Choice of formulation in Cylindrical Algebraic ... - Bathdjw42/talks/DJW-CAIMS2012.pdf · Lifts back to Rn. Implemented in Qepcad B. UWO3 Algorithm: Generates a cylindrical decomposition

Background Motivation Results When to reformulate Further Work Summary

Choice of formulation in Cylindrical AlgebraicDecomposition problems

David J. WilsonUniversity of Bath

CAIMS Meeting, June 2012

David J. Wilson University of Bath Choice of formulation for CAD problems


Acknowledgements

Work conducted at the University of Bath:

Professor James H. DavenportDr. Russel J. BradfordDavid J. Wilson

Part of EPSRC project:“Real Geometry and Connectedness”

Submitted to Calculemus, CICM 2012:Preprint: http://opus.bath.ac.uk/29509/


http://opus.bath.ac.uk/29509/


Talk Outline

1 Background

2 Motivation

3 Results

4 When to reformulate

5 Further Work

6 Summary



Outline

1 BackgroundGrobner TechniquesCAD

2 Motivation

3 Results


5 Further Work

6 Summary



Grobner Bases and Reduction

A set {g1, . . . , gk} ⊆ I such that

∀f ∈ I ∃i such that LT (gi )|LT (f ).

Note: Relies on a pre-decided monomial order

Ideal-related questions are reduced to computations

Explicit algorithms: initially expensive to compute, improvedalgorithms make computation reasonably cheap

Example: Grobner Basis (w.r.t lexicographic ordering x � y):

I := 〈2 x2 y + 3 x y − 1, x y2 + y x + y〉then

G = 〈1 + 3 y + 4 y2,−4 y + 1 + 2 x〉










I := 〈2 x2 y + 3 x y − 1, x y2 + y x + y〉then

G = 〈1 + 3 y + 4 y2,−4 y + 1 + 2 x〉










I := 〈2 x2 y + 3 x y − 1, x y2 + y x + y〉then

G = 〈1 + 3 y + 4 y2,−4 y + 1 + 2 x〉










I := 〈2 x2 y + 3 x y − 1, x y2 + y x + y〉then

G = 〈1 + 3 y + 4 y2,−4 y + 1 + 2 x〉



Definition of CAD

For a set of polynomials F in n real variables, an F-invariantCylindrical Algebraic Decomposition is:

A partition D of Rn

Each cell is semi-algebraic:

{y ∈ Rn | ∀p ∈ P, p(y) = 0 and ∀q ∈ Q, q(y) > 0}.

The partition is cylindrical: for every pair of cells, theprojections onto the first k variables are equal or disjoint.Each cell is F-invariant: for every f ∈ F and every cell D ∈ D,the sign of f does not change on D.

Can’t avoid doubly-exponential complexity1.

1

[Davenport,Heintz, 1984]



Definition of CAD


A partition D of Rn


{y ∈ Rn | ∀p ∈ P, p(y) = 0 and ∀q ∈ Q, q(y) > 0}.



1




Definition of CAD


A partition D of Rn


{y ∈ Rn | ∀p ∈ P, p(y) = 0 and ∀q ∈ Q, q(y) > 0}.



1




Definition of CAD


A partition D of Rn


{y ∈ Rn | ∀p ∈ P, p(y) = 0 and ∀q ∈ Q, q(y) > 0}.

The partition is cylindrical: for every pair of cells, theprojections onto the first k variables are equal or disjoint.

Each cell is F-invariant: for every f ∈ F and every cell D ∈ D,the sign of f does not change on D.


1




Definition of CAD


A partition D of Rn


{y ∈ Rn | ∀p ∈ P, p(y) = 0 and ∀q ∈ Q, q(y) > 0}.



1




Definition of CAD


A partition D of Rn


{y ∈ Rn | ∀p ∈ P, p(y) = 0 and ∀q ∈ Q, q(y) > 0}.



1[Davenport,Heintz, 1984]David J. Wilson University of Bath Choice of formulation for CAD problems


Cylindrical Algebraic Decomposition — Algorithms

Collins’2 Algorithm:

Projects polynomials down univariate problem,Lifts back to Rn.Implemented in Qepcad B.

UWO3 Algorithm:

Generates a cylindrical decomposition of Cn,Converts this to a CAD of Rn.Implemented in Maple.

Many applications: Quantifier Elimination, Branch CutAnalysis, Robotic Motion Planning, Optimizing HybridSystems.

2[Collins,1975]3

[Chen et al, 2009]





Projects polynomials down univariate problem,

Lifts back to Rn.Implemented in Qepcad B.

UWO3 Algorithm:



2[Collins,1975]3

[Chen et al, 2009]





Projects polynomials down univariate problem,Lifts back to Rn.

Implemented in Qepcad B.

UWO3 Algorithm:



2[Collins,1975]3

[Chen et al, 2009]






UWO3 Algorithm:



2[Collins,1975]3

[Chen et al, 2009]






UWO3 Algorithm:



2[Collins,1975]3[Chen et al, 2009]






UWO3 Algorithm:

Generates a cylindrical decomposition of Cn,

Converts this to a CAD of Rn.Implemented in Maple.








