tensor-structured methods in electronic structure calculations
TRANSCRIPT
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solution
Tensor-structured methods in electronic structure
calculations
Venera KhoromskaiaGAMM 2010, Karlsruhe, March 22-26, 2010.
Max-Planck Institute for Mathematics in the Sciences, Leipzig
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solution
Outline of the talk
1 Tensor-structured (TS) methods in Rd
Basic tensor formatsTensor product operationsBest Tucker approximationMultigrid accelerated BTA
2 Accurate TS computation of integral operators in Rd , d ≥ 3
The Hartree-Fock equationTS Computation of the Coulomb matrixComputation of the exchange matrix
3 TS numerical solution of the Hartree-Fock equation3D nonlinear EVP solver by TS methods
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionBasic tensor formats Tensor product operations Best Tucker
Rank-structured representation of higher order tensors
A tensor of order d , A = [ai1...id : iℓ ∈ Iℓ] ∈ RI1×...×Id ,
Iℓ = {1, ..., nℓ}, ℓ = 1, . . . , d ,for nℓ = n, N = nd .
tensor product of vectors u(ℓ) = {u(ℓ)iℓ
}niℓ=1 ∈ R
Iℓ forms thecanonical rank-1 tensor
A(1) ≡ [ui]i∈I = u(1) ⊗ . . . ⊗ u(d) with entries ui = u(1)i1
· · · u(d)id
,
storage: dn << nd .the canonical format, CR,n
A ≈ A(R) =∑R
k=1cku
(1)k ⊗ . . . ⊗ u
(d)k , ck ∈ R, (1)
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionBasic tensor formats Tensor product operations Best Tucker
Tucker tensor format
The rank-(r1, . . . , rd) orthogonal Tucker approximation[Tucker, De Lathauwer].
A ≈ A(r) =∑r1
ν1=1. . .∑rd
νd=1βν1,...,νd
v (1)ν1
⊗ . . . ⊗ v (d)νd
. (2)
A(r) = β ×1 V (1) ×2 V (2) ×3 . . . ×d V (d).
Vectors in V (ℓ) = [v(ℓ)1 . . . v
(ℓ)rℓ ] form the orthonormal basis.
The core tensor β = [βν1,...,νd] ∈ R
r1×...×rd .r = max
ℓ{rℓ} is the (maximal) Tucker rank.
Storage: rd + drn ≪ nd .
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionBasic tensor formats Tensor product operations Best Tucker
Mixed TC tensor format
A(r) =
(
R∑
k=1
bku(1)k ⊗ . . . ⊗ u
(d)k
)
×1 V (1) ×2 V (2) ×3 . . . ×d V (d).
A
r3
I3
I2
r2
(3)
(2)
I1
I
I
I
2
3
1r2
r3
r1
+ . . . +1(3) (3)
r
1
1(2)
(1)r2
r3
r1
+ d r nr1 rdnd
(1)
u
u
u u
b1 bR (2)
ru
u(1)r
2R < = rV
V
V β
β β
Advantages of the Tucker tensor decomposition:
1. Robust algorithm for decomposing full format tensors of size nd .2. Rank reduction of the rank-R canonical tensors with large R.
3. Very efficient for 3D tensors since r3 is small.
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionBasic tensor formats Tensor product operations Best Tucker
Multilinear operations in CR,n tensor format
A1 =
R1∑
k=1
cku(1)k ⊗ . . . ⊗ u
(d)k , A2 =
R2∑
m=1
bmv (1)m ⊗ . . . ⊗ v (d)
m .
Euclidean scalar product (complexity O(dR1R2n) ≪ nd ),
〈A1, A2〉 :=
R1∑
k=1
R2∑
m=1
ckbm
d∏
ℓ=1
⟨
u(ℓ)k , v (ℓ)
m
⟩
.
Hadamard product of A1, A2
A1 ⊙ A2 :=
R1∑
k=1
R2∑
m=1
ckbm
(
u(1)k ⊙ v (1)
m
)
⊗ . . . ⊗(
u(d)k ⊙ v (d)
m
)
.
