Mathematics for large scale tensor computations

Mathematics for large scale

tensor computations

Jose M. Martın-Garcıa

Institut d’Astrophysique de Paris,

Laboratoire Universe et Theories

Meudon, October 2, 2009

What we consider

Tensors in a single vector space V (general. to several spaces). Use indexed notations. Numeric component indices (general.

to several frames). Einstein convention for contractions.

There is a single invertible metric gab, symmetric orantisymmetric (spinor calculus). Indices can always be raisedand lowered. Beware of non-compatible derivatives:

gab∂cvb 6= ∂cva, gab(∂v)bc = (∂v)ac

Tensors have symmetries. Most general expression we consider: an arbitrary polynomial

in any number of tensors, with homogeneous free indices.

· · · + FabcTbdT c

d − 7Feff Taduevd + · · ·

What we consider

Tensor canonicalization

0. Key question in computer algebra:How do we know whether two tensor expressions are equal?Convert expressions into a canonical form and then compare.

1. Simplification:

F antisym(1↔2) (Fabcua + Fb

dcud)vb

1) Expand all parenthesis Fabcuavb + Fb

dcudvb

2) Rename dummies p, q, . . . Fpqcupvq + Fp

3) Canonicalize terms separately Fpqcupvq − Fpqcu

4) Add equal terms 0

⇒ Need only canonicalization of tensor monomials.

1. Simplification:

dcud)vb

dcudvb

1. Simplification:

dcud)vb

dcudvb

2. Simplification: Reorder tensors in a monomial using namesand ranks; form an equivalent tensor with inherited properties.

T adFabcTbe → FTTabc

antisym(1↔2), sym(4↔6,5↔7). Anticommuting objects?

⇒ Need only canonicalization of single tensors.

3. Tensor properties (“symmetries”):

Slot-syms: relations among slots, independently of indices.

Index-syms: relations among indices, independently of slots.

Contractions, metric and components

Dimension-dependent identities.

4. Two types of slot-symmetries:

Permutation (or monoterm) symmetries:

Rbacd = −Rabcd Rcdba = Rabcd

Need permutation group theory.

Multiterm symmetries:

Rabcd + Racdb + Radbc = 0

Need permutation group algebra.

5. Canonical ordering of indices:

First symbolic free indices, alphabetically.

Then symbolic dummy pairs, alphabetically. Up-indices first.

Then component indices, numerically. Up-indices first.

Example:

T a1db2b

1ac3 → [c , d , a, a, b, b, 1, 1, 2, 3]

Rough idea: Approach this “as much as possible”, unless 0.

Example:

T a1db2b

1ac3 → [c , d , a, a, b, b, 1, 1, 2, 3]

Example:

T a1db2b

1ac3 → [c , d , a, a, b, b, 1, 1, 2, 3]

Example:

T a1db2b

1ac3 → [c , d , a, a, b, b, 1, 1, 2, 3]

Permutation Symmetries

Permutation slot-symmetries

Riemann tensor Rabcd . Three equivalence classes:

Rabcd = −Rbacd = −Rabdc = Rbadc = Rcdab = −Rdcab = −Rcdba = Rdcba

Racbd = −Rcabd = −Racdb = Rcadb = Rbdac = −Rdbac = −Rbdca = Rdbca

Radbc = −Rdabc = −Radcb = Rdacb = Rbcad = −Rcbad = −Rbcda = Rcbda

Symmetry group S of order 8 of permutations of slots:

(1234),−(2134),−(1243), (2143), (3412),−(4312),−(3421), (4321)

and two left cosets gS , with g=(1324) or g=(1423).

Canonical representatives (choose a permutation ordering)

(1234), (1324), (1423)

Algorithm: given T with slot-symmetry S and a configurationg find the canonical representative of the left coset gS .

(1234),−(2134),−(1243), (2143), (3412),−(4312),−(3421), (4321)

(1234), (1324), (1423)

(1234),−(2134),−(1243), (2143), (3412),−(4312),−(3421), (4321)

(1234), (1324), (1423)

Index-symmetries

Tensor Tabcd with no slot-symmetries.

