Algorithmic analysis of overdetermined PDE · Algorithms for overdetermined PDE Outline...

Algorithms for overdetermined PDE

Algorithmic analysis of overdetermined PDESymmetries, separation and historical perspectives

Greg Reid

Applied Mathematics, University of Western OntarioLecture at S4 conference, University of Minnesota, September 19, 11:45 am, 2010

In honor of Willard Miller


Acknowledgements

I feel deeply fortunate to have been introduced in my PhD tothe wonderful work of Kalnins-Miller on symmetry and over-determined systems.

Like a profound work of art the viewer sees different things init, some even un-intended by the creators.

It was the beginning of my love affair with structure, symmetry,geometry and over-determined differential systems. My lifelonginterest in algorithms, implicit and approximate forms was born.


Theme - algorithmic compatibility analysis ofoverdetermined systems of PDE

Compatibility analysis is an important for integrability approachessuch as determination of super-integrability and separation ofvariables.

There have been tremendous advances in theory and algorithmsfor compatibility analysis though more is needed due to its com-plexity.


One PDE:


Prolongation once:


Prolongation twice:

Total degree does not change, Bezout # increases exponentially.


Two PDEs:


Integrability Cond.

Need prolongation & elimination to get all integrability condns.


Outline

Retrospective 70’s - 80’s

symmetry & separation of variables (algebra-geometry-analysis)Computer algebra & other dark influences (numerics)Formula for enveloping algebra characterization

Symmetry algorithms for over-determined PDE 90’s

Structure of finite-dim symmetry groups of PDEStructure of infinite Lie pseudo symmetry groups of PDE

Current directions 00’s

Numerical Algebraic Geometry in Jet SpaceSimulation Environments

Summary and Future Directions


Retrospective 70’s - 80’s

Willard MillerLie Theory and Special Functions ’68Symmetry Groups and Their Applications ’72Symmetry and Separation of Variables ’77

Ernie KalninsSeparation of Variables for Spaces of Constant Curvature ’86


Symmetry and Separation of Variables (70’s - 80’s)

Kalnins-Miller Program: classification of separable coordinatesystems for equations of Math-Physics and related features.

PDE in coordinates z i −→ ODE in separable coordinates x i .This classification problem requires a blend of techniques:

Analytic (e.g. in orthogonal sep. coords g ii = ψi1

ψ where ψij(x i )is a Stackel matrix)

Geometric: ds2 = gijdx idx j (e.g. in flat space Rijk` = 0)

Algebraic: Connection with symmetry (symmetries of Rieman-nian space put restrictions on separable coordinate systems)


My (initial) PhD problem: classification of separable systemson C5 for

∇2Ψ = E Ψ

Instead I classified separable systems on Rn for

∇2Ψ + i∂tΨ = E Ψ

The geometric and algebraic problems lead to tedious compu-tations.


Fortunately:

Symbolic language Macsyma (early 80’s)

Automated many tedious computations (e.g. production ofseparation equations and Rijk` = 0)

Motivated derivation of explicit formula for higher order sym-metries characterizing separable coordinates

led to complete classification in lower dimensions motivatingthe classification on Rn


Uses of computer algebra here

These computations are typical of classification problems

Checking hand computations and automating tedious compu-tations

Doing low dimensional calculations (provides data)

Use data to infer results in n dimensions

Classification Theorems and (human) written proofs for the ndimensional case


The n-dimensional Cases

Kalnins & Miller (1986), Separation of variables on n-dimensionalRiemannian manifolds 1. The n-sphere Sn and Euclidean n-space Rn, J. Math. Phys.

Kalnins, Miller & Reid (1984), Separation of Variables for Com-plex Riemannian Spaces of Constant Curvature I., Proc. Roy.Soc.

Kalnins (1986), Separation of Variables for Spaces of ConstantCurvature, Pitman.

Reid (1986), R-separation for heat and Schrodinger equations,SIAM J. Math. Anal.


How much progress since 80’s in computer algebramethods for this problem?

