ADI for Tensor Structured Equations

MAX PLANCK INSTITUTE

FOR DYNAMICS OF COMPLEX

TECHNICAL SYSTEMS

MAGDEBURG

83rd GAMM Annual Scientific ConferenceDarmstadt, 28 March 2012

ADI for Tensor Structured Equations

Thomas Mach and Jens Saak

Max Planck Institute for Dynamics of Complex Technical SystemsComputational Methods in Systems and Control Theory

Max Planck Institute Magdeburg Thomas Mach, Jens Saak, Tensor-ADI 1/24

ADI ADI for Tensors Numerical Results and Shifts Conclusions

Classic ADI [Peaceman/Rachford ’55]

Developed to solve linear systems related to Poisson problems

−∆u = f in Ω ⊂ Rd , d = 1, 2

u = 0 on ∂Ω.

uniform grid size h, centered differences, d = 1,

⇒ ∆1,hu = h2 f

∆1,h =

2 −1−1 2 −1

. . .. . .

−1 2 −1−1 2

Developed to solve linear systems related to Poisson problems

−∆u = f in Ω ⊂ Rd , d = 1, 2

u = 0 on ∂Ω.

uniform grid size h, 5-point difference star, d = 2,

⇒ ∆2,hu = h2 f

∆2,h =

K −I−I K −I

. . .. . .

−I K −I−I K

and K =

4 −1−1 4 −1

. . .. . .

−1 4 −1−1 4

Observation

∆2,h = (∆1,h ⊗ I )︸︷︷︸=:H

+ (I ⊗∆1,h)︸︷︷︸=:V

Solve ∆2,hu = h2 f =: f exploiting structure in H and V .

For certain shift parameters perform

(H + pi I ) ui+ 12

= (pi I − V ) ui + f ,

(V + pi I ) ui+1 = (pi I − H) ui+ 12

until ui is good enough.

Observation

∆2,h = (∆1,h ⊗ I )︸︷︷︸=:H

+ (I ⊗∆1,h)︸︷︷︸=:V

(H + pi I ) ui+ 12

= (pi I − V ) ui + f ,

(V + pi I ) ui+1 = (pi I − H) ui+ 12

Observation

∆2,h = (∆1,h ⊗ I )︸︷︷︸=:H

+ (I ⊗∆1,h)︸︷︷︸=:V

(H + pi I ) ui+ 12

= (pi I − V ) ui + f ,

(V + pi I ) ui+1 = (pi I − H) ui+ 12

ADI and Lyapunov Equations [Wachspress ’88]

Lyapunov Equation

FX + XFT = −GGT

Vectorized Lyapunov Equation[(I ⊗ F )︸︷︷︸

+ (F ⊗ I )︸︷︷︸=:VF

]vec(X ) = −vec(GGT )