T aabb = Ta

abb = T a

abb = Ta

abb = T b

baa = Tb

baa = T b

baa = Tb

New group D of permutations of indices (in canonical order).

We need (start from [a, a, b, b]):

Exchange dummies: (3412)

Swap metric (2134) and (1243). Add sign with spinors

Symmetry under exchange of repeated components

T111 → D = S3

These three groups do not intersect and always commute.

Index-symmetries

Tensor Tabcd with no slot-symmetries.

T aabb = Ta

abb = T a

abb = Ta

abb = T b

baa = Tb

baa = T b

baa = Tb

New group D of permutations of indices (in canonical order).

We need (start from [a, a, b, b]):

Exchange dummies: (3412)

Swap metric (2134) and (1243). Add sign with spinors

Symmetry under exchange of repeated components

T111 → D = S3

These three groups do not intersect and always commute.

Both together

A tensor configuration Rabba will be given by g = (1342):

[a, a, b, b]g

−→ [a, b, b, a]

being equivalent to the double coset DgS :

[a, a, b, b]d

−→ [a, a, b, b]g

−→ [a, b, b, a]s

−→ −[a, b, a, b]

Algorithm: given T with slot-symmetry S and index-symmetryD, and a configuration g , find the representative of the doublecoset DgS . If there is p such that p,−p ∈ DgS , then rep=0.

Both together

A tensor configuration Rabba will be given by g = (1342):

[a, a, b, b]g

−→ [a, b, b, a]

being equivalent to the double coset DgS :

[a, a, b, b]d

−→ [a, a, b, b]g

−→ [a, b, b, a]s

−→ −[a, b, a, b]

Algorithm: given T with slot-symmetry S and index-symmetryD, and a configuration g , find the representative of the doublecoset DgS . If there is p such that p,−p ∈ DgS , then rep=0.

Example 1

R Riemann and T symmetric. Canonical index order:

[a, d , e, g , b, b, c , c , f , f ]

R R Ta b c db c

e ff g

Example 1

[a, d , e, g , b, b, c , c , f , f ]

R R Ta b c db c

e ff gR R Ta b c d

b ce f

Example 1

[a, d , e, g , b, b, c , c , f , f ]

R R Ta b c db c

e ff gR R Ta b c d

b ce f

f gR R Ta b c db c

e ff g

Example 1

[a, d , e, g , b, b, c , c , f , f ]

R R Ta b c db c

e ff gR R Ta b c d

b ce f

f gR R Ta b c db c

e ff gR R Ta b c d

b ce f

Example 1

[a, d , e, g , b, b, c , c , f , f ]

R R Ta b c db c

e ff gR R Ta b c d

b ce f

f gR R Ta b c db c

e ff gR R Ta b c d

b ce f

f gR R Ta b c db c

e ff g

Example 1

[a, d , e, g , b, b, c , c , f , f ]

R R Ta b c db c

e ff gR R Ta b c d

b ce f

f gR R Ta b c db c

e ff gR R Ta b c d

b ce f

f gR R Ta b c db c

e ff gR R Ta b c d

b ce f

Example 1

[a, d , e, g , b, b, c , c , f , f ]

R R Ta b c db c

e ff gR R Ta b c d

b ce f

f gR R Ta b c db c

e ff gR R Ta b c d

b ce f

f gR R Ta b c db c

e ff gR R Ta b c d

b ce f

R R Tfb ce

Example 2

Antisymmetric tensor Fba = −Fab.

FabFbc symmetric

n... F ha = 0 if number n of tensors is odd.

Timings (in seconds)

Example 2

Antisymmetric tensor Fba = −Fab.

FabFbc symmetric

n... F ha = 0 if number n of tensors is odd.

Timings (in seconds)

n |D |, |S | MathTensor xAct

1 2 0 0

3 48 0.01 0.01

5 3840 0.02 0.03

7 645120 1.12 0.05

9 185794560 350 0.07

11 8.2 1010

107745 0.09

19 6.4 1022

? 0.28

29 4.7 1039

? 0.94

39 1.1 1058

49 3.4 1077

59 8.0 1097

? 13.7 10 20 30 40 500.001

100000

Second

Minute

Example 3

Random Riemann scalar monomials. Example R7:

Rbjdk Rncel R

hgia Rk

gde R l

fjc Rmaf

Takes 19596s in MathTensor, and 0.031s in xTensor.