UWO3 Algorithm:

Generates a cylindrical decomposition of Cn,Converts this to a CAD of Rn.

Implemented in Maple.








UWO3 Algorithm:









UWO3 Algorithm:






Outline

1 Background

2 MotivationCombining Grobner Bases and CADBuchberger-Hong ’91

3 Results


5 Further Work

6 Summary



Can we combine the two?

If we have a conjunction of equalities:

k∧i=1

(pi (x) = 0),

what happens if we take a compatible plex Grobner basis, pi

of the pi before running the CAD algorithm?

If we also have inequalities

k∧i=1

(pi (x) = 0) ∧l∧

j=1

(qj(x) > 0),

what happens if we reduce the qj with respect to the pi?



Can we combine the two?

If we have a conjunction of equalities:

k∧i=1

(pi (x) = 0),

what happens if we take a compatible plex Grobner basis, pi

of the pi before running the CAD algorithm?

If we also have inequalities

k∧i=1

(pi (x) = 0) ∧l∧

j=1

(qj(x) > 0),

what happens if we reduce the qj with respect to the pi?



Previous Work

Buchberger-Hong4 considered the case of equalities

Showed that in certain cases it can reduce computation time

Brief analysis but no further research

Could only consider the Collins algorithm as the U.W.O.algorithm had not been discovered

4[Buchberger, Hong, 1991]David J. Wilson University of Bath Choice of formulation for CAD problems


Previous Work







Previous Work







Previous Work







Outline

1 Background

2 Motivation

3 ResultsGrobner Basis and CADGrobner Normal Form


5 Further Work

6 Summary



Buchberger-Hong’s Results

We checked Buchberger-Hong’s results still held with moderncomputers and the Qepcad algorithm:

For their 5 examples (each with 2 orderings) the results weresimilarA reduction in cells and execution time for most examplesMaximum benefit: 25-fold reduction in cells, 202-foldreduction in timeIn one example pre-conditioning rendered a feasible probleminfeasible

Unlike in 1991, the calculation of Grobner bases is nowrelatively inconsequential compared to the CAD calculation





For their 5 examples (each with 2 orderings) the results weresimilar

A reduction in cells and execution time for most examplesMaximum benefit: 25-fold reduction in cells, 202-foldreduction in timeIn one example pre-conditioning rendered a feasible probleminfeasible






For their 5 examples (each with 2 orderings) the results weresimilarA reduction in cells and execution time for most examplesMaximum benefit: 25-fold reduction in cells, 202-foldreduction in time

In one example pre-conditioning rendered a feasible probleminfeasible






For their 5 examples (each with 2 orderings) the results weresimilarA reduction in cells and execution time for most examplesMaximum benefit: 25-fold reduction in cells, 202-foldreduction in timeIn one example pre-conditioning rendered a feasible probleminfeasible




Summary of New Results

Worked with three example banks 5: Buchberger-Hong, Chenet al, Intersecting Spheres & Cylinders.Assumed variable ordering was given.

Concentrated on the Maple algorithm:

22 examples of which 16 showed an improvement, 4 showed aworsening, 2 remained infeasibleMaximum improvement: 65-fold reduction in cells, 759-foldreduction in timeWorst case: 3.5-fold increase in cells, 6-fold increase in time

Improvement was split between the two stages of thealgorithm: CylindricalDecompose andMakeSemiAlgebraicPreconditioning seems to shift the dominating part of thealgorithm from MakeSemiAlgebraic toCylindricalDecompose

5available at http://opus.bath.ac.uk/29503/David J. Wilson University of Bath Choice of formulation for CAD problems






22 examples of which 16 showed an improvement, 4 showed aworsening, 2 remained infeasible

Maximum improvement: 65-fold reduction in cells, 759-foldreduction in timeWorst case: 3.5-fold increase in cells, 6-fold increase in time








22 examples of which 16 showed an improvement, 4 showed aworsening, 2 remained infeasibleMaximum improvement: 65-fold reduction in cells, 759-foldreduction in time

Worst case: 3.5-fold increase in cells, 6-fold increase in time

















Improvement was split between the two stages of thealgorithm: CylindricalDecompose andMakeSemiAlgebraic

Preconditioning seems to shift the dominating part of thealgorithm from MakeSemiAlgebraic toCylindricalDecompose












Summary of Data



Specific Example: Spheres and Cylinder

Si (x) = 0 ∧ Si+1(x) = 0 ∧ C (x) ∗ 0 ∗ ∈ {<, >,≤,≥, =, 6=}

Calculate a Grobner basis of Si and Si+1, then reduce C withrespect to this basis

Reduction always resulted in an improvementMaximum improvement from Grobner basis problem: 6-foldimprovement in cells, 14-fold improvement in timeMaximum improvement from original problem: 57-foldimprovement in cells, 419-fold improvement in time




