What is the problem? Why should I care? Results HOW? Our Methods How to Integrate a Polynomial over a Convex Polytope: Combinatorics and Algorithms Jes´ us A. De Loera, UC Davis September 19, 2012
What is the problem? Why should I care? Results HOW? Our Methods Theorems are joint work with
What is the problem? Why should I care? Results HOW? Our Methods Software LattE integrale was developed with help by several smart students. Most notably
What is the problem? Why should I care? What we want Results Reality check HOW? Our Methods There is hope! Picking up the pieces.... Our Problem Background and Motivation
What is the problem? Why should I care? What we want Results Reality check HOW? Our Methods There is hope! Picking up the pieces.... Our Wishes Given P be a d -dimensional rational polytope inside R n and let f ∈ Q [ x 1 , . . . , x n ] be a polynomial with rational coefficients. � Compute the EXACT value of the integral P f dm ?
What is the problem? Why should I care? What we want Results Reality check HOW? Our Methods There is hope! Picking up the pieces.... Example If we integrate the monomial x 17 y 111 z 13 over the three-dimensional ∆ x 17 y 111 z 23 dxdydz equals exactly � standard simplex ∆. Then 1 317666399137306017655882907073489948282706281567360000
What is the problem? Why should I care? What we want Results Reality check HOW? Our Methods There is hope! Picking up the pieces.... Why compute integrals over polytopes? Integration over polyhedra is useful!!
What is the problem? Why should I care? What we want Results Reality check HOW? Our Methods There is hope! Picking up the pieces.... Why compute integrals over polytopes? Integration over polyhedra is useful!! Physical simulation: Realistic animation and geometric design must both pay attention to the physics implied by the first moments, the volume, center of mass, and inertia frame of the objects they manipulate.
What is the problem? Why should I care? What we want Results Reality check HOW? Our Methods There is hope! Picking up the pieces.... Why compute integrals over polytopes? Integration over polyhedra is useful!! Physical simulation: Realistic animation and geometric design must both pay attention to the physics implied by the first moments, the volume, center of mass, and inertia frame of the objects they manipulate. Tomography and Inverse problems: The X-rays of a polytope can be used to estimate the moments of the underlying mass distribution. One can reconstruct of any convex polytope, from knowledge of its moments.
What is the problem? Why should I care? What we want Results Reality check HOW? Our Methods There is hope! Picking up the pieces.... Why compute integrals over polytopes? Integration over polyhedra is useful!! Physical simulation: Realistic animation and geometric design must both pay attention to the physics implied by the first moments, the volume, center of mass, and inertia frame of the objects they manipulate. Tomography and Inverse problems: The X-rays of a polytope can be used to estimate the moments of the underlying mass distribution. One can reconstruct of any convex polytope, from knowledge of its moments. Probability and Statistics: marginal likelihood integrals in model selection.
What is the problem? Why should I care? What we want Results Reality check HOW? Our Methods There is hope! Picking up the pieces.... Why compute integrals over polytopes? Integration over polyhedra is useful!! Physical simulation: Realistic animation and geometric design must both pay attention to the physics implied by the first moments, the volume, center of mass, and inertia frame of the objects they manipulate. Tomography and Inverse problems: The X-rays of a polytope can be used to estimate the moments of the underlying mass distribution. One can reconstruct of any convex polytope, from knowledge of its moments. Probability and Statistics: marginal likelihood integrals in model selection. But, why EXACT integration? Numeric Integration is successful, right? My point: Exact integration useful for calibration!!!!
What is the problem? Why should I care? What we want Results Reality check HOW? Our Methods There is hope! Picking up the pieces.... VOLUMES: a few reasons to compute them (for algebraic geometers) If P is an integral d -dimensional polytope, then d ! times the volume of P is the degree of the toric variety associated to P .
What is the problem? Why should I care? What we want Results Reality check HOW? Our Methods There is hope! Picking up the pieces.... VOLUMES: a few reasons to compute them (for algebraic geometers) If P is an integral d -dimensional polytope, then d ! times the volume of P is the degree of the toric variety associated to P . (for computational algebraic geometers) Let f 1 , . . . , f n be polynomials in C [ x 1 , . . . , x n ]. Let New ( f j ) denote the Newton polytope of f j , If f 1 , . . . , f n are generic, then the number of solutions of the polynomial system of equations f 1 = 0 , . . . , f n = 0 with no x i = 0 is equal to the normalized mixed volume n ! MV ( New ( f 1 ) , . . . , New ( f n )) .