Convolution, for d = 3 (O(R1R2n log n) ≪ n3 log n (3D FFT))
A1 ∗ A2 =
R1∑
k=1
R2∑
m=1
ckbm(u(1)m ∗ v
(1)k ) ⊗ (u(2)
m ∗ v(2)k ) ⊗ (u(3)
m ∗ v(3)k )
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionBasic tensor formats Tensor product operations Best Tucker
Tucker tensor approximation
A0 ∈ Vn : f (A) := ‖A − A0‖2 → min over A ∈ T r. (3)
The minimisation problem [De Lathauwer et al., 2000] (3) is equivalentto the maximisation problem
g(V (1), ..., V (d)) :=∥
∥
∥A0 ×1 V (1)T × ... ×d V (d)T
∥
∥
∥
2
→ max (4)
over the set of orthogonal matrices V (ℓ) ∈ Rnℓ×rℓ .
For given V (ℓ), the core β that minimises (3) is
β = A0 ×1 V (1)T × ... ×d V (d)T ∈ Rr1×...×rd . (5)
[De Lathauwer et al, 2000]: Algorithm BTA (Vn → T r,n).Complexity for d = 3: WF→T = O(n4 + n3r + n2r2 + nr3).
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionBasic tensor formats Tensor product operations Best Tucker
BTA of function related tensors
2 4 6 8 10 12 14
10−10
10−5
100
Tucker rank
erro
r
Slater function, AR=10, n = 64
EFN
EFE
EC 0
5
10
05
10
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Orthogonal vectors, r=6
y−axis
Slater function g(x) = exp(−α‖x‖) with x = (x1, x2, x3)T ∈ R
3
EFN =‖A0 − A(r)‖
||A0||EFE =
‖A0‖ − ‖A(r)‖||A0||
, EC :=maxi∈I |a0,i − ar,i |
maxi∈I |a0,i |.
[Khoromskij,VKH. ’07]
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionBasic tensor formats Tensor product operations Best Tucker
MGA BTA for 3D periodic structures
2 4 6 8 10 1210
−15
10−10
10−5
100
Tucker rank
Multicentered Slater funct.with 1000 samp.
EFN
, n=129
EFE
, n=129
−5
0
5
−5
0
50
0.2
0.4
0.6
The “multi-centered Slater potential“ obtained by displacing a singleSlater potential with respect to the m × m × m spatial grid of sizeH > 0, (here m = 10)
g(x) =
m∑
i=1
m∑
j=1
m∑
k=1
e−α√
(x1−iH)2+(x2−jH)2+(x3−kH)2.
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionBasic tensor formats Tensor product operations Best Tucker
Canonical to Tucker Approximation
[B.N. Khoromskij, VKH ’09]
Theorem
Let A ∈ CR,n. Then the minimisation problem
A ∈ CR,n ⊂ Vn : A(r) = argminV∈Tr,n‖A − V ‖Vr , (6)
[Z (1), ..., Z (d)] = argmaxV (ℓ)∈Mℓ
∥
∥
∥
∥
∥
R∑
ν=1
cν
(
V (1)T u(1)ν
)
⊗ ... ⊗(
V (d)T u(d)ν
)
∥
∥
∥
∥
∥
2
Vr
,
Init. guess: RHOSVD Z(ℓ)0 ⇒ the truncated SVD of U(ℓ) = [u
(ℓ)1 ...u
(ℓ)R
].Error bounds for RHOSVD
‖A − A0(r)‖ ≤ ‖c‖
dX
ℓ=1
(
min(n,R)X
k=rℓ+1
σ2ℓ,k )
1/2, where ‖c‖
2=
RX
ν=1
c2ν .
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionBasic tensor formats Tensor product operations Best Tucker
Canonical to Tucker Approximation
The maximizer Z (ℓ) = [z(ℓ)1 ...z
(ℓ)rℓ ] ∈ R
n×rℓ is obtained by ALS of thematrix unfolding
B(q) ∈ Rn×r̄q , r̄q = r1...rq−1rq+1...rd . (7)
(c) The minimiser in (6) is then
A(r) = β ×1 Z (1) ×2 Z (2) ×3 . . . ×d Z (d).