I would now use modern tools to automatically reduce the (non-linear) over-determined PDE (for gij) for finite n

For example Rijk` = 0 together with PDE equivalent condition

for a Stackel matrix: ∂jk log ψψi1 = ∂j log ψ

ψi1∂k log ψψi1 + · · ·

Current computer algebra can execute in some cases classifica-tion problems in finitely many variables using techniques suchas Grobner Bases, differential Bases, etc.

But an outstanding problem, still requiring human work, wouldbe the n dimensional analysis


Challenge Problem for current Computer Algebra

A key challenge is to develop symbolic tools and theory to ex-ecute automatic computations in n variables (n ≥ 1) arbitraryinteger for such geometric classification problems


Early Influences

Ellipsoidal separated solutions highly non-elementary −→ I hadto use numerics

Interest in general structural (algebraic, geometric) featuresgrowing


Myriad Symmetries and over-determined PDE

Theory and computation of local and nonlocal (potential) sym-metries of PDE with Bluman

Down to serious business with over-determined PDE; muchhand computation

Then started implementation of ‘advanced’ symbolic methodin symbolic language Macsyma


Historical Influences

Analytic: Riquier - Janet - Thomas

Algebraic : Ritt - Kolchin - Seidenberg

Geometric : Cartan - Kuranishi - Spencer - Pommaret

It’s important to note that many of these approaches implicitlyhad Grobner Bases in them, long before Grobner Bases had a name!


90’s Tremendous Advances in Algorithmic Analysis ofOver-determined PDE

Early packages: Schwarz 84 (spde); Wolf (crack); Champagne, Win-ternitz & Hereman (symmgrp.max).

Some highlights (restricted to those taking polynomial input, andwith implementations) −→ differentiations/eliminations −→ nice forms

Mansfield 91 - Differential Grobner Bases

Hartley and Tucker EDS

Boulier 94, BLOP 95: RosenfeldGroebner; Hubert 04

Seiler 94 & 10 - Cartan-Kuranishi Algorithm

Reid, Lisle, Wittkopf and Boulton 94 - Reduced involutive form(RIF)- analytic nonlinear PDE

Note Maple automatically calls symmetry analysis in its exact solvers(due to Edgardo Cheb Terrab). Another significant recent addition toMaple, has been Anderson’s DifferentialGeometry package.


Maple Demo – worksheet

Mention use of package automatically in Maple ...Don’t discuss theory here: but its combination of analytic (Riquier);

algebraic and Geometric (CK) ideas


Computation of the structure of finite dim. LieSymmetry Algebras of PDE

Linear Lie Algebra Defining SystemLinearizing the group transformations about the identity sym-

metry yields s linear homogeneous symmetry defining system withsolutions ζ which generate symmetry vector fields

L = ζ1(y)∂

∂y1+ · · ·+ ζk(y)

∂

∂yk

where L ∈ L a Lie Algebra. In the finite dimensional case, we havea basis

[Li ,Lj ] = ckij Lk

where the ckij are the structure constants of L


Computing structure constants without knowingsolutions ζ

Key idea Reid, Boulton, Wittkopf, Lisle 92: Our differentialelimination algorithms yield canonical forms and an associated initialdata map

ID : S −→ I Solns −→ Initial Data

Computation in the soln space S is not algorithmic, but it is algo-rithmic in the initial data space I.

Suppose solutions ζ and ρ have initial data ID(ζ) = a andID(ρ) = b. At a regular point y0 compute

[L(a),L(b)]I := ID ([Lζ ,Lρ]S)


Group (nonlinear) P −→ R ↓

Algebra (linear) R −→ rif(R) −→ [Li ,Lj ] = ckij Lk


Computation of the structure of infinite dim. Liepseudo groups of PDE (struct. trans. case)

G P −→ R ↓

L R −→ rif(R) −→ dωk = akiρπ

ρ ∧ ωi − 12ck

ij ωj ∧ ωj


Computation of the structure of infinite dim. Liepseudo groups of PDE (struct. intrans. case)

G P −→ R model −→ dωk = akiρπ

ρ ∧ ωi − 12ck

ij ωj ∧ ωj

↓ ↑L R −→ rif(R)

Also see Peter Olver & Juha for a fully infinitesimal methodfor the intransitive case. See Kamran & Hsu for early breakthroughwork on Cartan’s method.


