Approximate symmetries of di erential equationscheviakov/bluman2014/talks/Reid.pdfThus care has to be taken with symbolic approaches ... approximate basis is obtained for the kernel

Introduction & Historical Comments Symmetries and Introductory Examples Prolongation and Projection References

Approximate symmetries of differential equations

Tuesday May 13, 2:00 - 2:45 pm

Greg Reid 1 Ian Lisle and Tracy Huang 2

Symmetry Methods, Applications and Related Fieldsa conference in honour of George Bluman

University of British Columbia

1Ontario Research Centre for Computer Algebra, Western University, Canada.

2University of Canberra


Table of contents

Introduction & Historical Comments

Symmetries and Introductory ExamplesDirect use of floating point numbers in exact differential elimina-

tion methodsDifficulties with symbolic approaches

Prolongation and Projectionyxx + 7.1 yyx + 5 y3 = 0Symbolic-numeric algorithm for structure constants

References


• Considerable progress in the theory and computer implementa-tion of symbolic computation algorithms to automatically de-termine and exploit exact symmetries of exact DE.

• Underlying such algorithms are differential elimination algo-rithms (e.g. RifSimp and diffalg).

• Often DE describing a model are only approximately known– they may contain parameters that are only known approxi-mately. Symbolic methods are unstable.

• In earlier work described a stable method for determining thesize of symmetry groups of approximate DE.

• Today we extend this numerical method to determining struc-ture of the Lie algebra.



























George Bluman


Symmetry defining system

The (exact) symmetries of a differential equation are usuallynot known a priori and have to be determined.

The Infinitesimal Lie symmetries of a DE ∆ = 0 are the lin-earized form of such symmetries about the identity transformation:

x = x + εξ(x , y) +O(ε2)

y = y + εη(x , y) +O(ε2)

(1)

that leave the ∆ invariant:

pr(L)(∆)|∆=0 = 0 (2)

and modulo certain regularity conditions result in a linear homoge-neous system of over-determined PDE for the functions ξ, η and η[5, 19], which are called the symmetry defining system.


Consider the much studied class of ODE

yxx = g(x , y , yx) (3)

The Symmetry Defining System for (3) is:

ξyx = 0, ηyx = 0,

−gyη + g (−3 yx ξy − 2 ξx)− gxξ (4)

+gyx(−yx ηy + yx

2ξy + yx ξx − ηx)

+ηx ,x + yx2ηy ,y − yx

3ξy ,y − 2yx2 ξx ,y + 2yx ηx ,y − yx ξx ,x = 0


In particular for our illustrative example we consider:

yxx + 6.708203932 yyx + 5 y3 = 0 (5)

The symmetry defining system of (5) is:

ξyy = 0,

ηyy − 2 ξxy + 13.41640787 yξy = 0, (6)

2 ηxy − ξxx + 15 y3ξy + 6.708203928 yξx + 6.708203932 η = 0,

ηxx − 5 y3ηy + 10 y3ξx + 6.708203932 yηx + 15 y2η = 0


Apply the differential-elimination packages RifSimp and DifferentialAlgebra

for yxx + 6.708203932 yyx + 5 y3 = 0.The defining system:

ξx = 0.0, ξy = 0.0, η = 0.0 (7)

which represents a 1 parameter translation symmetry in x for (5).

Convert the defining system to rationals:

ξx = 0, ξy = 0, η = 0, (8)

Convert the ODE to rationals, then create the defining system:

ξxx = 0, ξy = 0, η = −yξx . (9)

Thus the symbolic simplification methods are not continuouswith respect to small changes in the coefficients.

Ideas, anyone?




ξx = 0.0, ξy = 0.0, η = 0.0 (7)

which represents a 1 parameter translation symmetry in x for (5).Convert the defining system to rationals:

ξx = 0, ξy = 0, η = 0, (8)


ξxx = 0, ξy = 0, η = −yξx . (9)


Ideas, anyone?