++++++

oooooooo

ooooooooo

ooooooo

ooooooooooooo

oooooo

ooooooooo

oooooooo

ooooooo

Second

Minute

0 10 20 30 40 50

number of Riemann tensors

A million random Riemann scalar monomials R10.Timings histogram:

CPU 1.7 GHz

bin: 0.0001 s

Zero: 424108

Nonzero: 575892

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.100

10 000

15 000

timings in seconds

All but 7 below 1s, 6 in 1-2s, and 1 with 4.9s. Worst case:

Ra1b1a2b2

Ra2b2a3b3

· · ·Ranbna1b1

Working with permutation groups: strong generators

Describe a group G through generators.S = s1, . . . , sm generating set: G = 〈S〉.Example: Sn = 〈(2134 . . . n), (234 . . . n1)〉.Difficult to extract information from S . Direct algorithms areexponential in n.

Better idea: use strong generators (Sims 1960’s). Take agroup G acting on the points Ω = 1, . . . , n. We need threedefinitions:

Orbit pG of a point p: pg , g ∈ G. Ω partitioned intodisjoing orbits.

Stabilizer Gp of a point p: g ∈ G , pg = p subgroup of G .G partitioned into disjoint cosets of Gp .

Base of G : B = [b1, . . . , bk ], none of the permutations of G

fixes all points of B. That is, Gb1...bk= id.

Examples: for Sn we need n − 1 points in the base. For theslot-symmetries of Riemann [1, 3] is a base, but [1, 2] is not.

Key observation 1: bijection between the orbit pG and the setof cosets G/Gp. Proof:

1) g ∈ Gp ⇔ pg = p

2) h′ ∈ Gp · h ⇒ g ∈ Gp, h′ = g · h ⇒ ph′ = pg ·h = ph

3) ph = ph′ ⇒ p = ph′·h−1⇒ h′ · h−1 ∈ Gp, same coset

In particular|G | = |Gp| · |p

Key observation 2: Base images identify permutations:

[bg1 , . . . , bg

k ] = [bg ′

1 , . . . , bg ′

k ] ⇔ g = g ′

Otherwise g ′g−1 would fix all points of B .

We want to work in the hierarchy of stabilizers

G > Gb1> Gb1b2

> · · · > Gb1...bk= id

Key observation 1: bijection between the orbit pG and the setof cosets G/Gp. Proof:

1) g ∈ Gp ⇔ pg = p

2) h′ ∈ Gp · h ⇒ g ∈ Gp, h′ = g · h ⇒ ph′ = pg ·h = ph

3) ph = ph′ ⇒ p = ph′·h−1⇒ h′ · h−1 ∈ Gp, same coset

In particular|G | = |Gp| · |p

Key observation 2: Base images identify permutations:

[bg1 , . . . , bg

k ] = [bg ′

1 , . . . , bg ′

k ] ⇔ g = g ′

Otherwise g ′g−1 would fix all points of B .

We want to work in the hierarchy of stabilizers

G > Gb1> Gb1b2

> · · · > Gb1...bk= id

Do we already have generators for the stabilizers?In general, no. Example:

G = S3 = 〈(213), (231)〉 with base B = [1, 2]

G1 = 〈(132)〉

G12 = id

However,

G = S3 = 〈(213), (231)〉 with base B = [3, 2]

G3 = 〈(213)〉

G32 = id

Definition: S = s1, . . . , sm is a strong generating set of G

relative to the base B = [b1, . . . , bk ] if S contains generatingsets for all the stabilizers Gb1...bi

, i = 1, . . . , k.

G = S3 = 〈(213), (231)〉 with base B = [1, 2]

G1 = 〈(132)〉

G12 = id

However,

G = S3 = 〈(213), (231)〉 with base B = [3, 2]

G3 = 〈(213)〉

G32 = id

, i = 1, . . . , k.