β =
R∑
ν=1
cν(Z (1)T u(1)ν ) ⊗ ... ⊗ (Z (d)T u(d)
ν ) ∈ CR,r.
Complexity for d = 3, n ≥ R : WC→T = O(nR2 + nr4) .
C2T ⇒ T2C: R → R ′ ≤ r2 ≪ R
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionBasic tensor formats Tensor product operations Best Tucker
MGA Tucker approximation for CR,n tensors
1 Sequence of nonlinear appr. problems for A = An ,n = nm := n02m,
m = 0, 1, . . . , M , on a sequence of refined grids ω3,nm.
2 Z(q)0 on grid ω3,nm
→ by linear interp. of Z (q) ∈ Rnm−1×rq from
ω3,nm−1 .
3 The restricted ALS iteration, is based on “most important fibers”(MIFs) of unfolding matrices.Positions of MIFs are extracted at the coarsest grid
β(q) = Z (q)TB(q) ∈ Rrq×rq . (8)
Location of MIFs corresponds to pr columns in β(q) with maximalEuclidean norms (maximal energy principle).
4 Complexity for d = 3: O(RrnM + p2r2nM).(Linear w.r.t. n and R !)
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionBasic tensor formats Tensor product operations Best Tucker
MGA C2T (CR,n)[B.N. Khoromskij, VKH ’08]
B(q) → unfolding of (B[q]), β(q) = Z (q)TB(q); n0 = 64, nM = 16384
+ ... +
n1 n nn
1 11
n
n3
2
rr
r12r r2
r33 r3 3r
r2 23
+ ... +
n3
n2
n1find MIFs
core
B[q]
[q]β
number 1
2 4 6 8 10 12 14
2
4
6
8
10
12
14
Number 2
2 4 6 8 10 12 14
2
4
6
8
10
12
14
Number 3
2 4 6 8 10 12 14
2
4
6
8
10
12
14
MIFs for H2O
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionBasic tensor formats Tensor product operations Best Tucker
MGA Tucker vs. single grid
200 400 600 800 1000 12000
10
20
30
40
50
60
70
80
90
rank
seco
nds
n=2048n=1024n=512n=256
1000 2000 3000 4000 5000 6000 7000 80000
20
40
60
80
100
120
univariate grid size
min
utes
Single grid vs. multigrid times for C2 H
6
C2T single gridC2T multigrid
Linear scaling in R and in n (left) for C2T algorithm.
Multigrid vs. single grid times for the C2T transform in computations of
VH for C2H6 (right).
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionBasic tensor formats Tensor product operations Best Tucker
MGA BTA for full format tensors
MGA best Tucker decomposition for full format tensors,
[B. Khoromskij,VKH’09], [VKH ’10, in progress]
WF→T = O(n3) for MGA BTA, instead of O(n4) for BTA.
2 4 6 8 10 1210
−14
10−12
10−10
10−8
10−6
10−4
10−2
100
Tucker rank
Slater function, MGA Tucker, EFN
(solid), EEN
(dashed)
n=512
n=256
n=128
n=64
n=512
n=256
n=128
n=64
2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
3.5
Tucker rank
min
ute
s
Times for MGA Tucker, Slater function
n=512
n=256
n=128
Full size tensor with number of entries 5123 , times for Matlab.
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionBasic tensor formats Tensor product operations Best Tucker
MGA BTA for full format tensors (Aluminium Cluster)
[T. Blesgen, V. Gavini and VKH, ’09]
−10
0
10
−15
−10
−5
0
5
10
15
0
0.05
−10−5
05
10
−10
−5
0
5
100
0.02
0.04
0.06
cluster1/sdv4 , initial ρ for Al, z=0
−10−5
05
10
−10
0
100
1
2
3
4
x 10−4
cluster1/subdiv4, abs. diff. for ρ, rT=20
−8 −6 −4 −2 0 2 4 6 8−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
u1(1)
u2(1)
u3(1)
u4(1)
u5(1)
u6(1)
5 10 15 20 2510
−5
10−4
10−3
10−2
10−1
100
Tucker rank
ρ for Al, Cluster1, Subdib3
Een
, n=201
EF, n=201
EF, n=101
Een
, n=101
FE (coarse graining) ⇒ computed on large supercomputer clusters∗ .
