From Isolated Points to Positive Dimensions & Back Again:
This presentation is the property of its rightful owner.
Sponsored Links
1 / 34

Charles Wampler General Motors R&D Center (Adjunct, Univ. Notre Dame) Including joint work with PowerPoint PPT Presentation


  • 100 Views
  • Uploaded on
  • Presentation posted in: General

From Isolated Points to Positive Dimensions & Back Again: A Brief History of Numerical Algebraic Geometry. Charles Wampler General Motors R&D Center (Adjunct, Univ. Notre Dame) Including joint work with Andrew Sommese, University of Notre Dame Jon Hauenstein, University of Notre Dame.

Download Presentation

Charles Wampler General Motors R&D Center (Adjunct, Univ. Notre Dame) Including joint work with

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Charles wampler general motors r d center adjunct univ notre dame including joint work with

From Isolated Points to Positive Dimensions & Back Again:A Brief History of Numerical Algebraic Geometry

Charles Wampler

General Motors R&D Center

(Adjunct, Univ. Notre Dame)

Including joint work with

Andrew Sommese, University of Notre Dame

Jon Hauenstein, University of Notre Dame


Charles wampler general motors r d center adjunct univ notre dame including joint work with

42 distinct co-authors


Outline

Outline

  • Introduction to continuation

  • Historical timeline

    • Algorithms for zero-dimensional solution sets

    • Algorithms for positive-dimensional solution sets

      • Creation of numerical algebraic geometry

        • Built on methods for zero-dimensional solving

      • Numerical irreducible decomposition

      • Intersecting algebraic sets

        • Diagonal homotopy

        • New: Regeneration method

    • Extensions of numerical algebraic geometry

      • Extracting real points

      • Computing the genus of a curve

      • Exceptional sets via fiber products

  • Using positive-dimensional methods to find isolated solution sets

    • Equation-by-equation solving using regeneration


Introduction to continuation

S0

S1

t=0

t

t=1

Introduction to Continuation

  • Basic idea: to solve F(x)=0 (N equations, N unknowns)

    • Define a homotopy H(x,t)=0 such that

      • H(x,1) = 0 has a known isolated solutions, S1

      • H(x,0) = F(x)

    • Track solution paths as t goes from 1 to 0

      • Paths satisfy the Davidenko o.d.e.

        • (dH/dx)(dx/dt) + dH/dt = 0

      • Endpoints of the paths are solutions of F(x)=0

      • Let S0 be the set of path endpoints

  • Concerns

    • Points where dH/dx is singular

      • Turning points

      • Path crossings

      • Multiple roots

    • Path divergence

    • Will S0 contain all isolated solutions of F(x)=0?

    • What happens if V(F) has positive dimensional components?

    • What if we have N equations, n unknowns, N≠n?

    • Numerical accuracy, Computational efficiency


Basic total degree homotopy

Basic Total-degree Homotopy

To find all isolated solutions to the polynomial system F: CN CN, i.e.,

form the linear homotopy

H(x,t) = (1-t)F(x) + tG(x)=0,

where


Brief timeline part 1 before sommese

Brief Timeline: Part 1 (Before Sommese)

  • Pre-computer age

    • Homotopy & continuity for existence proofs

  • 1960s: early computational methods

    • Real number field, heuristic methods

      • “Bootstrap method” in kinematics – Roth 1963

  • 1970s, early 80s: Algorithmic methods

    • Complex number field, Geometric arguments

      • Fixed-point continuation (T.Y.Li 1976)

      • Specialization to polynomial systems

        • All isolated solutions, 1977 (Garcia & Zangwill; Drexler)

        • Total degree homotopy, 1978 (Chow, Mallet-Paret & Yorke)

      • Numerical path tracking (Allgower & Georg, book ’80)

      • Use of projective space (Wright, 1985)

      • Application to 6R robot kinematics (Tsai & Morgan, 1985)

      • Morgan’s book on polynomial continuation 1987


Addressing the concerns

S0

S1

t=0

t

t=1

Addressing the concerns

Concerns

  • Points where dH/dx is singular

    • Turning points

    • Path crossings

    • Multiple roots

  • Path divergence

  • Will S0 contain all isolated solutions of F(x)=0?

  • Computational efficiency

  • Numerical accuracy

  • What happens if V(F) has positive dimensional components?