G = S3 = 〈(213), (231)〉 with base B = [1, 2]

G1 = 〈(132)〉

G12 = id

However,

G = S3 = 〈(213), (231)〉 with base B = [3, 2]

G3 = 〈(213)〉

G32 = id

, i = 1, . . . , k.

The Schreier-Sims algorithm constructs a strong-generatingset from a generating set of G , for a freely specifiable base.

Sims 1970: O(n6) Knuth 1991: O(n5) Today:

Deterministic: O(n2 log3 |G |)Randomized: O(n log n log4 |G |)

Things which can be computed in polynomial time: Group order and membership. Coset enumeration and representatives.

Things which cannot be computed in polynomial time: Intersection of groups. Double coset enumeration and representatives.

General strategy: backtrack search, usually quite fast, butexponential in the worst case.

Back to tensors

Canonicalization of tensor expressions with permutationsymmetries entails finding double coset representatives.

Permutation groups can be efficiently manipulated usingstrong generating sets.

Butler 1984: Use of backtrack search with SGS to find doublecoset representatives of any two subgroups.

Portugal 2001: Tensorial case: general S but special D.

Implementations (both GPL free software):

Portugal 2003: Maple

JMM-G 2003: Mathematica and C (∼ 100 times faster)

Solves the general case of permutation symmetries, withtimings of seconds up to 100 indices (minutes in worst cases).

Back to tensors

Multiterm Symmetries

In particular,

where is the dimension of the vector space?

Sources of multiterm symmetries

Intrinsic cyclic symmetries:

The Bianchi identity:

∇aRbcde + ∇bRcade + ∇cRabde = 0

The Ricci identity (more general concept of symmetry):

∇a∇bvc = ∇b∇av

c + Rabdcvd

Crystallographic symmetries

Dimensionally-dependent identities

The Cayley-Hamilton theorem

1. A square n × n matrix A satisfies its own characteristicequation:

p(x) := det(x I − A) = xn + c1xn−1 + · · · + cn−1x + cn

p(A) = 0

c1 = −tr(A) = −Aaa

2(tr(A)2 − tr(A2)) = A[a

c3 = · · · = −A[aaA

The Cayley-Hamilton theorem is simply

A[a1a1A

a2a2 · · ·A

ananδ

a]b = 0

The Cayley-Hamilton theorem

1. A square n × n matrix A satisfies its own characteristicequation:

p(x) := det(x I − A) = xn + c1xn−1 + · · · + cn−1x + cn

p(A) = 0

c1 = −tr(A) = −Aaa

2(tr(A)2 − tr(A2)) = A[a

c3 = · · · = −A[aaA

The Cayley-Hamilton theorem is simply

A[a1a1A

a2a2 · · ·A

ananδ

a]b = 0

2. This is a Lovelock-type (dimensionally-dependent) identity:

A[a1a2a3b1b2

Ba4a5b3· · ·Hanan+1]an+2

3. Gover 1997: Lovelock identities exhaust all dim-dep tensorrelations.

4. Obvious general algorithm (“brute force”): Take all possible canonical monomials appearing in a problem. Construct all possible linear multiterm relations among them.

For instance, in dim 4, antisymmetrize over all subsets of 5indices and perm-canonicalize again.

Introduce an order for those canonical monomials. Solve for the largest canonical monomials in terms of the

smaller monomials. Store all multiterm relations in a database.

Sounds crazy but works if perm-canonicalization is fastenough, and for moderate dimension.

A[a1a2a3b1b2

Young tableaux

Schur-Weyl duality. Tensor transformation under M ∈ GL(n):

(MT )a1...ak= Ma1

bi · · ·Mak

bk Tb1...bk

Tensor transformation under p ∈ Sk :

(pT )a1...ak= Ta′1...a

k, a′j = ajp

(pMT )a1...ak= Ma′1

bi · · ·Ma′k

bk Tb1...bk

= Ma′1

i · · ·Ma′k

k Tb′

1...b′

= Ma1bi · · ·Mak

bk (pT )b1...bk

= (MpT )a1...ak

Linear transformations and permutations commute on tensors!