New methods/theory algorithms needed

Computational evidence (e.g. geometric classifications in in ar-bitrary (n) dimensions) indicate that these pessimistic estimatesare not obtained for some nice subclasses of problems

Natural strategies based on such evidence is to include moresymmetry and structure of the problem in the methods (e.g.Geometric Integration, Olver et al, ... )

Gain inspiration from polynomial subclass


PDE in Jet Space

Consider q-th order PDE system R = (R1, ...,Rk) = 0 with in-dep vars x = (x1, x2, ..., xn) and dep vars u(x) = (u1(x), u2(x), ..., um(x)).

Denoting ur

as the formal (jet) variables corresponding to r -th

order partial derivatives of u(x) the jet variety is

V (R) := (x , u, u1, ..., u

q) ∈ Jq : R(x , u, u

1, ..., u

q) = 0

Here Rk : Jq → C, Jq = CNq and Nq = # jet variables ≤ q.


Basic operations of Jet Geometry:

The formal total derivative is: Dxj = ∂∂xj

+∑

l ulxj

∂∂ul + ...

a single symbolic prolongation of the system is:

D(R) := (x , u, u1, ..., u

q+1) ∈ Jq+1 : Rk = 0, Dxi R

k = 0.

A single geometric projection is:

π(R) := (x , u, u1, ..., u

q−1) ∈ Jq−1 : Rk(x , u, u

1, ..., u

q−1, uq

) = 0.

Note D and π are coordinate-independent.


Numerical Algebraic Geometry

Is a new area introduced by Sommese and Wampler

Has favorable complexity properties

Represents components by points

Hybrid symbolic-numeric method of Reid, Verschelde, Wittkopf& Wu 05 [1]

Method for PDE for fully approximate systems Wu and Reid06.


Background on Polynomial Decomposition

Consider a system of polynomials p(v) = (p1(v), p2(v), ..., pk(v))in the ring C[v ] with v = (v1, ..., vN) in CN :

V (p) := v ∈ CN : p(v) = 0decompose by dimension

= ZN−1 ∪ ... ∪ Z1 ∪ Z0

decompose into irreducible components

=⋃j∈Ij

Zi ,j


Example of Polynomial Decomposition

p =

[(w − v2)(u2 + v2 + w2 − 1)

u(u − v3)(u2 + v2 + w2 − 1)

]= 0

The decomposition is:

V (p)



p =

[(w − v2)(u2 + v2 + w2 − 1)

u(u − v3)(u2 + v2 + w2 − 1)

]= 0


V (p) = Z2



p =

[(w − v2)(u2 + v2 + w2 − 1)

u(u − v3)(u2 + v2 + w2 − 1)

]= 0


V (p) = Z2 ∪ Z1



p =

[(w − v2)(u2 + v2 + w2 − 1)

u(u − v3)(u2 + v2 + w2 − 1)

]= 0


V (p) = Z2 ∪ Z11 ∪ Z12


Roots of start system q(v) at τ = 0:


Homotopy paths from τ = 0 to τ = 1:


Complexity Issues

For bounds on the number of steps in a linear homotopy see

Blum, Shub, Cucker & Smale 98. Complexity and Real Com-putation, Springer.

Shub & Smale 94. Complexity of Bezout’s Theorem V. Poly-nomial Time. Theo. Comp. Sci.

On average an approximate root of a generic polynomial systemcan be found in polynomial time.

Also see Ilie 05, Ilie, Corless & Reid 06 for poly. cost complexitymodel of differential equation solving.


Witness Points

Defn: Each k-dimensional component of a variety is repre-sented by a

witness set ≡ k random hyperplanes ∪d witness points

whered witness points ≡ component ∩ random hyperplanes

and d is the degree of the component.


3 dimensional example of witness setsRecall the example

p =

[(w − v2)(u2 + v2 + w2 − 1)

u(u − v3)(u2 + v2 + w2 − 1)

]= 0

The variety and list of witness sets are now displayed


Fundamental Problem

Find numerically stable methods to identify all hidden con-straints without prolongation explosion and complicated differentialelimination?

Our approaches to this problem:

stable methods for general systems

fast methods for certain generic systems

Special in our approach - geometry.