Same structure ⇒ apply ADI

(F + pi I )Xi+ 12

= −GGT − Xi

(FT − pi I

)(F + pi I )Xi+1 = −GGT − XT

(FT − pi I

Lyapunov Equation

FX + XFT = −GGT

+ (F ⊗ I )︸︷︷︸=:VF

(F + pi I )Xi+ 12

= −GGT − Xi

(FT − pi I

)(F + pi I )Xi+1 = −GGT − XT

(FT − pi I

Lyapunov Equation

FX + XFT = −GGT

+ (F ⊗ I )︸︷︷︸=:VF

(F + pi I )Xi+ 12

= −GGT − Xi

(FT − pi I

)(F + pi I )Xi+1 = −GGT − XT

(FT − pi I

)Max Planck Institute Magdeburg Thomas Mach, Jens Saak, Tensor-ADI 4/24

Generalizing Matrix Equations

∆2,hvec(X ) = vec(B)(

I ⊗ I ⊗

I ⊗ ∆1,h︸︷︷︸=H

I ⊗ I ⊗

∆1,h ⊗ I︸︷︷︸=V

+ I ⊗ ∆1,h ⊗ I ⊗ I︸︷︷︸=R

+ ∆1,h ⊗ I ⊗ I ⊗ I︸︷︷︸=Q

)vec(X )︸︷︷︸

= vec(B)︸︷︷︸=f

d Xabcd

a∆µa

b∆µb

c ∆µc

d ∆µd

+ = Ba

Generalizing Matrix Equations

∆4,hvec(X ) = vec(B)(I ⊗ I ⊗ I ⊗ ∆1,h︸︷︷︸

+ I ⊗ I ⊗ ∆1,h ⊗ I︸︷︷︸=V

+ I ⊗ ∆1,h ⊗ I ⊗ I︸︷︷︸=R

+ ∆1,h ⊗ I ⊗ I ⊗ I︸︷︷︸=Q

)vec(X )︸︷︷︸

= vec(B)︸︷︷︸=f

Xabcd Xabcd

a∆µa

b∆µb

c ∆µc

d ∆µd

+ = Babcd

Generalizing ADI

(I ⊗ ∆1,h︸︷︷︸

+ ∆1,h ⊗ I︸︷︷︸=V

)vec(X )︸︷︷︸

= vec(B)︸︷︷︸=f

(H + I ⊗ pi ,1I )Xi+ 12

= (pi ,1I − V )Xi + B

(V + pi ,2I ⊗ I )Xi+ 12

= (pi ,2I − H)Xi+ 12

(I ⊗ I ⊗ I ⊗ ∆1,h︸︷︷︸

+ I ⊗ I ⊗ ∆1,h ⊗ I︸︷︷︸=V

+ I ⊗ ∆1,h ⊗ I ⊗ I︸︷︷︸=R

+ ∆1,h ⊗ I ⊗ I ⊗ I︸︷︷︸=Q

)vec(X )︸︷︷︸

= vec(B)︸︷︷︸=f

(H + I ⊗ I ⊗ I ⊗ pi ,1I )Xi+ 14

= (pi ,1I − V − R − Q)Xi + B

(V + I ⊗ I ⊗ pi ,2I ⊗ I )Xi+ 12

= (pi ,2I − H − R − Q)Xi+ 14

(R + I ⊗ pi ,3I ⊗ I ⊗ I )Xi+ 34

= (pi ,3I − H − V − Q)Xi+ 12

(Q + pi ,4I ⊗ I ⊗ I ⊗ I )Xi+1 = (pi ,4I − H − V − R)Xi+ 34

Generalizing ADI

(I ⊗ ∆1,h︸︷︷︸

+ ∆1,h ⊗ I︸︷︷︸=V

)vec(X )︸︷︷︸

= vec(B)︸︷︷︸=f

(H + I ⊗ pi ,1I )Xi+ 12

= (pi ,1I − V )Xi + B

(V + pi ,2I ⊗ I )Xi+ 12

= (pi ,2I − H)Xi+ 12

(I ⊗ I ⊗ I ⊗ ∆1,h︸︷︷︸

+ I ⊗ I ⊗ ∆1,h ⊗ I︸︷︷︸=V

+ I ⊗ ∆1,h ⊗ I ⊗ I︸︷︷︸=R

+ ∆1,h ⊗ I ⊗ I ⊗ I︸︷︷︸=Q

)vec(X )︸︷︷︸

= vec(B)︸︷︷︸=f

(H + I ⊗ I ⊗ I ⊗ pi ,1I )Xi+ 14

= (pi ,1I − V − R − Q)Xi + B

(V + I ⊗ I ⊗ pi ,2I ⊗ I )Xi+ 12

= (pi ,2I − H − R − Q)Xi+ 14

(R + I ⊗ pi ,3I ⊗ I ⊗ I )Xi+ 34

= (pi ,3I − H − V − Q)Xi+ 12

(Q + pi ,4I ⊗ I ⊗ I ⊗ I )Xi+1 = (pi ,4I − H − V − R)Xi+ 34

Solve AX = B

A = I ⊗ I ⊗ · · · ⊗ I ⊗ I ⊗ A1 +

I ⊗ I ⊗ · · · ⊗ I ⊗ A2 ⊗ I +

. . . +

Ad ⊗ I ⊗ · · · ⊗ I ⊗ I ⊗ I

B is given in tensor train decomposition⇒ X is sought in tensor train decomposition.