  • What if we have N equations, n unknowns, N≠n?

complex homotopy 1977

projective space 1985

1977

Total degree 1978


Kinematic milestone

6R Serial-Link Inverse Kinematics

Formulation

Pieper, 1968 & earlier

32nd degree polynomial

Duffy & Crane, 1980

16 solutions shown by continuation

Tsai & Morgan, 1985

Total degree homotopy with 256 paths

16th degree polynomial

Li & Liang, 1988

Kinematic Milestone


Brief timeline part 2 after sommese

Brief Timeline: Part 2 (After Sommese)

  • Late 80s, early 90s: Use of algebraic geometry

    • Still finding all isolated solutions

    • Focus on reducing the number of paths

      • Homotopies adapted to the structure of the target system

      • Multihomogeneity & the “γ trick” (Morgan & Sommese, 1987)

      • Parameterized systems: f(x;p)=0

        • Cheater’s homotopy (Li, Sauer & Yorke, 1988)

        • Coefficient parameter homotopy (Morgan & Sommese, 1989)

      • Polytope homotopies (a.k.a., polyhedral, mixed volume, or BKK)

        • Verschelde, Verlinden & Cools, ’94; Huber & Sturmfels, ’95

        • T.Y. Li with various co-authors, 1997-present

    • Computing singular endpoints

      • Morgan, Sommese, & Wampler (1991,1992)

    • Kinematics applications

      • Tutorial for mechanical engineers (MSW 1990)

      • Complete solution of the 9-point four-bar, (MSW 1992)


The trick

The “γ trick”

  • Simple but useful consequence of algebraicity

  • Instead of the homotopy

    H(x,t) = (1-t)f(x) + tg(x)

    use

    H(x,τ) = (1-τ)f(x) + gτg(x)


Parameter continuation

Parameter Continuation

initial parameter space

target parameter space

Start system easy in initial parameter space

Root count may be much lower in target parameter space

Initial run is 1-time investment for cheaper target runs


Kinematic milestone1

9-Point Path Generation for Four-bars

Problem statement

Alt, 1923

Bootstrap partial solution

Roth, 1962

Complete solution

Wampler, Morgan & Sommese, 1992

m-homogeneous continuation

1442 Robert cognate triples

Kinematic Milestone


Nine point four bar summary

Nine-point Four-bar summary

  • Symbolic reduction

    • Initial total degree≈1010

    • Roth & Freudenstein, tot.deg.=5,764,801

    • Our reformulation, tot.deg.=1,048,576

    • Multihomogenization 286,720

    • 2-way symmetry 143,360

  • Numerical reduction (Parameter continuation)

    • Nondegenerate solutions4326

    • Roberts cognate symmetry1442

  • Synthesis program follows 1442 paths


Parameter continuation 9 point problem

Parameter Continuation: 9-point problem

2-homogeneous systems with symmetry:

143,360 solution pairs

9-point problems*:

1442 groups of 2x6 solutions

*Parameter space of 9-point problems is 18 dimensional (complex)


Addressing the concerns1

S0

S1

t=0

t

t=1

Addressing the concerns

  • Points where dH/dx is singular

    • Turning points

    • Path crossings

    • Multiple roots: Singular endgames 1991-92

  • Path divergence

  • Will S0 contain all isolated solutions of F(x)=0?

  • Computational efficiency

    • Ab initio:

      • Multihomogeneous 1987

      • Polytopes 1994-present

    • Parameter homotopy 1989

  • Numerical accuracy

  • What happens if V(F) has positive dimensional components?

  • What if we have N equations, n unknowns, N≠n?


Brief timeline part 3 numerical alg geom

Sommese & Wampler

(FoCM 1995 , publ. 1996)

Brief Timeline: Part 3 (Numerical Alg.Geom.)

  • “Numerical Algebraic Geometry” founded

    • Treatment of positive dimensional sets by slicing

    • Treatment of N≠n by randomization

  • Cascade algorithm for slicing

    • Sommese & Verschelde, 2000

  • Numerical irreducible decomposition

    • Factoring by sampling & fitting (Sommese, Verschelde, & Wampler, 2001)

      • Coinage of “witness set”

    • Use of monodromy (SVW 2001 )

    • Use of symmetric functions (SVW 2002)

  • Intersection of components

    • Diagonal homotopy (SVW 2004)

  • Book by Sommese & Wampler, 2005

  • Local methods

    • Deflation of μ>1 isolated points (Leykin, Verschelde, & Zhao, 2006)

    • Multiplicity structure (Dayton & Zeng 2005, Bates, Sommese, Peterson 2006)

    • Deflation of nonreduced positive dimensional components (S&W 2005)

  • Curves

    • Real curves (Lu, Bates, S&W, 2007), Curve genus (Bates, Peterson, S&W, preprint)

  • Equation-by-equation solving

    • Using diagonal homotopy (SVW 2008)

    • Using regeneration (Hauenstein, S, & W, in preparation)

Up to

positive

dimensions

And back again!


