Tutorial: Numerical Algebraic Geometry

Transcription

1 Tutorial: Numerical Algebraic Geometry Back to classical algebraic geometry... with more computational power and hybrid symbolic-numerical algorithms Anton Leykin Georgia Tech Waterloo, November 2011

2 Outline Homotopy continuation predictor-corrector numerical methods, Newton s method, (global) homotopy continuation scenarios Singular isolated solutions regularization of singular solutions, deflation, dual spaces/inverse systems Positive dimension witness sets, numerical irreducible decomposition, numerical primary decomposition Certified homotopy tracking numerical zeros, α-theory of Smale, heuristic vs. rigorous path-tracking

3 Computational algebraic geometry What is the game? Level 0: Given a system of polynomial equations in K[x 1,..., x n ] with finitely many solutions, ( K could be Q, Z/pZ, R, C,... ) SOLVE. Level 1+: Describe positive-dimensional solutions (curves, surfaces,...) Classical methods generalize linear algebra: Gröbner basis: a generalization of Gaussian reduction; Resultant: a generalization of determinant. These methods are symbolic.

4 Linear Algebra Numerical Linear Algebra Algebraic Geometry Numerical Algebraic Geometry

5 Applications Robotics: Stewart-Gough platforms. Griffis-Duffy platform: the solution contains a curve of degree 28. s 2 l 4 p l 3 l 2 s 1 l 1 Enumerative algebraic geometry: solutions of Schubert problems.... control theory, optimization, computer vision, math biology, real algebraic geometry, algebraic curves...

6 Polynomial homotopy continuation Target system: n equations in n variables, F (x) = (f 1 (x),..., f n (x)) = 0, where f i R = C[x] = C[x 1,..., x n ] for i = 1,..., n. Start system: n equations in n variables: G(x) = (g 1 (x),..., g n (x)) = 0, such that it is easy to solve. Homotopy: for γ C \ {0} consider H(x, t) = (1 t)g(x) + γtf (x), t [0, 1].

7 Example target start f 1 = x 4 1x 2 + 5x 2 1x x g 1 = f 2 = x 2 1 x 1 x 2 + x 2 8 g 2 = x x Start solutions target solutions: H(x, t) = 0 implies dx ( ) 1 H dt = H x t.

8 Example target start f = x g = x 2 1 The solution of the homotopy equation H(x, t) = (1 t)g(x) + tf(x) = (1 2t)x t = 0 is singular for t = 1/3.

9 Randomization Note: the complement of a complex algebraic variety is connected. f g space of polynomials For all but finite number of γ C the homotopy is regular for 0 t < 1. H(x, t) = (1 t)g(x) + γtf (x).

10 Global picture Optimal homotopy: the continuation paths are regular; the homotopy establishes a bijection between the start and target solutions. Possible singular scenarios: non-generic diverging paths multiple solutions

11 Numerical algebraic geometry Sommese, Verschelde, and Wampler, Introduction to Numerical AG (2005) Sommese and Wampler, The numerical solution of systems of polynomials (2005) Software: PHCpack (Verschelde); HOM4PS (group of T.Y.Li); Bertini (group of Sommese); NAG4M2: Numerical Algebraic Geometry for Macaulay2 (L.). and more, e.g.: Maple s ROOTFINDING[HOMOTOPY].

12 Possible improvements Parallel computation: Paths are mutually independent linear speedups. Minimize the number of diverging paths: Optimal homotopies: Total degree: Number of start solutions = product of degrees of equations (Bézout bound). Polyhedral homotopies: Number of start solutions = mixed volume of sparse system (BKK bound). Cheater s homotopy; Special homotopies: e.g., Pieri homotopy.

13 Multiple solutions In general, with probability 1, the picture looks like this: Singular end games [Morgan, Sommese, Wampler (1991)]: power-series method; Cauchy integral method; trace method. Deflation: regularizes an isolated singular solution; restores quadratic convergence of the Newton s method. How to describe a singularity?

14 Cauchy integral endgame An implication of Cauchy residue theorem: Let y : U C holomorphic on a simply connected U C, a C, and C C S 1 be a contour winding I(C, a) times around a. Then y(a) = 1 y(z) 2πiI(C, a) C z a dt, H(x, t) = 0 defines (a possibly multivalued function) x = x(t) in a neighborhood of t = 1. Idea: as the homotopy tracker approaches a singular x = x(1) use Cauchy integral to compute x staying away from x.