Tensor Trains [Oseledets, Tyrtyshnikov ’09]

T (i1, i2, . . . , id) =

r1,...,rd−1∑α1,...,αd−1=1

G1(i1, α1)G2(α1, i2, α2)

· · ·Gj(αj−1, ij , αj) · · ·Gd−1(αd−2, id−1, αd−1)Gd(αd−1, id).

G1(i1, α1) α1 G2(α1, i2, α2) α2 · · · Gd(αd−1, id)

Tensor trains are

computable, and

require only O(dnr2) storage, with TT-rank r and T ∈ Rnd .

Canonical representation

T (i1, i2, . . . , id) =∑α

G1(i1, α) · · ·Gd(id , α)

Tucker decomposition

T (i1, i2, . . . , id) =∑

α1,...,αd

C (α1, . . . , αd)G1(i1, α1) · · ·Gd(id , αd)

(I ⊗ · · · ⊗ I ⊗ A1)

G1(i1, α1)

A1(β, i1)

α1 G2(α1, i2, α2) α2 · · · Gd(αd−1, id)

= G1(β, α1) = A1G1

T (i1, i2, . . . , id)×1 A1

α1,...,αd−1

∣∣β,i1G1(i1, α1)G2(α1, i2, α2)

· · ·Gd−1(αd−2, id−1, αd−1)Gd(αd−1, id)

(I ⊗ · · · ⊗ I ⊗ A1)

G1(i1, α1)

A1(β, i1)

α1 G2(α1, i2, α2) α2 · · · Gd(αd−1, id)

= G1(β, α1) = A1G1

T (i1, i2, . . . , id)×1 A1

α1,...,αd−1

∣∣β,i1G1(i1, α1)G2(α1, i2, α2)

· · ·Gd−1(αd−2, id−1, αd−1)Gd(αd−1, id)

(I ⊗ · · · ⊗ I ⊗ A1)

G1(i1, α1)

A1(β, i1)

α1 G2(α1, i2, α2) α2 · · · Gd(αd−1, id)

= G1(β, α1) = A1G1

T (i1, i2, . . . , id)×1 A1

α1,...,αd−1

∣∣β,i1G1(i1, α1)G2(α1, i2, α2)

· · ·Gd−1(αd−2, id−1, αd−1)Gd(αd−1, id)

(I ⊗ · · · ⊗ I ⊗ A1)

G1(i1, α1)

A1(β, i1)

α1 G2(α1, i2, α2) α2 · · · Gd(αd−1, id)

= G1(β, α1) = A1G1

T (i1, i2, . . . , id)×1 A1

α1,...,αd−1

∣∣β,i1G1(i1, α1)G2(α1, i2, α2)

· · ·Gd−1(αd−2, id−1, αd−1)Gd(αd−1, id)

(I ⊗ · · · ⊗ I ⊗ A1)−1T

G1(i1, α1)

A1(β, i1)

α1 G2(α1, i2, α2) α2 · · · Gd(αd−1, id)

= G1(β, α1) = A1\G1

T (i1, i2, . . . , id)×1 A1−1 =

∑α1,...,αd−1

A1−1∣∣β,i1G1(i1, α1)G2(α1, i2, α2)

· · ·Gd−1(αd−2, id−1, αd−1)Gd(αd−1, id)

Algorithm

Input: A1, . . . ,Ad, tensor train B, accuracy εOutput: tensor train X , with AX = Bforall j ∈ 1, . . . , d do

X(0)j := zeros(n, 1, 1)

while∥∥r (i)

∥∥ > ε do

Choose shift piforall k ∈ 1, . . . , d do

X (i+ kd ) :=

(B+piX

(i+ k−1d )−

d∑j=1j 6=k

X (i+ k−1d ) ×j Aj

)×k (Ak + pi I )

r (i) := Bforall j ∈ 1, . . . , d do

r (i) := r (i) − Xi ×j Aj

(I ⊗ I ⊗ · · · ⊗ I ⊗ Aj ⊗ I ⊗ · · · ⊗ I )Xi+ k−1d

Algorithm

X(0)j := zeros(n, 1, 1)

while∥∥r (i)