Si (x) = 0 ∧ Si+1(x) = 0 ∧ C (x) ∗ 0 ∗ ∈ {<, >,≤,≥, =, 6=}






Si (x) = 0 ∧ Si+1(x) = 0 ∧ C (x) ∗ 0 ∗ ∈ {<, >,≤,≥, =, 6=}


Reduction always resulted in an improvement

Maximum improvement from Grobner basis problem: 6-foldimprovement in cells, 14-fold improvement in timeMaximum improvement from original problem: 57-foldimprovement in cells, 419-fold improvement in time




Si (x) = 0 ∧ Si+1(x) = 0 ∧ C (x) ∗ 0 ∗ ∈ {<, >,≤,≥, =, 6=}


Reduction always resulted in an improvementMaximum improvement from Grobner basis problem: 6-foldimprovement in cells, 14-fold improvement in time

Maximum improvement from original problem: 57-foldimprovement in cells, 419-fold improvement in time




Si (x) = 0 ∧ Si+1(x) = 0 ∧ C (x) ∗ 0 ∗ ∈ {<, >,≤,≥, =, 6=}





Outline

1 Background

2 Motivation

3 Results

4 When to reformulatePrevious metrics - td and sotdNew metric - TNoI

5 Further Work

6 Summary



How do we know when to reformulate?

Not always beneficial — rendered one feasible probleminfeasible

Important to identify when preconditioning might bedetrimental

If possible, identifying how beneficial preconditioning could bewould be useful















Will a previous metric suffice?

Dolzmann et al6 looked at various metrics to help decideoptimal variable ordering, including:

td — the sum of the total degree of each polynomialsotd — the sum of the total degree of each monomial in eachpolynomial

Would this be useful as an indicator of whetherpreconditioning would be beneficial?

Seemingly not. Although initially promising it produced astring of contradictory results to our preconditioning

6[Dolzmann et al, 2004]David J. Wilson University of Bath Choice of formulation for CAD problems























A new idea

When preconditioning we are looking to simplify (for somedefinition) the set of polynomials

Often applying the Grobner basis algorithm eliminatesvariables (but may produce extra polynomials)

Created a new metric — Total Number of Indeterminates

TNoI(A) =∑p∈A

NoI(p)

where NoI is the number of indeterminates in a polynomial.

Easy to calculate!



A new idea




TNoI(A) =∑p∈A

NoI(p)


Easy to calculate!



A new idea




TNoI(A) =∑p∈A

NoI(p)


Easy to calculate!



A new idea




TNoI(A) =∑p∈A

NoI(p)


Easy to calculate!



A new idea




TNoI(A) =∑p∈A

NoI(p)


Easy to calculate!



Correlation

When TNoI decreased: always a benefit gained frompreconditioning

When TNoI increased: preconditioning usually detrimental(one false-positive — 1.5-fold improvement)

For our (small) data set there was explicit correlation:

log(TNoI) change against log(time) change gave a correlationcoefficient r = 0.821log(TNoI) change against log(cells) change gave a correlationcoefficient r = 0.829



Correlation







Correlation







Correlation







Possible Causation

Correlation does not imply causation! Especially on arelatively small data set.

When might TNoI decrease?

1 The number of polynomials in a specific set of variables isdecreased

2 At least one variable is eliminated from a specific polynomial3 A polynomial in a large number of variables is replaced with a

few polynomials in much fewer variables

This needs to be formalised.



Possible Causation









Possible Causation









Possible Causation









Outline

1 Background

2 Motivation

3 Results


5 Further WorkGeneralised Pseudo-divisionVariable Orderings

6 Summary



Combining with Phisanbhut

In her Ph.D thesis Phisanbut7 considered applying CAD tobranch cuts

Her problems were exclusively of the form f = 0 ∧ g > 0

She applied reduction to g akin to pseudo-division allowing formultiplication by a square of a monomial to facilitatereduction by f

We may be able to adapt this form of reduction for ourpreconditioning

7[Phisanbut, 2011]David J. Wilson University of Bath Choice of formulation for CAD problems























Does ‘best’ variable ordering = ‘good’ reformulation?

Dolzmann et al8 looked at the best variable ordering for agiven problem

Does the ‘best’ variable ordering before preconditioningcorrespond to the ‘best’ variable ordering afterpreconditioning?

Early indications suggest this is not the caseWe are yet to look deeply into why












Early indications suggest this is not the case

We are yet to look deeply into why









Outline

1 Background

2 Motivation

3 Results


5 Further Work

6 Summary



Summary

Preconditioning input to a CAD algorithm by calculating aGrobner Basis can be highly beneficial

Grobner reduction of inequalities with respect to equalities hasnever, on our examples, made things worse

The metric TNoI seems a good predictor for when toprecondition

Pseudo-division reduction and variable orderings still need tobe investigated

Thank You



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Documents

Choice of formulation in Cylindrical Algebraic ... - Bathdjw42/talks/DJW-CAIMS2012.pdf · Lifts back to Rn. Implemented in Qepcad B. UWO3 Algorithm: Generates a cylindrical decomposition