That implies the existence of a common labelling of therepresentations of the groups GL(n) and Sk . The labels arethe integer partitions of k in at most n parts, representable asYoung diagrams of k boxes with at most n rows.Example: 12=4+3+3+2

This is a possible irreducible symmetry for a rank-12 tensor indimension 4 or larger. Other examples:

Lanczos Riemann ∇Riemann

That implies the existence of a common labelling of therepresentations of the groups GL(n) and Sk . The labels arethe integer partitions of k in at most n parts, representable asYoung diagrams of k boxes with at most n rows.Example: 12=4+3+3+2

This is a possible irreducible symmetry for a rank-12 tensor indimension 4 or larger. Other examples:

Lanczos Riemann ∇Riemann

Permutation symmetry in a tableau: columns are antisymmetries symmetry under exchange of columns of equal length

Dimensions of Sk representation: number of standardtableaux. Riemann:

a cb d

a bc d but not

a bd c

Dimensions of GL(n) representation: number of semistandardtableaux. Riemann in n = 4:

1 12 2

1 12 3

1 12 4

1 13 3

1 13 4

1 14 4

1 22 3

1 22 4

1 23 3

1 23 4

1 24 4

1 32 4

1 33 4

1 34 4

2 23 3

2 23 4

2 24 4

2 33 4

2 34 4

3 34 4 but not

1 24 3

a cb d

a bc d but not

a bd c

1 12 2

1 12 3

1 12 4

1 13 3

1 13 4

1 14 4

1 22 3

1 22 4

1 23 3

1 23 4

1 24 4

1 32 4

1 33 4

1 34 4

2 23 3

2 23 4

2 24 4

2 33 4

2 34 4

3 34 4 but not

1 24 3

a cb d

a bc d but not

a bd c

1 12 2

1 12 3

1 12 4

1 13 3

1 13 4

1 14 4

1 22 3

1 22 4

1 23 3

1 23 4

1 24 4

1 32 4

1 33 4

1 34 4

2 23 3

2 23 4

2 24 4

2 33 4

2 34 4

3 34 4 but not

1 24 3

Cyclic symmetry in a tableau: Garnir hooks. Riemann: twoequivalent Garnir hooks:

a cb d +

a bd c −

a bc d = 0

Symmetry of a tensor product: For different tensors, use the Littlewood-Richardson rule

× = + + +

For equal tensors, use plethysms

ˆ = + + +

Cyclic symmetry in a tableau: Garnir hooks. Riemann: twoequivalent Garnir hooks:

a cb d +

a bd c −

a bc d = 0

Symmetry of a tensor product: For different tensors, use the Littlewood-Richardson rule

× = + + +

For equal tensors, use plethysms

ˆ = + + +

Each standard tableau defines a projector. For Riemann:

12P = (1234) + (2134) − (3214) − (1432)

+(1243) − (4132) − (2314) − (3241)

−(1423) − (4123) + (4312) + (3421)

−(2341) + (2143) + (3412) + (4321)

with P2 = P . Using permutations symmetries we get

Rabcd = (PR)abcd =2

3Rabcd +

3Racbd −

3Radbc

Agacy 1998, Peeters 2005: replace Rabcd by this expression,and use only permutation symmetries then. That implementsall symmetries of Riemann!

Each standard tableau defines a projector. For Riemann:

12P = (1234) + (2134) − (3214) − (1432)

+(1243) − (4132) − (2314) − (3241)

−(1423) − (4123) + (4312) + (3421)

−(2341) + (2143) + (3412) + (4321)

with P2 = P . Using permutations symmetries we get

Rabcd = (PR)abcd =2

3Rabcd +

3Racbd −

3Radbc

Agacy 1998, Peeters 2005: replace Rabcd by this expression,and use only permutation symmetries then. That implementsall symmetries of Riemann!

Riemann invariants

History

Algebraic inv RabcdRabef Rcdef , or differential inv Rab∇c∇dRacbd .

Used in the classification of spacetimes, GR Lagrangian expansions,QG renormalization, etc.

Haskins 1902: 14 independent scalar invs in 4d.