FE data∗ ⇒ 3D Cartesian grid ⇒ Tucker decomposition
∗[V. Gavini, J. Knap, K. Bhattacharya, and M. Ortiz,’07]
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionThe Hartree-Fock equation TS Computation of the Coulomb matrix
The Hartree-Fock equation
The Hartree-Fock equation
Fϕi (x) = λi ϕi (x), i = 1, ..., Norb, x ∈ R3
F := −1
2∆ − Vc + VH −K,
the Hartree potential
VH(x) :=
∫
R3
ρ(y)
|x − y | dy , ρ(y) = 2
Norb∑
i=1
(ϕi (y))2
where ϕi =R0∑
k=1
ci ,kgk(x).
The exchange operator
(Kϕ) (x) :=
∫
R3
τ(x , y)
|x − y | ϕ(y)dy , τ(x , y) =
Norb∑
i=1
ϕi (x)ϕi (y).
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionThe Hartree-Fock equation TS Computation of the Coulomb matrix
O(n) computation of the Hartree potential on huge 3D grids
VH is computed on large n × n × n Cartesian grids, with n ∼ 103 ÷ 104
ρ(x) = 2
Norb∑
i=1
(ϕi )2, ϕi =
R0∑
k=1
ci ,kgk(x), x = (x1, x2, x3) ∈ R3
gk(x) = gk,1(x1) gk,2(x2) gk,3(x3),
gk,ℓ = (xℓ − Ak,ℓ)βk,ℓe−λk (xℓ−Ak,ℓ)
2
,
gk,ℓ → u(ℓ)k ; xℓ ∈ [−b, b],
b = 6 ÷ 10◦
A.
H
H
H
H
H
H
C
C
z
y
x
R0(CH4) = 55, R0(C2H6) = 96, R0(H2O) = 41, Rρ = R0(R0+1)2 .
ρ ≈ Fρ =∑Rρ
k=1cku
(1)k ⊗ u
(2)k ⊗ u
(3)k , ck ∈ R, u
(ℓ)k ∈ R
n.
⇒ ρ is presented in the computation box [−b, b]3 with Rρ ∼ 103 ÷ 104.
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionThe Hartree-Fock equation TS Computation of the Coulomb matrix
O(n) computation of the Hartree potential on huge 3D grids
Rρ(CH4) = 1540, Rρ(C2H6) = 4656, Rρ(H2O) = 861
Next : MGA C2T + T2C transformations to reduce the rank:
Rρr∼ 102.
RN ∼ 20 ÷ 30 canonical tensor G for the Newton kernel 1‖x‖
(using sinc-quadratures) [C. Bertoglio, B. Khoromskij ’08]).
The rank-Rρrcanonical tensor F for ρ:
VH = F ∗ G =
Rρr∑
m=1
RN∑
k=1
ckbm
(
uk1 ∗ vm
1
)
⊗(
uk2 ∗ vm
2
)
⊗(
uk3 ∗ vm
3
)
Coulomb matrix: tensor inner products in GTO basis {gk}R0
k=1
Jkm :=
∫
R3
gk(x)gm(x)VH(x)dx , k , m = 1, . . .R0.
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionThe Hartree-Fock equation TS Computation of the Coulomb matrix
Tensor-product convolution vs. 3D FFT
The cost of computation of VH by tensor product convolution (1D FFT):
NC∗C = O(RρrRNn log n)
instead of O(n3 log n) for 3D FFT.
n3 1283 2563 5123 10243 20483 40963 81923 163843
FFT3 4.3 55.4 582.8 ∼ 6000 – – – ∼ 2 yearsC ∗ C 0.2 0.9 1.5 8.8 20.0 61.0 157.5 299.2C2T 4.2 4.7 5.6 6.9 10.9 20.0 37.9 86.0
CPU time (in sec) for the computation of VH for H2O.
(3D FFT time for n ≥ 1024 is obtained by extrapolation).
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionThe Hartree-Fock equation TS Computation of the Coulomb matrix
High accuracy computations of VH
(Calculation over the number of entries N ≈ 4.3 · 1012).