15 Winding numbers 1 t = ε ( )(6 7 8)(9 10)

16 Cauchy integral endgame 1. Pick a point on x = x( t), a solution to H(x, t) = 0 for t = t R; let ε = 1 t. 2. Track the path where Ĩ > 0 is such that C = {x(1 εe iθĩ) θ [0, 2π]}, x = x(1 εe iθĩ) θ {0, 2π}. 3. Let y(z) = x(1 zĩ), then y(z) is holonomic for z < ε (if ε 1). 4. Find numerically the integral x(1) = y(0) = 1 y(z) 2π z z =ε dz = 1 x(1 εe iθĩ) dθ. 2π [0,2π] (Note: one may use samples made when tracking the path C.)

17 Newton s method: x (n+1) = x (n) f(x (n)) f (x (n) ) Example 1: f(x) = x(x 1) 3, x (0) = 0.1 x (1) = x (2) = x (3) = x (4) = x (5) = x (6) = Example 2: f(x) = x 2 (x 1) 3, x (0) = 0.1 x (1) = x (2) = x (3) = x (4) = x (5) = x (6) =

18 Deflation method Let f(x) = (f ( 1 (x), )..., f N (x)), N n, f i (x) C[x] = C[x 1,..., x n ]. fi Let A(x) = C N n be the Jacobian matrix. x j Given: an approximation x (0) of an exact isolated solution x, which is singular, i.e., corank A(x ) = n rank A(x ) > 0. Newton s method in homotopy continuation loses quadratic convergence around x. Is there a way to restore the convergence? Want: a symbolic procedure that makes x regular. Rules: New variables are allowed. Assume that the numerical rank of A(x (0) ) equals A(x ).

19 Deflation step: create an augmented system in C[x, a] 1. Introduce n new variables a; 2. Add equations coming from A(x)a = 0; Example. Let f 1 = x x 1x 2 2, f 2 = x 1x x 3 2, f 3 = x 2 1x 2 + x 1x 2 2 and x = f 1 3x x 2 2 2x 1x 2 [ ] f 2 x 2 2 2x 1x 2 + 3x 2 a1 2 = 0. a f 3 2x 1x 2 + x 2 2 x x 1x 2 3. Compute the rank r of A(x ); (r = 0 for our example) 4. Add n r random linear equations. 5. Find the solution (x, a ) of the augmented system; (8 equations) 6. Repeat if (x, a ) is singular. (2 steps for the example) Theorem (L., Verschelde, Zhao) The multiplicity of (x, a ) in the augmented system is smaller than that of x in the original system.

20 Multiplicity Staircases for I = f 1, f 2, f 3, where f 1 = x x 1 x 2 2, f 2 = x 1 x x 3 2, f 3 = x 2 1x 2 + x 1 x 2 2. x ω = ( 1, 2) x x 1x ω = ( 2, 1) x 1x x x 1x x x 2 1x 2 + x 1x x 2 1x 2 + x 1x x x 1x x Multiplicity is the number of integer points under the staircase.

21 Dual space (local inverse system) For x C n, let β x : R C be a linear functional, β x(f) = ( β f)(x) = β f (x), f R. β For an ideal I, the dual space D x [I] is the subspace of Span C { β x} of the functionals that annihilate I. Filter by order: D (0) x [I] D (1) x [I] D (2) x [I]... where D (d) x [I] is the set of functionals of order at most d.

22 Macaulay array For I = f 1,..., f m, the deflation matrix A (d) I (x) is a part of the Macaulay array, the infinite matrix with entries ( ) x β (xα f i ) where α < d and β d. For example, for I = f 1, f 2 C[x, y], A (2) I = ((i,α),β) id x y x 2 x y y 2 f 1 f 2 xf 1 x (xf 2 1 ) xf 2 yf 1 yf 2

23 Dual spaces and deflation For x V (I), get D x (d) [I] by computing ker A (d) I (x). For the running example, D 0 [I] = Span{ (3,0) (2,1) (1,2) + (0,3), (2,0), (1,1), (0,2), (1,0), (0,1), (0,0) }. The leading terms with respect to ω = (2, 1) correspond to the monomials under the staircase for the standard basis for ω = ( 2, 1). x x 1x ω = ( 2, 1) x 1x x x 2 1x 2 + x 1x x 4 1 1

24 Deflation continued... Idea of proof of termination for deflation: Deflation "deflates the staircase"; the multiplicity of becomes 1 after a finite number of steps. Related work: Dual spaces: Macaulay (1916),..., Stetter (1993), Mourrain (1997), Dayton and Zeng (2005), Krone (2011). Deflation: Lyapunov (1900?),..., Ojika et al (1987), Lecerf (2002), L. et al (2006),..., Lihong Zhi et al (2010).