Narlikar and Karmarkar 1948: first list of 14 scalar invs:(R , R2, W 2, R3, W 3, R2W , W 4, R4, R2W 2,W 5, R2W 3, R4W , R4W 2, R4W 3).

Other invariants are (possibly nonpolynomial) functions of them.

Completeness: polynomic basis for all invariants. Relations?

Sneddon 1999: 38 a-invs (deg 11). Complete basis. (Rotorcalculus.)

Carminati-Lim 2007: graph-based construction of a-inv relations.

Fulling et al 1992: bases for d-invs up to 10 derivatives and R6.

History

Riemann invariants up to degree 7 7 Riemann → 28 indices → 28! ≃ 3.0 · 1029 a-invs

Use monoterm symmetries: 16352 invariants

Use cyclic property: 1639 invariants

Antisymmetrise over sets of 5 indices in 4d: 4311 integer equations

Gauss elimination: Intermediate swell problem:

Original integers: |ci | ≤ 384 Final integers: |cf | ≤ 4978120 Largest intermediate integer:

4128432088811462631210560847258796357727760565961955312109033719974522122912026393839047919079833136293326586851425207976783555836309071365640456573585413951344201540916373441818847340888359206516826545838550350981924166243854816386358011913196352080821446913112544207777945582431400973483457511551249427165040578011486377996431957110887574044723612095478539444181734311327487135158150147408144620915351339939806272116543186970020596936859101026073658967889992306803277195043926510784936890214764598221917466623055176060658271638645490139036389024466375930303586688550738508615214422459534528028026604339296211874545398934176513452668215589707310886321265833677782947671903197039133298738083435857983747685083659334680222681616686514058699829948476521738772411701178283002256312672449814493504188768078308059566617955048754211100127225300485494079978006577938025856377710049543176142178387315401497328

Required 8Gb RAM and 384-bytes integers (BigNum: GMP).

4128432088811462631210560847258796357727760565961955312109033719974522122912026393839047919079833136293326586851425207976783555836309071365640456573585413951344201540916373441818847340888359206516826545838550350981924166243854816386358011913196352080821446913112544207777945582431400973483457511551249427165040578011486377996431957110887574044723612095478539444181734311327487135158150147408144620915351339939806272116543186970020596936859101026073658967889992306803277195043926510784936890214764598221917466623055176060658271638645490139036389024466375930303586688550738508615214422459534528028026604339296211874545398934176513452668215589707310886321265833677782947671903197039133298738083435857983747685083659334680222681616686514058699829948476521738772411701178283002256312672449814493504188768078308059566617955048754211100127225300485494079978006577938025856377710049543176142178387315401497328

Required 8Gb RAM and 384-bytes integers (BigNum: GMP).

We get the 27 invs of Sneddon’s basis up to degree 7 (6 dual), plus allpolynomial expression of any other invariant. Invar package: CPC 2007.

Database of 645 625 relations up to 12 metric derivatives. CPC 2008.

Results for algebraic Riemann invariants

Degree A A∗ B B∗ C C∗ D D∗

1 1 1 1 0 1 0 1 02 3 4 2 1 2 1 2 13 9 27 5 6 3 2 3 24 38 232 15 40 4 1 3 15 204 2582 54 330 5 2 3 26 1613 35090 270 3159 8 2 4 27 16532 558323 1639 – 7 (1) 3 (1)8 217395 – 13140 – (9) (1) (2) (1)9 3406747 – – – (11) (1) (3) (1)10 – – – – (9) (1) (1) (1)11 – – – – (9) (0) (1) (0)12 – – – – (9) (0) (0) (0)

A: Permutation symmetries, B: Cyclic symmetry,C: Dim-dep identities, D: Products of duals (∗).

Conclusions

1. We have a fast algorithm for canonicalization underpermutation symmetries, both of slot and index type. Basedon the use of strong generating sets.

2. It is possible to use brute force to implement multitermsymmetries, in particular Lovelock identities, for moderatecases. We have shown the example of Riemann invariants.

3. It is possible to use Young tableaux to work with irreduciblemultiterm symmetries. Algorithm not complete, though.

Mathematics for large scale tensor computations

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