ξx = 0.0, ξy = 0.0, η = 0.0 (7)

which represents a 1 parameter translation symmetry in x for (5).Convert the defining system to rationals:

ξx = 0, ξy = 0, η = 0, (8)


ξxx = 0, ξy = 0, η = −yξx . (9)


Ideas, anyone?


Symbolic substitution strategyFor example embed the given ode in the one parameter class

of ode:yxx + αyyx + 5y3 = 0 (10)

Application of RifSimp and diffalg yields:Case 1 (α2 6= 45, dimG = 2):

ξxx = 0, ξy = 0, η = −yξx (11)

Case 2 (α2 = 45, dimG = 8):

η =2

15ξxxxy + 2 y2ξxy − yξx −

2

15y αξxxy +

1

15αξxx −

1

3αξyy

3,

ξxxxx =30 y2

αξxxxy −

30 y

αξxxx ,

ξyy = 0. (12)

Indeed second order ode are linearizable [20] if and only if they havean eight dimensional group. Hence the ode (5) is very close to alinearizable ode.


Difficulties with symbolic approaches

Thus care has to be taken with symbolic approaches

• Produce unstable results when used with numeric coefficients.

• Round-off errors can mean that the rational approximation doesnot inherit desired properties of the original system.

• Symbolic replacement, although powerful in certain situations,can be impractical due to their greater complexity.

• It’s useful to integrate symbolic and numeric methods, and suchmethods need to consider close-by systems.

The goal of the talk is to stably seek close-by systems admittinglarge symmetry groups in the presence of round off errors from theoriginal system.


Prolongation

A q-th order system R = {R1 = 0, ...,Rs = 0} where its j-thequation has order dj ≤ q has prolongation to order q + k :

Dk(R) = {∂KRj : K ∈ Nn with |K |+ dj ≤ (q + k)} (13)

So the k-th prolongation consists of the set of all possible partialderivatives of the equations of R of order ≤ (q + k).

Symmetry defining systems of form of (differential) order q areconsidered as a system R in matrix form:

A(q)(z)v (q) = 0 (14)

where v (q) is a column vector of all partial derivatives of infinitesi-mals of order ≤ q.


Prolongations DR, D2R, . . . yield the sequence of matrix systems:

A(q)(z)v (q) = 0, A(q+1)(z)v (q+1) = 0, . . . (15)

Then substitute a random point z = z0 in the space of independentvariables.This leads to a sequence of constant matrix systems:

A(q)(z0)v (q) = 0, A(q+1)(z0)v (q+1) = 0, . . . (16)

Note that v (q) has N(n, q,m) = m(q+n

q

)coordinates and kerA(q)(z0)

is a subspace of Jq ≈ RN(n,q,m).


Projection

Define projection π by on v (q) ∈ Jq by πv (q) ∈ Jq−1 whereπv (q) is obtained by deleting coordinates in v (q) of order q.

Successive projections are defined by iteration: π2v (q) = ππv (q)

by deleting coordinates of order q and q − 1, etc. Define

π`R := {π`v (q) ∈ Jq−k : A(q)(z0)v (q) = 0} (17)

The main step of our geometric involutive form algorithm computes

π`DkR (18)

Symbolically: just compute symbolic prolongations and after subsi-tuting z = z0 use symbolic Gauss elimination to obtain the projec-tions. But this is not numerically stable.


Numeric computation of π`DkR

Apply the SVD to the matrix of each prolongation. Then anapproximate basis is obtained for the kernel of the matrix by settingsingular values to zero below a user input tolerance.

To numerically compute projections coordinates correspondingto higher order derivatives are deleted in the basis, and this yields aspanning set for the projection as described in [9]. These spanningsets can be converted to a basis by another SVD calculation.

These systems are tested for the property of projective involu-tivity which as shown in [9] is equivalent to Cartan involutivity.