∥∥ > ε do

X (i+ kd ) :=

(B+piX

(i+ k−1d )−

d∑j=1j 6=k

X (i+ k−1d ) ×j Aj

)×k (Ak + pi I )

r (i) := Bforall j ∈ 1, . . . , d do

r (i) := r (i) − Xi ×j Aj

(I ⊗ I ⊗ · · · ⊗ I ⊗ Aj ⊗ I ⊗ · · · ⊗ I )Xi+ k−1d

Algorithm

X(0)j := zeros(n, 1, 1)

while∥∥r (i)

∥∥ > ε do

X (i+ kd ) :=

(B+piX

(i+ k−1d )−

d∑j=1j 6=k

X (i+ k−1d ) ×j Aj

)×k (Ak + pi I )

r (i) := Bforall j ∈ 1, . . . , d do

r (i) := r (i) − Xi ×j Aj

(I ⊗ I ⊗ · · · ⊗ I ⊗ Aj ⊗ I ⊗ · · · ⊗ I )Xi+ k−1d

Eigenvalues

A = I ⊗ · · · ⊗ I ⊗A1 + I ⊗ · · · ⊗ I ⊗A2 ⊗ I + . . .+ Ad ⊗ I ⊗ · · · ⊗ I

Stephanos’ theorem:

⇒ λi (A) = λi1(A1) + λi2(A2) + · · ·+ λid (Ad),

with i = i1 + i2n1 + · · ·+ idd−1∏j=1

AX = B ⇔d∑

j=1X ×j Aj = B

A is regular ⇔ λi (A) 6= 0 ∀i ⇐ Ai Hurwitz ∀i

mk = argmaxj∈1,...,nk : Im (λi (Ak ))≥0

|λj(Ak)|

Eigenvalues

A = I ⊗ · · · ⊗ I ⊗A1 + I ⊗ · · · ⊗ I ⊗A2 ⊗ I + . . .+ Ad ⊗ I ⊗ · · · ⊗ I

Stephanos’ theorem:

⇒ λi (A) = λi1(A1) + λi2(A2) + · · ·+ λid (Ad),

with i = i1 + i2n1 + · · ·+ idd−1∏j=1

AX = B ⇔d∑

j=1X ×j Aj = B

A is regular ⇔ λi (A) 6= 0 ∀i ⇐ Ai Hurwitz ∀i

mk = argmaxj∈1,...,nk : Im (λi (Ak ))≥0

|λj(Ak)|

Lemma [Grasedyck ’04]

The tensor equation ∑dj=1 X ×j Aj = B

with Ak Hurwitz ∀k has the solution

X = −∫∞

0 B ×1 exp(A1t)×2 · · · ×d exp(Ad t)dt

Z (t) = B ×1 exp(A1t)×2 · · · ×d exp(Ad t)

Z (t) =d∑

Z (t)×j Aj Z (∞)− Z (0) =

∫ ∞0

Z (t)dt,

0− B =d∑

∫ ∞0

Z (t)dt ×j Aj

Theorem

A1, . . . ,Ad ⇒ A, Λ(A) ⊂ [−λmax,−λmin]⊕ ı [−µ, µ] ⊂ C−.Let k ∈ N and use the quadrature points and weights:

hst := π√k, tj := log

(e jhst +

√1 + e2jhst

), wj := hst√

1+e−2jhst.

Then the solution X can be approximated by

X (i1, i2, . . . , id) = −r1,...,rd−1∑

α1,...,αd−1=1H1(i1, α1) · · ·Hd(αd−1, id),

with Hp(αp−1, ip, αp) :=∑k

j=−k2wj

∑βpe

2tjλmin

Ap∣∣ip,βp

Gp (αp−1, βp, αp)

with the approximation error

‖X − X‖2 ≤ Cstπλmin

e2µλ−1

π−π√k∮

∥∥∥(λI − 2A/λmin)−1∥∥∥

2dΓλ ‖B‖2 .

extending [Grasedyck ’04] (X and B of low Kronecker rank) to low TT-rank

B (i1, i2, . . . , id) =∑

α1,...,αd−1

G1(i1, α1)G2(α1, i2, α2)

· · ·Gj(αj−1, ij , αj) · · ·Gd−1(αd−2, id−1, αd−1)Gd(αd−1, id).