25 Higher-dimensional solution sets Let I = (f 1,..., f N ) be an ideal of C[x 1,..., x n ]. Goal: Understand the variety X = V(I) = {x C n f I, f(x) = 0}. A witness set for an equidimensional component Y of X: a generic slicing plane L with dim L = codim Y witness points w Y,L = Y L (generators of I)

26 Numerical irreducible decomposition Homotopy mapping w Y,L w Y,L : ( f(x) H L,L,γ(x, t) = (1 t)l(x) + γtl (x) ), t [0, 1]. L L Monodromy action: a permutation on w Y,L is produced by homotopy H L,L,γ followed by H L,L,γ for random γ, γ C. Irreducible decomposition: a partition of the witness set w Y,L stable under this action. Linear trace test: the average of the points in a witness set of an irreducible component behaves linearly.

27 Example with an embedded component Example I = (f 1, f 2 ), where f 1 = x 2 (y + 1) and f 2 = xy(y + 1). solution set of { x x(y + 1) = 0; y x(y + 1) = 0. y + 1 = 0 Numerical irreducible decomposition* sees two 1-dimensional components... x = 0 *NID reference: Sommese, Verschelde, Wampler Numerical decomposition of the solution sets... (2001)

28 Example with an embedded component Example I = (f 1, f 2 ), where f 1 = x 2 (y + 1) and f 2 = xy(y + 1). solution set of y + 1 = 0 x = 0 { x x(y + 1) = 0; y x(y + 1) = 0. Numerical irreducible decomposition* sees two 1-dimensional components... *NID reference: Sommese, Verschelde, Wampler Numerical decomposition of the solution sets... (2001)

29 Example with an embedded component Example I = (f 1, f 2 ), where f 1 = x 2 (y + 1) and f 2 = xy(y + 1). { x x(y + 1) = 0; solution set of y x(y + 1) = 0. (0, 0) Numerical irreducible decomposition* sees two 1-dimensional components... y + 1 = 0... but does not discover the embedded point. x = 0 *NID reference: Sommese, Verschelde, Wampler Numerical decomposition of the solution sets... (2001)

30 Deflation matrix, system, and ideal Example (Deflation ideal of order d = 2) original system f(x, y) deflation matrix A (2) I (x, y) x y x 2 x y y 2 f 1 f 2 xf 1 6x(y + 1) xf 2 yf 1 yf 2 2 (xf 1) x 2 Deflation ideal I (2) = f, D (2) f C[x, y, a]

31 Deflation matrix, system, and ideal Example (Deflation ideal of order d = 2) original system f(x, y) deflation matrix A (2) I (x, y) deflation system D (2) f(x, y, a) x y x 2 x y y 2 f 1 f 2 xf 1 6x(y + 1) xf 2 yf 1 yf 2 2 (xf 1) x 2 Deflation ideal I (2) = f, D (2) f C[x, y, a] a x a y a xx a xy a yy =: D (2) f(x, y, a) C[x, y, a] 6

32 Deflation matrix, system, and ideal Example (Deflation ideal of order d = 2) original system f(x, y) deflation matrix A (2) I (x, y) deflation system D (2) f(x, y, a) deflation ideal I (2) x y x 2 x y y 2 f 1 f 2 xf 1 6x(y + 1) xf 2 yf 1 yf 2 2 (xf 1) x 2 Deflation ideal I (2) = f, D (2) f C[x, y, a] a x a y a xx a xy a yy =: D (2) f(x, y, a) C[x, y, a] 6

33 Deflated variety For the given variety X = V (I) = {x f(x) = 0 for all f I} define deflated variety of order d: X (d) = V (I (d) ) C B(n,d), where B(n, d) = n 1 + ( ) n+d+1 d. Projection: π d : C B(n,d) C n, π d (x, a) x; π d X (d) = X. A component Y X is called visible at order d if Y = π d Z for an isolated component Z X (d). (0, 0) Theorem (L.) Every component is visible at some order d. Example For the running example d = 1 is sufficient.