In summary in the finite d dimensional case we get d approxi-mate basis vectors for a projective involutive system. These are usedin the numerical computation of the structure constants outlined inSection 2.


yxx + 6.708203932 yyx + 5 y3 = 0The symmetry defining system of our illustrative ODE above in

matrix form with z = (x , y) is:1 0 0 0 0 0 0 0 0 0 0 0

0 1 −2 0 0 0 13.416y 0 0 0 0 0

0 0 0 0 0 1 0 −5.0y3 10.0y3 6.708y 0 15.0y2

0 0 0 2 −1 1 15.0y3 0 6.708y 0 0 6.708

v (2)=0

where 0 = [0, 0, 0, 0]T and

v (2) = [ξyy , ηyy , ξxy , ηxy , ξxx , ηxx , ξy , ηy , ξx , ηx , ξ, η]T

Table 1: dim π`DkR for (5)

k = 0 k = 1 k = 2` = 0 8 8 8` = 1 6 8 8` = 2 6 8` = 3 6




0 1 −2 0 0 0 13.416y 0 0 0 0 0

0 0 0 0 0 1 0 −5.0y3 10.0y3 6.708y 0 15.0y2

0 0 0 2 −1 1 15.0y3 0 6.708y 0 0 6.708

v (2)=0

where 0 = [0, 0, 0, 0]T and



k = 0 k = 1 k = 2` = 0 8 8 8` = 1 6 8 8` = 2 6 8` = 3 6




0 1 −2 0 0 0 13.416y 0 0 0 0 0

0 0 0 0 0 1 0 −5.0y3 10.0y3 6.708y 0 15.0y2

0 0 0 2 −1 1 15.0y3 0 6.708y 0 0 6.708

v (2)=0

where 0 = [0, 0, 0, 0]T and



k = 0 k = 1 k = 2` = 0 8 8 8` = 1 6 8 8` = 2 6 8` = 3 6


We get the table of dimensions:


k = 0 k = 1 k = 2

` = 0 8 8 8` = 1 6 8 8` = 2 6 8` = 3 6




k = 0 k = 1 k = 2

` = 0 8 8 8

` = 1 6 8 8

` = 2 6 8` = 3 6

For finite type systems the symbol is involutive iff

dim(S π`DkR)) = 0 ⇐⇒ dim π`(DkR) = dim D`+1(DkR)

So πDR, πD2R and π2D2R have involutive symbols.




k = 0 k = 1 k = 2

` = 0 8 8 8

` = 1 6 8 8

` = 2 6 8` = 3 6

The system involutive if its symbol is involutive and

dim π`+1(Dk+1R) = dim D`(DkR)

So πDR is an involutive system and we expect approximately an8 dimensional solution space and symmetry group. This coincideswith the dimension for the case α =

√45 ≈ 6.708203932.


yxx + 7.1 yyx + 5 y3 = 0

Applying the symbolic-numeric method to case α = 7.1 yields

Table 2: dim π`DkR for yxx + 7.1 yyx + 5 y3 = 0

k = 0 k = 1 k = 2 k = 3 k = 4 k = 5

` = 0 8 8 6 3 2 2` = 1 6 8 6 3 2 2` = 2 6 6 3 2 2` = 3 4 3 2 2

` = 4 3 2 2` = 5 2 2` = 6 2

We conclude that π`DkR is approximately involutive for k = 4and ` = 4 (the boxed entry in Table 2), and we expect approximatelya 2 dimensional solution space and symmetry group.


ode k ` dim(G)