What is the problem? Why should I care? What we want Results Reality check HOW? Our Methods There is hope! Picking up the pieces.... VOLUMES: a few reasons to compute them (for algebraic geometers) If P is an integral d -dimensional polytope, then d ! times the volume of P is the degree of the toric variety associated to P . (for computational algebraic geometers) Let f 1 , . . . , f n be polynomials in C [ x 1 , . . . , x n ]. Let New ( f j ) denote the Newton polytope of f j , If f 1 , . . . , f n are generic, then the number of solutions of the polynomial system of equations f 1 = 0 , . . . , f n = 0 with no x i = 0 is equal to the normalized mixed volume n ! MV ( New ( f 1 ) , . . . , New ( f n )) . (for Combinatorialists ) Volumes count things! CR m = { ( a ij ) : � i a ij = 1 , � j a ij = 1 , with a ij ≥ 0 but a ij = 0 when j > i + 1 } , then NV ( CR m ) = product of first ( m − 2) Cat alan numbers . (D. Zeilberger). Many Other applications...
What is the problem? Why should I care? What we want Results Reality check HOW? Our Methods There is hope! Picking up the pieces.... A running example � Suppose we wish to integrate pentagon f ( x , y ) dxdy (1,3) (0,2) (3,1) (0,0) (2,0) We teach undergraduates to decompose the integral into boxes: � 1 � x +2 � 2 � − x +4 � 3 � − x +4 f ( x , y ) dydx + f ( x , y ) dydx + f ( x , y ) dydx 0 0 1 0 2 x − 2
What is the problem? Why should I care? What we want Results Reality check HOW? Our Methods There is hope! Picking up the pieces.... Hey! I took calculus already!! For f ( x ) = f ( x 1 , . . . , x d ) a polynomial function calculus books say THINK BOXES, ITERATION!!! For a full-dimensional polytope P = { A x ≤ b } ⊆ R d � b 1 � b 2 ( x 1 ) � b 3 ( x 1 , x 2 ) � b d ( x 1 ,..., x d − 1 ) � � f ( x ) d x = . . . f ( x ) d x P a 1 a 2 ( x 1 ) a 3 ( x 1 , x 2 ) a d ( x 1 ,..., x d − 1 ) boxes
What is the problem? Why should I care? What we want Results Reality check HOW? Our Methods There is hope! Picking up the pieces.... Hey! I took calculus already!! M. Schechter, American Mathematical Monthly 105 (1998), 246–251. For f ( x ) = f ( x 1 , . . . , x d ) a polynomial function calculus books say THINK BOXES, ITERATION!!! For a full-dimensional polytope P = { A x ≤ b } ⊆ R d � b 1 � b 2 ( x 1 ) � b 3 ( x 1 , x 2 ) � b d ( x 1 ,..., x d − 1 ) � � f ( x ) d x = . . . f ( x ) d x P a 1 a 2 ( x 1 ) a 3 ( x 1 , x 2 ) a d ( x 1 ,..., x d − 1 ) boxes To handle the parametric limits of integration: Need Fourier–Motzkin projection – exponential time BAD even for simplices
What is the problem? Why should I care? What we want Results Reality check HOW? Our Methods There is hope! Picking up the pieces.... Context and Prior work: mostly bad news... It is # P -hard to compute the volume of a vertex presented polytopes (Dyer and Frieze 1988, Khachiyan 1989).
What is the problem? Why should I care? What we want Results Reality check HOW? Our Methods There is hope! Picking up the pieces.... Context and Prior work: mostly bad news... It is # P -hard to compute the volume of a vertex presented polytopes (Dyer and Frieze 1988, Khachiyan 1989). It is # P -hard to compute the volume of a d -dimensional polytope P represented by its facets. (Brightwell and Winkler 1992) Hard to compute the volume of zonotopes (Dyer, Gritzmann 1998).
What is the problem? Why should I care? What we want Results Reality check HOW? Our Methods There is hope! Picking up the pieces.... Context and Prior work: mostly bad news... It is # P -hard to compute the volume of a vertex presented polytopes (Dyer and Frieze 1988, Khachiyan 1989). It is # P -hard to compute the volume of a d -dimensional polytope P represented by its facets. (Brightwell and Winkler 1992) Hard to compute the volume of zonotopes (Dyer, Gritzmann 1998). Number of digits necessary to write the volume of a rational polytope P cannot always be bounded by a polynomial on the input size. (J. Lawrence 1991).
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