Key Ideas

We identify a certain class of pde (called square t-dominatedsystems):

Only prolongations w.r.t one variable are needed.

No elimination is needed.

Connection to Riquier Basis −→ algebraic interpretation.

Genericity– any system is t-dominated after a random changeof coordinates.


5. Ranking

We introduce ranking here only for theory and algebraic inter-pretation. In computation we use implicit form without elimination,so it is stable.

Definition (Ranking see (Rust 98) - 1 indep variable case)

A positive ranking ≺ of the set of all derivatives Ω is a totalordering on Ω:

U ≺ V ⇒ DtU ≺ DtV (1)

U ≺ DtU (2)


6. Signature Matrix of t-Dominated Systems

Consider the simple case of two independent vars (t, x). Foreach uj , we choose a (partial) ranking which satisfies:

uj ≺ ujx ≺ uj

xx ≺ · · · ≺ ujt ≺ uj

tx ≺ · · · (3)

Determine the leading derivative ld(Ri , uj) of each equation Ri

w.r.t. each uj using the ranking ≺ given in (3).


6. Signature Matrix of t-Dominated Systems

For a partial derivative w ∈ Ω (the set of all partial derivatives)define |w | := #t derivs in w (i.e. the differential order wrt t).

For sufficiently small ε > 0 define ϕ on Ω by:

ϕ(w) :=

|w |, if w is not a mixed derivative|w |+ ε, if w is a mixed derivative

(4)

For example ϕ(uttt) = 3, ϕ(uxtt) = 2 + ε.


Define the signature matrix of R is defined by

(σi ,j)(R) :=

ϕ(ld(Ri , u

j)), if Ri depends on uj ;−∞, otherwise .

(5)

See Pryce 01 for the ode case.

Definition (t-dominated system)

We say R is dominated by pure derivatives in the independentvariable t if there is no ε appearing in (σi ,j)(R).

Our method for determining missing constraints is applicableto such t dominated systems. How common are they?


7. Generalizing Pryce’s Method to PDE

Let R be a square t-dominated system. Consider R as ode(the only independent variable is t). Suppose Ri needs to be dif-ferentiated ci times (ci ≥ 0 ) to find hidden local constraints. Thenew system after differentiation is denoted by Dc

t R.

Suppose the highest order of uj appearing in Dct R is dj .

From the definition of (σi ,j), clearly dj is the largest of ci +σij ,which implies that dj − ci ≥ σij , for all i , j .

There are∑

dj + m jet variables and∑

ci + m equations inDc

t R (only count pure t-derivatives). If each equation drops thedimension of the zero set by one, then the dimension of Dc

t R is∑dj −

∑ci . To find all the constraints means to minimize the

dimension of Dct R (Geometric Interpretation).


8. Formulation of Linear Programming Problem

This can be formulated as an integer linear programming problem(LLP) in the variables c = (c1, ..., cm) and d = (d1, ..., dm): Minimize z =

∑dj −

∑ci ,

where dj − ci ≥ σij ,ci ≥ 0

(6)

The computation of c and d which only involves the information on dif-ferential order is consequently very fast. This problem is dual to the as-signment problem Pryce 01.

Eric Schost pointed out that assignment problems can be done inO(m3) by using the Hungarian Method (Harold W. Kuhn, 1955).


9. Jacobian Criterion for Termination

Assume c1 ≥ c2 ≥ · · · ≥ cm, and let kc = c1. Then we can partitionDc

t R into kc + 1 parts,

B0 B1 · · · Bkc−1 Bkc

R(0)1 R

(1)1 · · · R

(c1−1)1 R

(c1)1

R(0)2 · · · R

(c2−1)2 R

(c2)2

......

...

R(0)m · · · R

(cm)m

Table: The triangular block structure of Dct R, Bi+1 involves more jet

variables than Bi .

For each Bi , 0 ≤ i ≤ kc , let Ui := lcd(Bi ) and define the JacobianMatrix

Jaci :=

(∂Bi

∂Ui

). (7)


9. Jacobian Criterion for Termination

To apply the Riquier Existence Theorem, we need to refine the partialranking (3) to a positive ranking.

