quantum stochastic integrals. The antipode is related to change of orientation in the simplicial domains of such integrals.

Solving nonlinear polynomial systems in the barycentric Bernstein basis

We present a method for solving arbitrary systems of N nonlinear polynomials in n variables over an n-dimensional simplicial domain based on polynomial representation in the barycentric Bernstein basis and subdivision. The roots are approximated to arbitrary precision by iteratively constructing a series of smaller bounding simplices. We use geometric subdivision to isolate multiple roots within a simplex. An algorithm implementing this method in rounded interval arithmetic is described and analyzed. We find that when the total order of polynomials is close to the maximum order of each variable, an iteration of this solver algorithm is asymptotically more efficient than the corresponding step in a similar algorithm which relies on polynomial representation in the tensor product Bernstein basis. We also discuss various implementation issues and identify topics for further study.

A unified subdivision approach for multi-dimensional non-manifold modeling

This paper presents a new unified subdivision scheme that is defined over a k -simplicial complex in n -D space with k ≤ 3 . We first present a series of definitions to facilitate topological inquiries during the subdivision process. The scheme is derived from the double ( k + 1 ) -directional box splines over k -simplicial domains. Thus, it guarantees a certain level of smoothness in the limit on a regular mesh. The subdivision rules are modified by spatial averaging to guarantee C 1 smoothness near extraordinary cases. Within a single framework, we combine the subdivision rules that can produce 1-, 2-, and 3-manifolds in arbitrary n -D space. Possible solutions for non-manifold regions between the manifolds with different dimensions are suggested as a form of selective subdivision rules according to user preference. We briefly describe the subdivision matrix analysis to ensure a reasonable smoothness across extraordinary topologies, and empirical results support our assumption. In addition, through modifications, we show that the scheme can easily represent objects with singularities, such as cusps, creases, or corners. We further develop local adaptive refinement rules that can achieve level-of-detail control for hierarchical modeling. Our implementation is based on the topological properties of a simplicial domain. Therefore, it is flexible and extendable. We also develop a solid modeling system founded on our subdivision schemes to show potential benefits of our work in industrial design, geometric processing, and other applications.

Inter-surface mapping

We consider the problem of creating a map between two arbitrary triangle meshes. Whereas previous approaches compose parametrizations over a simpler intermediate domain, we directly create and optimize a continuous map between the meshes. Map distortion is measured with a new symmetric metric, and is minimized during interleaved coarse-to-fine refinement of both meshes. By explicitly favoring low inter-surface distortion, we obtain maps that naturally align corresponding shape elements. Typically, the user need only specify a handful of feature correspondences for initial registration, and even these constraints can be removed during optimization. Our method robustly satisfies hard constraints if desired. Inter-surface mapping is shown using geometric and attribute morphs. Our general framework can also be applied to parametrize surfaces onto simplicial domains, such as coarse meshes (for semi-regular remeshing), and octahedron and toroidal domains (for geometry image remeshing). In these settings, we obtain better parametrizations than with previous specialized techniques, thanks to our fine-grain optimization.

Evaluation and approximate evaluation of the multivariate Bernstein-Bézier form on a regularly partitioned simplex

Polynomials of the total degree d in m variables have a geometrically intuitive representation in the Bernstein-Be´zier form defined over an m -dimensional simplex. The two algorithms given in this article evaluate the Bernstein-Be´zier form on a large number of points corresponding to a regular partition of the simplicial domain. The first algorithm is an adaptation of isoparametric evaluation. The second is a subdivision algorithm. In contrast to de Casteljau's algorithm, both algorithms have a cost of evaluation per point that is linear in the degree regardless of the number of variables. To demonstrate practicality, implementations of both algorithms on a triangular domain are compared with generic implementations of six algorithms in the literature.

On the numerical condition of algebraic curves and surfaces 1. Implicit equations

The numerical stability of algebraic curves and surfaces represented by implicit equations is investigated. The condition number at a point of a curve or surface is defined as the ratio of the maximum normal displacement of that point to the relative magnitude ϵ of the random perturbations in the curve or surface coefficients, in the limit ϵ → 0. Closed-form expressions for such condition numbers are presented, and the singular points of implicitly defined curves and surfaces are shown to be inherently ill-conditioned. Condition numbers for curve and surface intersections may be expressed in terms of those of the participant entities at the given point and certain geometric factors determined by the normal directions there. Tangential intersections are also seen to be inherently ill-conditioned. The dependence of condition numbers on the chosen multivariate polynomial basis is then examined. In particular, we compare power expansions about a given center, barycentric Bernstein bases over simplicial domains, and tensor-product Bernstein bases over rectangular domains. Configurations are enumerated in which one of these bases provides better conditioning than another at each point of every curve or surface in a given domain. The subdivision and degree elevation of multivariate Bernstein forms (barycentric or tensor-product) exhibit such behavior.