Slicing witness sets

Slicing & Witness Sets

  • Slicing theorem

    • Andegree N reduced algebraic set of dimension m in n variables hits a general (n-m)-dimensional linear space in N isolated points

  • Witness generation

    • Slice at every dimension

    • Use continuation to get sets that contain all isolated solutions at each dimension

      • “Witness supersets”

  • Irreducible decomposition

    • Remove “junk”

    • Monodromy on slices finds irreducible components

    • Linear traces verify completeness


Membership test

Membership Test


Linear traces

Linear Traces

Track witness paths as slice translates parallel to itself.

Centroid of witness points for an algebraic set must move on a line.


Further reading

Further Reading

World Scientific 2005


Example griffis duffy platform

Example: Griffis-Duffy Platform

Special Stewart-Gough platform

Degree 28 motion curve (in Study coordinates)

  • if legs are equal & plates congruent:

    • factors as 6+(6+6+6)+4

Studied by:

Husty & Karger, 2000


Real points on a complex curve

Real Points on a Complex Curve

  • Go to Griffis-Duffy movie…


Addressing the concerns2

S0

S1

t=0

t

t=1

Addressing the concerns

  • Points where dH/dx is singular

    • Turning points

    • Path crossings

    • Multiple roots

  • Path divergence

  • Will S0 contain all isolated solutions of F(x)=0?

  • What happens if V(F) has positive dimensional components?

  • What if we have N equations, n unknowns, N≠n?

  • Computational efficiency

    • Isolated points

    • Positive dimensional components

      • Cascade goes like total degree

    • Working equation-by-equation helps! (esp., regeneration)

  • Numerical accuracy – a talk for another day

Using traditional homotopies


Equation by equation solving

Intersect

Intersect

Intersect

  • Special case:

    • N=n

    • nonsingular solutions only

    • results are very promising

Equation-by-Equation Solving

N equations, n variables

f1(x)=0  Co-dim 1

f2(x)=0  Co-dim 1

Co-dim 1,2

f3(x)=0  Co-dim 1

Co-dim 1,2,3

Theory is in place for μ>1 isolated and for full witness set generation.

Similar intersections

Final Result

Co-dim 1,2,...,N-1

Co-dim 1,2,...,min(n,N)

fN(x)=0  Co-dim 1


Working equation by equation

Working Equation-by-Equation

  • Basic step


Regeneration step 1

Regeneration: Step 1

Union of sets

move slice dk times


Regeneration step 2

Regeneration: Step 2

Linear homotopy


Software for polynomial continuation

Software for polynomial continuation

  • PHC (first release 1997)

    • J. Verschelde

    • First publicly available implementation of polytope method

    • Used in SVW series of papers

    • Isolated points

      • Multihomogeneous & polytope method

    • Positive dimensional sets

      • Basics, diagonal homotopy

  • Hom4PS-2.0, Hom4PS-2.0para (recent release)

    • T.Y. Li

    • Isolated points:

      • Multihomogeneous & polytope method

  • Bertini (ver1.0 released Apr.20, 2008)

    • D. Bates, J. Hauenstein, A. Sommese, C. Wampler

    • Isolated points

      • Multihomogeneous

    • Positive dimensional sets

      • Basics, diagonal homotopy, & regeneration


Test run 1 6r robot inverse kinematics

Test Run 1: 6R Robot Inverse Kinematics

*All runs in Bertini


Test run 2 9 point four bar problem

Test Run 2: 9-point Four-bar Problem

1442 Roberts cognates


Test run 3 lotka volterra systems

Test Run 3: Lotka-Volterra Systems

  • Discretized (finite differences) population model

    • Order n system has 8n sparse bilinear equations

Work Summary

+ mixed volume

Total degree = 28n

Mixed volume = 24n is exact


Lotka volterra systems cont

Lotka-Volterra Systems (cont.)

  • Time Summary

xx = did not finish

All runs on a single processor


Algebraic vs geometric approaches

Algebraic vs. Geometric Approaches


Summary

Summary

  • Polynomials arise in applications

    • Especially kinematics

  • Continuation methods for isolated solutions

    • Highly developed in 1980’s, 1990’s

  • Numerical algebraic geometry

    • Builds on the methods for isolated roots

    • Treats positive-dimensional sets

    • Witness sets (slices) are the key construct

  • Regeneration: equation-by-equation approach

    • Uses slice moves to regenerate each new equation

    • Captures much of the same structure as polytope methods, without a mixed volume computation

    • Most efficient method yet for large, sparse systems

  • Thanks, Andrew, for 2 decades of fun collaboration!


  • Login