Proposition

Let lcd(R) = θ1, ..., θm and let B be the set of all the otherderivatives of R. Then there exists a positive ranking ≺ which satisfiesthe partial ranking (3) and θ1 ≺ θ2 ≺ · · · ≺ θm and each θi is greaterthan any b ∈ B.

By Implicit Function Theorem and properties of analytic functions,we can show the Analytic Interpretation:

Theorem (Jacobian Criterion)

Let R be a square analytic t-dominated system of pde and Dct R be

the system of t-prolongation by solving LPP (6). If Jackc is nonsingularat some point p in Z (Dc

t R), then Dct R is an implicit Riquier Basis at p

with a ranking given by Proposition 1.


But method fast only for some systems.Does it work often enough to be useful in practise?Try systems generated by simulation environments (the MapleSim

environment for us, see Niloo Mani’s Thesis).


Simplification

Systems from MapleSim models look like:

X13(t) = 0,X17(t) = 0,X2(t)− X10(t) = 0, . . .

X1(t)−d

dtX3(t) = 0,X4−

d

dtX6(t) = 0, . . .

C1X34(t)) + C2 cos(X25(t))− X22(t) = 0, . . .

They contain many 1-term and 2-term eqns of form u = 0 andu − v = 0 or w − dz

dt = 0

Classical symbolic simplification of the 2-term eqns w.r.t rank-ing so that u ≺ v , w ≺ dz

dt .

Execute this symbolic simplification using Maple.


Result:

MapleSim Model Before simplifn. After simplifn.#Eqns×#Vars #Eqns×#Vars

2D Slider Crank 23×23 4×43D Slider Crank 24×24 7×7

Rotating Pendulum 26×26 2×2Gimbal 27×27 3×3

RLC-circuit 32×32 1×1Furuta Pendulum 34×34 2×2

Triplets 39×39 3×3Heating transfer system 41×41 1×1

DC-motor 57×57 1×1Non-linear Spring Damper 57×57 2×2

Five-Pendulums 65×65 5×5One-loop-Pendulum 69×69 8×8

Slider crank+DC-motor 72×72 5×5Centrifuge 100×94 3×3

Two-loops-Pendulum 102×102 14×143-D Pendulum 117×96 3×3

Ice Tank 186×186 5×5

Table: Simplification Result for various MapleSim Models.


Result:

MapleSim Model Before simplifn After simplifn.#Eqns×#Vars #Eqns×#Vars Time (s)

2D Slider Crank 23×23 4×4 0.0163D Slider Crank 24×24 7×7 0.016

Rotating Pendulum 26×26 2×2 0.015Gimbal 27×27 3×3 0.015

RLC-circuit 32×32 1×1 0.016Furuta Pendulum 34×34 2×2 0.031

Triplets 39×39 3×3 0.001Heating transfer system 41×41 1×1 0.016

DC-motor 57×57 1×1 0.015Non-linear Spring Damper 57×57 2×2 0.001

Five-Pendulums 65×65 5×5 0.031One-loop-Pendulum 69×69 8×8 0.046

Slider crank+DC-motor 72×72 5×5 0.001Centrifuge 100×94 3×3 0.016

Two-loops-Pendulum 102×102 14×14 0.0623-D Pendulum 117×96 3×3 0.006

Ice Tank 186×186 5×5 0.016

Table: Simplification Result for various MapleSim Models.


Summary - future work

Compatibility analysis is an important for integrability approachessuch as determination of super-integrability and separation ofvariables.

There have been tremendous advances in theory/algoirithms/symbolicsoftware for compatibility analysis though more is needed dueto its complexity.

We briefly described the new are of numerical algebraic geom-etry which is a radical change in approach to exact problems.Expect further blurring of the boundaries between exact andnumerical methods.

Leads to a new area - Numerical Jet Geometry.


Acknowledgements

I feel deeply fortunate to have been introduced in my PhD tothe wonderful work of Kalnins-Miller on symmetry and over-determined systems.

Like a profound work of art the viewer sees different things init, some even un-intended by the creators.

It was the beginning of my love affair with structure, symmetry,geometry and over-determined differential systems. My lifelonginterest in algorithms, implicit and approximate forms was born.


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