Computing the Volume of Compact Semi-Algebraic Sets

Lairez, Pierre; Mezzarobba, Marc; Din, Mohab Safey El

doi:10.1145/3326229.3326262

Cited by 15 publications

(25 citation statements)

References 60 publications

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“…As already mentioned, the theory of D-modules comes with a vast range of applications. Among them are the high-precision computation of periods [145] or the volume of compact semi-algebraic sets [146], and the holonomic gradient method (HGM). This method is a numerical scheme for the evaluation of holonomic functions [178].…”

Section: Partial Differential Equations By Anna-laura Sattelbergermentioning

confidence: 99%

Nonlinear algebra and applications

Breiding¹,

Çelik²,

Duff³

et al. 2023

NACO

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<p style='text-indent:20px;'>We showcase applications of nonlinear algebra in the sciences and engineering. Our review is organized into eight themes: polynomial optimization, partial differential equations, algebraic statistics, integrable systems, configuration spaces of frameworks, biochemical reaction networks, algebraic vision, and tensor decompositions. Conversely, developments on these topics inspire new questions and algorithms for algebraic geometry.</p>

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Section: Partial Differential Equations By Anna-laura Sattelbergermentioning

confidence: 99%

Nonlinear algebra and applications

Breiding¹,

Çelik²,

Duff³

et al. 2023

NACO

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“…In some cases, these efforts have concentrated solely on computing the volume of semi-algebraic sets. For example, in Lairez et al [35], and with a focus on arbitrary-precision calculations in algebraic geometry, the authors develop a method to compute volumes of compact semi-algebraic sets; the methods apply a dimension-reduction process to reformulate the volume computation as a solution of a high-order Picard-Fuchs differential equation, which in turn relies on highprecision ODE solvers. As another example, Henrion et al [28] show how the volumes and moments of basic semi-algebraic sets can be approximated using a hierarchy of semi-definite programming problems, with guaranteed lower and upper bounds on these estimates, though the accuracy of these methods appear limited.…”

Section: Introductionmentioning

confidence: 99%

High-Order Quadrature on Multi-Component Domains Implicitly Defined by Multivariate Polynomials

Saye

2021

Preprint

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A high-order quadrature algorithm is presented for computing integrals over curved surfaces and volumes whose geometry is implicitly defined by the level sets of (one or more) multivariate polynomials. The algorithm recasts the implicitly defined geometry as the graph of an implicitly defined, multi-valued height function, and applies a dimension reduction approach needing only one-dimensional quadrature. In particular, we explore the use of Gauss-Legendre and tanh-sinh methods and demonstrate that the quadrature algorithm inherits their high-order convergence rates. Under the action of h-refinement with q fixed, the quadrature schemes yield an order of accuracy of 2q, where q is the one-dimensional node count; numerical experiments demonstrate up to 22nd order. Under the action of q-refinement with the geometry fixed, the convergence is approximately exponential, i.e., doubling q approximately doubles the number of accurate digits of the computed integral. Complex geometry is automatically handled by the algorithm, including, e.g., multi-component domains, tunnels, and junctions arising from multiple polynomial level sets, as well as self-intersections, cusps, and other kinds of singularities. A variety of accompanying numerical experiments demonstrates the quadrature algorithm on two-and three-dimensional problems, including randomly generated geometry involving multiple high curvature pieces; challenging examples involving high degree singularities such as cusps; adaptation to simplex constraint cells in addition to hyper-rectangular constraint cells; and boolean operations to compute integrals on overlapping domains.

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“…Nevertheless, there are notable exceptions that consider algorithms for computing the volume of compact (basic) semi-algebraic sets. For example, [29] that exploits the periods of rational integrals. In the same setting, [24,28] introduce numerical approximation schemes for volume computations, which rely on the moment-based algorithms and semi-definite programming.…”

Section: Introductionmentioning

confidence: 99%

Sampling the feasible sets of SDPs and volume approximation

Chalkis

Fisikopoulos

Repouskos

et al. 2020

ACM Commun. Comput. Algebra

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We present algorithmic, complexity, and implementation results on the problem of sampling points in the interior and the boundary of a spectrahedron, that is the feasible region of a semidefinite program. Our main tool is random walks. We define and analyze a set of primitive geometric operations that exploits the algebraic properties of spectrahedra and the polynomial eigenvalue problem and leads to the realization of a broad collection of efficient random walks. We demonstrate random walks that experimentally show faster mixing time than the ones used previously for sampling from spectrahedra in theory or applications, for example Hit and Run. Consecutively, the variety of random walks allows us to sample from general probability distributions, for example the family of log-concave distributions which arise frequently in numerous applications. We apply our tools to compute (i) the volume of a spectrahedron and (ii) the expectation of functions coming from robust optimal control. We provide a C++ open source implementation of our methods that scales efficiently up to dimension 200. We illustrate its efficiency on various data sets.

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Computing the Volume of Compact Semi-Algebraic Sets

Cited by 15 publications

References 60 publications

Nonlinear algebra and applications

Nonlinear algebra and applications

High-Order Quadrature on Multi-Component Domains Implicitly Defined by Multivariate Polynomials

Sampling the feasible sets of SDPs and volume approximation

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