1 y′′ − y2 = 0 5 5 2

2 y′′ − 6 y2 − x4 = 0 7 7 0

3 y′′ − 6 y2 − x = 0 7 7 0

4 y′′ − 6 y2 + 4 y = 0 6 6 1

5 y′′ + y2 + 2 x + 3 = 0 7 7 0

6 y′′ − 2 y3 − yx + 1 = 0 5 5 0

7 y′′ − y3 = 0 4 4 2

8 y′′ − 2 y3 + 4 yx − 2 = 0 5 5 0

9 y′′ + 4 + 2 yx + 3 y + y3 = 0 5 5 0

10 y′′ + 4 + 2 y2 + 3 y + y3 = 0 4 4 1

11 y′′ + x4y9 = 0 4 4 112 y′′ y − 1 = 0 4 4 213 y′′ y = 0 1 0 8

14 y′′ y − x2 = 0 4 4 1

15 2(1 + x2)y′′ − xy′2(x + 4y′) + 2(x + y′)y′ − 2y = 0 4 4 0

16 y′′′ −(

1− x3y′)3

= 0 2 2 1

17 y′′ − y′2

y+4− y′

x+2= 0 1 0 8

18 y′ − y = 0 0 0 ∞19 y′′′′ + 9 y = 0 2 2 6

20 y′′ − 3 y′ − y2 − 2 y = 0 6 6 1

21 y′′ − 7 y′ − y3 + 12 y = 0 4 4 1

22 y′′ + 5 y′ − 6 y2 + 6 y = 0 5 5 2

23 y′′ − 3 y′ − 2 y3 + 2 y = 0 4 4 2

24 y′′ − 72y′ − 45

16y(y9 − 1

)= 0 4 4 1

25 y′′ + y′ + 2 y8 = 0 4 4 1

26 y′′ + y′ + 2 x3y8 = 0 5 5 0

27 x4y′′ + y8 = 0 4 4 1

28 x4y′′ − x(x2 + 2 y

)y′ + 4 y2 = 0 4 4 2

29 x4y′′ − x2 (x + y′

)y′ + 4 y2 = 0 4 4 1

30 x4y′′ +(xy′ − y

)3 = 0 1 0 8

31 y′′ + yy′ − y3 + y = 0 4 4 1

32 y′′ + (y + 3) y′ − y3 + y2 + 2 y = 0 4 4 2

33 x4y′′ +(x4y + 3 x3

)y′ − x4y3 + x3y2 + 2 x2y = 0 4 4 1

34 y′′ + 2 yy′ + xy′ + y = 0 5 5 0

35 y′′ + 2 yy′ + x(y′ + y2

)− 1 = 0 5 5 0

36 y′′ + 3 yy′ + y3 + yx − 1 = 0 1 0 8

37 y′′ + (3 y + x) y′ + y3 + xy2 = 0 1 0 8

38 y′′ − 3 yy′ − 3 y2 − 4 y − 2 = 0 4 4 1

39 y′′ − (3 y + x) y′ + y3 + xy2 + x2y + x4 = 0 1 0 840 y′′ − 2 yy′ = 0 4 4 2

41 y′′ + yy′ + 2 y3 = 0 4 4 2

42 y′′ + x2y′ + x3 = 0 1 0 8

43 y′′ + y′2 + 2 y = 0 4 4 1

44 y′′ + y′2 + 2 y′ + 3 y = 0 4 4 1

45 y′′ + y′2 + 2 y′ + 3 y3 = 0 4 4 1

46 y′′ + y′2 + 2 x3y = 0 5 5 0

47 y′′ + y′2 + 2 = 0 1 0 8

48 y′′ + yy′2 + 2 y = 0 4 4 1

49 y′′ + x2y′2 + y′ = 0 4 4 1

50 4 x2y′′ − x4y′2 + 4 y = 0 4 4 1


Symbolic-numeric algorithm for structure constants

• Input a DE. Compute its symmetry defining system.

• Make the defining system involutive.

• Use SVD for numerical stability.

• Read-off the ckij from the commutation relations.


Make the defining system involutive

• We suppose we have identified an involutive finite-type systemπ`Dk ′R is where Dk ′R has matrix form A(q)(z0).

• The q′ = q − ` order involutive system π`Dk ′R means thatlocal values of derivatives of solutions are uniquely determinedby π`Dk ′R up to order q′.

• To determine the structure constants in the comutation rela-tions we need derivatives up to order q′ + 1 to be uniquelydetermined. This means that we need a pair of neighboringinvolutive systems π`Dk ′R and π`−1Dk ′R.
































SVD for stability

• Input a tolerance ε.

• At random point z0, compute the SVD A(q)(z0) = UΣV T . Sin-gular values below an input tolerance are replaced with zeroes,giving a nearby matrix

A′ = UΣ′V T =[U1 U0

] [Σ1 00 0

] [V T

1

V T0

]where Σ1 contains the numerically nonzero singular values.

• See Trefethen & Bau 06 and Akritas 06.