Approximation Accuracy

0 5 10 15 20 25 30

Iteration

le constant truncation errortightened truncation error

10−20

10−14

10−8

10−2

orε i

Example: Laplace – Ai = ∆1, 111

Ai = ∆1, 111

B =[0 0 . . . 0 1

Shifts:pi := e1(∗1) + . . .+ ed(∗d) — random chosen eigenvalue

Numerical Results – Ai = ∆1, 111

d t in s residual mean(#it)

2 3.887 e−01 7.015 e−10 112.85 5.398 e+00 7.467 e−10 45.88 6.007 e+00 6.936 e−10 12.8

10 3.662 e+00 7.685 e−10 6.825 3.142 e+01 2.437 e−10 5.050 2.268 e+02 2.049 e−10 5.075 7.192 e+02 4.036 e−10 5.0

100 1.700 e+03 1.864 e−10 5.0150 5.538 e+03 1.801 e−10 5.0200 1.280 e+04 1.472 e−10 5.0250 2.499 e+04 1.816 e−10 5.0300 4.298 e+04 2.535 e−10 5.0500 1.952 e+05 2.039 e−10 5.0

sparse densed TADI \ MESS Penzl’s sh. \ lyap

2 0.310 0.0006 0.024 0.003 0.0003 0.00054 3.130 0.1695 0.011 0.049 6.331 0.0126 8.147 — 0.076 0.094 — 7.178 5.458 — 5.863 1.097 — 13 698.2

10 5.306 — 3 445.523 249.464 — —

10 100 30010−2

10−1

Dimension d

Tensor ADI

sparse \MESSPenzl’s shifts

dense \lyap

10 100 30010−2

10−1

Dimension d

Tensor ADI

sparse \MESSPenzl’s shifts

dense \lyap

Single Shift and Convergence

A = I ⊗ · · · ⊗ I ⊗A1 + I ⊗ · · · ⊗ I ⊗A2 ⊗ I + . . .+ Ad ⊗ I ⊗ · · · ⊗ I

We assume Λ(Ak) ⊂ R−.

Error Propagation, Single Shift

‖G1‖2 ≤ maxλk∈Λ(Ak ),k=1,...,d

∣∣∣∣∣∣d∏

p −∑k

λk + λl

p + λl

∣∣∣∣∣∣ =

∣∣∣∣∣∣d∏

p + λl

∣∣∣∣∣∣ .If ‖G1‖2 < 1, then the ADI iteration converges.

p < 0 and p > −∞

p < λi (A) =∑d

k=1 λk(Ak) ∀iLyapunov case (Ak = A0 ∀k): p < d−2

2 λmin(A0)

A = I ⊗ · · · ⊗ I ⊗A1 + I ⊗ · · · ⊗ I ⊗A2 ⊗ I + . . .+ Ad ⊗ I ⊗ · · · ⊗ I

‖G1‖2 ≤ maxλk∈Λ(Ak ),k=1,...,d

∣∣∣∣∣∣d∏

p −∑k

λk + λl

p + λl

∣∣∣∣∣∣ =

∣∣∣∣∣∣d∏

p + λl

p < 0 and p > −∞p < λi (A) =

∑dk=1 λk(Ak) ∀i

Lyapunov case (Ak = A0 ∀k): p < d−22 λmin(A0)

A = I ⊗ · · · ⊗ I ⊗A1 + I ⊗ · · · ⊗ I ⊗A2 ⊗ I + . . .+ Ad ⊗ I ⊗ · · · ⊗ I

‖G1‖2 ≤ maxλk∈Λ(Ak ),k=1,...,d

∣∣∣∣∣∣d∏

p −∑k

λk + λl

p + λl

∣∣∣∣∣∣ =

∣∣∣∣∣∣d∏

p + λl

p < 0 and p > −∞p < λi (A) =

Lyapunov case (Ak = A0 ∀k): p < d−22 λmin(A0)