34 Example: I = (f 1, f 2 ), f 1 = x 2 (y + 1), f 2 = xy(y + 1) Isolated components of X = V (I): V (y + 1), V (x). Additional equations A (1) I a = 0: [ ] [ ] 2x(y + 1) x 2 ax = 0. y(y + 1) x(2y + 1) Isolated components of X (1) = V (I (1) ): a y V (y + 1, a y ), V (x, y), V (x, a x ), V (x, y + 1) The first three project onto the components of X, the last one (a pseudo-component ) projects onto a singular point.

35 Witness sets For an irreducible subvariety Y X = V (I), where I R is an ideal, a witness set consists of a generic slicing plane L with dim L = codim Y witness points w = Y generators of I L

36 Witness sets For an irreducible subvariety Y X = V (I), where I R is an ideal, a generalized witness set consists of a generic slicing plane L with dim L = codim Y (d) witness points w = Y (d) L and their projections via π d generators of I (d) Definition: Y (d) is an isolated irreducible component of the deflated variety X (d) mapping onto Y under projection π d.

37 The algorithm and NPD concept Idea of the algorithm: Use numerical irreducible decomposition of a deflated variety to find (generalized) witness sets representing components. Definition: Numerical primary decomposition is a collection of such witness sets, one per component. Deficiencies: There is an apriori bound on the order d of needed to make all components visible, however it is not practical. Pseudocomponents (projections of components of X (d) that are not components of X) are hard to eliminate.

38 Local global? Local knowledge: D x [I] describes the local ring (R/I) x = R x /I x. A generic point on a component Y is a smooth point of Y that does not belong to any component not containing Y properly. A generic point x Y together with the algorithm for computing D (d) x [I] describe Y. Global knowledge: Given a NPD (as a collection of witness sets), mark one generic point on each component. The set of marked points describe the ideal I.

39 Path switching Applications in mathematics where path certification is desirable: Numerical irreducible decomposition algorithms Galois group computation based on monodromy Problems where the root count is impossible by other methods

40 Newton s method and approximate zeros Given f C[x], consider the Newton operator associated to f, N(f)(x) = x Df(x) 1 f(x), where Df(x) is the n n derivative (Jacobian) matrix of f at x C n. Definition: x C n is an approximate zero of f with associated zero η C n if N(f) l (x) η x η 2 2l 1, l 0. γ-theorem(smale): Let x C n, η f 1 (0), and x η 3 7 2γ(f, x), where γ(f, x) = sup k 2 Df(x) 1 D k f(x) k! 1 k 1. Then x is an approximate zero of f associated to η. Hauenstein, Sottile: alphacertify, software for certification of regular solutions.

41 Smale s α theory α-theorem: Let β(f, x) = x N(f)(x) = Df(x) 1 f(x). Then α(f, x) = β(f, x)γ(f, x) < certifies that x is an approximate zero of f. "robust" theorem: Let x C n with α(f, x) < If x y < 1 20γ(f, x), then y is an approximate zero associated to the same zero as x.

42 Newton s operator attraction basins

43 Certified regions

44 Robust regions

45 Linear and segment homotopy Let H (d) be the space of systems of homogeneous polynomials of fixed degrees (d) = (d 1,..., d n ) (with the Bombieri-Weyl norm). Consider f, g S = {f H (d) : f = 1} H (d). Using α-theory we design certified homotopy tracking (CHT) algorithm that tracks a linear homotopy on S (assuming BSS model of computation). The robust α-theory leads to the robust CHT algorithm (Beltrán, L.): f Take input f, g with coefficients in Q[i]. Use the segment homotopy: g t h t = (1 t)g + tf, t = [0, 1]. All computations use exact linear algebra over Q[i].

46 Future General methods (Numerical) local ring structure (Numerical) primary decomposition Real solutions, real homotopy continuation Certification Generalizations to higher order methods Certification of singular isolated solutions Certification of (irreducible/primary) decomposition Upcoming events SI(AG) 2 : SIAM activity group in algebraic geometry IMA PI summer program for graduate students on Algebraic Geometry for Applications June 18th July 6th at Georgia Tech