SVD for stability



A′ = UΣ′V T =[U1 U0

] [Σ1 00 0

] [V T

1

V T0




SVD for stability



A′ = UΣ′V T =[U1 U0

] [Σ1 00 0

] [V T

1

V T0




SVD for stability



A′ = UΣ′V T =[U1 U0

] [Σ1 00 0

] [V T

1

V T0




SVD for stability



A′ = UΣ′V T =[U1 U0

] [Σ1 00 0

] [V T

1

V T0




Project to subspace for involutive system

• Project the (column) vectors of V0 to get V0 = π`−1V0 ={π`−1(v) : v ∈ V0} by simply deleting the relevant coordinatesin these column vectors.

• So V0 is a spanning set for the involutive system π`−1Dk ′R

at z = z0. Convert it to a basis V(q′+1)0 of π`−1Dk ′R. Then

dim V0 = dimL. Application of one more projection to this

basis gives a basis V(q′)0 for π`Dk ′R.

• Each basis vector v(q′+1)i ∈ V0 uniquely determines local sym-

metry vector field of form X = ξj ∂∂x j

and these vector fieldsclose under the commutator bracket:

[U,U ′] =(ξj∂ξ′i

∂x j− ξ′j ∂ξ

i

∂x j

) ∂

∂x i(19)











[U,U ′] =(ξj∂ξ′i


i

∂x j

) ∂

∂x i(19)











[U,U ′] =(ξj∂ξ′i


i

∂x j

) ∂

∂x i(19)











[U,U ′] =(ξj∂ξ′i


i

∂x j

) ∂

∂x i(19)











[U,U ′] =(ξj∂ξ′i


i

∂x j

) ∂

∂x i(19)











[U,U ′] =(ξj∂ξ′i


i

∂x j

) ∂

∂x i(19)


The structure constants ckij are given by

[Xi ,Xj ] =r∑

k=1

ckijXk , 1 ≤ i , j ≤ d (20)

If Y = [U,U ′] has components ηi (z) then by (19) we have ηi =

ξj ∂ξ′i


i

∂x j.

By repeated symbolic differentiation one obtains any derivative∂Iηi as a sum of bilinear skew terms of the form

b(∂Jξi∂J

′ξ′j − ∂Jξ′i∂J′ξj

)with |J|+ |J ′| = |I |+ 1 and coefficients b being integers. Computethese derivatives for order 0 ≤ |I | ≤ q′.


• Substitution of v(q′+1)i , v

(q′+1)j ∈ V0 for ∂Iηi gives a vector

w(q′)ij of jet variable values uniquely determining Lie bracket.

• Since the solution vector fields are to form a Lie algebra, these

vectors w(q)ij should lie in the numerical nullspace of π`Dk ′R:

w(q′)ij =

∑k

ckij v(q′)k

• Strategy 1 to determine ckij : Projecting the vectors v(q′+1)i ∈ V0

yields {π2v (q′+1)} which is a basis of d vector correspondingto the Lie algebra vector. Convert this to an orthonormal basis

V(q′−1)0 . As a consequence:

ckij = w(q′−1)ij (v

(q′−1)k )T

• Strategy 2 to determine ckij : Prolong the Lie vector fields toorder q′ + 1. Since they have the same commutation relationsthere, one can exploit the orthogonality of the basis at orderq′ + 1 directly to compute ckij .


Concluding Remarks

• We extended an algorithm that computes structure in the exactcase to the approximate case.

• Exact methods lack properties of continuity, so we needed toconsider nearby problems.

• Recast as matrix problem, using geometric techniques from Lin-ear algebra (SVD) and geometry of differential equations.

• Many new unexplored approximate problems (approximate equiv-alence, ... )

• Increasingly mixture of algebra, analysis, geometry, ... is needed

• Other methods for approximate symmetries – Gaziov & Ibragi-mov, ...

• Looking into the future will everything be done numerically?


Thank you al and thank you George!


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Approximate symmetries of di erential equationscheviakov/bluman2014/talks/Reid.pdfThus care has to be taken with symbolic approaches ... approximate basis is obtained for the kernel