A = I ⊗ · · · ⊗ I ⊗A1 + I ⊗ · · · ⊗ I ⊗A2 ⊗ I + . . .+ Ad ⊗ I ⊗ · · · ⊗ I

‖G1‖2 ≤ maxλk∈Λ(Ak ),k=1,...,d

∣∣∣∣∣∣d∏

p −∑k

λk + λl

p + λl

∣∣∣∣∣∣ =

∣∣∣∣∣∣d∏

p + λl

p < 0 and p > −∞p < λi (A) =

Lyapunov case (Ak = A0 ∀k): p < 2−22 λmin(A0) = 0

Shifts

Min-Max-Problem

minp1,1,...,p`,d⊂C

maxλk∈Λ(Ak ) ∀k

∣∣∣∣∣∏i=0

d∏k=0

pi ,k −∑

j 6=k λj

pi ,k + λk

∣∣∣∣∣

Min-Max-Problem, Lyapunov case (Ak = A0 ∀k , A0 Hurwitz)

maxλk∈Λ(A0) ∀k

∣∣∣∣∣∏i=0

d∏k=0

−∑

j 6=k λj

∣∣∣∣∣λk = λ0 ∀kPenzl’s idea: p1, . . . , p` ⊂ (d − 1)Λ(A0)

Shifts

Min-Max-Problem

minp1,1,...,p`,d⊂C

∣∣∣∣∣∏i=0

d∏k=0

pi ,k −∑

j 6=k λj

pi ,k + λk

∣∣∣∣∣Min-Max-Problem, Lyapunov case (Ak = A0 ∀k , A0 Hurwitz)

minp1,1,...,p`,d⊂C

∣∣∣∣∣∏i=0

d∏k=0

pi ,k −∑

j 6=k λj

pi ,k + λk

∣∣∣∣∣

λk = λ0 ∀kPenzl’s idea: p1, . . . , p` ⊂ (d − 1)Λ(A0)

Shifts

Min-Max-Problem

minp1,1,...,p`,d⊂C

∣∣∣∣∣∏i=0

d∏k=0

pi ,k −∑

j 6=k λj

pi ,k + λk

minp1,...,p`⊂C

∣∣∣∣∣∏i=0

d∏k=0

−∑

j 6=k λj

∣∣∣∣∣

λk = λ0 ∀kPenzl’s idea: p1, . . . , p` ⊂ (d − 1)Λ(A0)

Shifts

Min-Max-Problem

minp1,1,...,p`,d⊂C

∣∣∣∣∣∏i=0

d∏k=0

pi ,k −∑

j 6=k λj

pi ,k + λk

minp1,...,p`⊂C

∣∣∣∣∣∏i=0

d∏k=0

−∑

j 6=k λj

∣∣∣∣∣λk = λ0 ∀kPenzl’s idea: p1, . . . , p` ⊂ (d − 1)Λ(A0)

Random Example

seed := 1;

R := rand(10);

R := R + R ′;

R := R − λmin + 0.1;

A0 = −R;

Λ(A0) = −0.1000,−0.2250,−1.1024,−1.7496,−2.0355,

−2.4402,−3.1330,−3.3961,−3.9347,−11.9713

⇒ The random shifts do not lead to convergence.

p0 = λ10(A0)(d − 1)

p1 = λ9(A0)(d − 1)

p2 = λ8(A0)(d − 1)

Numerical Results – Ai = −R

d t in s residual #it

2 2.7673 9.1353 e−09 219.05 7.8942 9.6503 e−09 98.08 18.9964 9.8650 e−09 84.0

10 18.4739 7.5746 e−09 58.015 27.5661 5.0619 e−09 40.020 32.2409 4.9971 e−09 32.025 40.2462 5.1732 e−09 29.050 76.3225 7.4093 e−09 14.075 159.6627 3.2629 e−09 10.0

100 436.6120 9.1137 e−09 11.0

Numerical Results – Ai = −R

d t in s tdmrg in s #it

2 2.7673 0.0148 219.05 7.8942 2.5576 98.08 18.9964 5.4536 84.0

10 18.4739 5.5852 58.015 27.5661 6.3068 40.020 32.2409 7.4044 32.025 40.2462 8.3371 29.050 76.3225 11.8840 14.075 159.6627 18.0581 10.0

100 436.6120 28.8515 11.0

Conclusions and Outlook

We have seen

a generalization of the ADI method,

capable of solving tensor Lyapunov and Sylvester equations,

producing solutions of low TT-rank.

Open questions:

more sophisticated shift strategies and

why is the dmrg solver so much faster?

Thank you for your attention.

We have seen

Open questions:

We have seen

Open questions:

ADI for Tensor Structured Equations

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