−6 −4 −2 0 2 4 6
10−4
10−3
hart
ree
Abs. approx. error, VH
for H2O
a) atomic units
n=4096n=8192Ri−4096−8192
2000 4000 6000 8000 10000 12000 14000 160000
1
2
3
4
5
6
univariate grid size n
min
utes
H2O , r
T=20
C−2−T time3D conv. time
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionThe Hartree-Fock equation TS Computation of the Coulomb matrix
High accuracy computations of the Coulomb matrix for H2O
(for calculation over the number of entries N ≈ 4.3 · 1012). Approx. error O(h3), h = 2b/n (h ≈ 0.0008◦
A).
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionThe Hartree-Fock equation TS Computation of the Coulomb matrix
High accuracy computations of the Coulomb matrix for C2H6
a) CPU times for VH : O(n log n) on n × n × n grids with max.:n = 8192.b) Absolute approximation error of the tensor-product computation ofthe Coulomb matrix for C2H6 molecule.
2000 4000 6000 80000
2
4
6
8
10
12
14
b) univariate grid size
min
utes
C−2−T time3D conv. time
−4 −2 0 2 4 6
10−6
10−4
10−2
100
a) atomic units
hart
ree
C2H
6, abs. appr. error for V
H
n=4096n=8192Ri−4096−8192
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionThe Hartree-Fock equation TS Computation of the Coulomb matrix
TS computation of the exchange matrix
[VKH ’09]
Kk,m :=
∫
R3
∫
R3
gk(x)τ(x , y)
|x − y |gm(y)dxdy , k , m = 1, . . .R0
1. Compute the discrete tensor product convolution
Wa,m(x) =
∫
R3
ϕa(y)gm(y)
|x − y | dy , cost:O(RNR0n log n) per element (×R0N)
2. Entries of the Exchange matrix for every orbital,
Vkm,a =
∫
R3
ϕa(x)gk(x)Wa,m(x)dx
with the cost O(RNR40nf ).
3. Entries of the Exchange matrix
Kk,m =
Norb∑
a=1
Vkm,a
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solutionThe Hartree-Fock equation TS Computation of the Coulomb matrix
TS computation of the exchange matrix on huge 3D grids
Abs. error for the Exchange matrices Kex forCH4 (Richardson extrapolation for n = 4096)
and H2O (Rich. for n = 16384).
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solution3D nonlinear EVP solver by TS methods
Solution of the Hartree-Fock on a sequence of 3D grids
[Flad, Khoromskij, VKH ’09]
Discretized GTO basis, ϕi (x) ≈R0∑
k=1
ckigk(x), x ∈ R3, i = 1, ..., Norb
Fϕi (x) = λi ϕi (x), ⇒ FU = λSU
F = H0 + J(C ) − K (C ), C = {cki} = [U1U2 . . . UNorb] ∈ R
R0×Norb
H0 is the stiffness matrix of the core Hamiltonian − 12∆ + Vc
Our iterative scheme
Initial guess for m = 0: J = 0, K = 0 ⇒ F0 = H0.
On a sequence of refined grids n = n0, 2n0, . . . , 2pn0,
solve EVP [H0 + Jm(C ) − Km(C )]U = λSU .
Fast update of Jm(C ) and Km(C ).
Grid dependent termination criteria εp = ε0 · 4−p.
Use DIIS [Pulay ’80] to provide fast convergence.
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solution3D nonlinear EVP solver by TS methods
HF for the pseudopotential case of CH4
Left: multilevel convergence of iterations for the pseudopotential case ofCH4.Right: convergence in effective iterations for finest grid.
2 4 6 8 1010
−4
10−3
10−2
10−1
100
abs. error, | λ − λn,it
|, CH4, pseudo
n=128, 115 s/it
n=64, 72 s/it
n=256, 260 s/it
n=512, 488 s/it
total time: 33 min.
0 1 2 3 410
−4
10−3
10−2
10−1
100
conv.in eff.iterations, CH4, pseudo, n=512
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solution3D nonlinear EVP solver by TS methods
Convergence in HF energy
Left: convergence in HF energy.Right: linear scaling of the iteration time in the univariate grid size.
200 400 600 800 100010
−5
10−4
10−3
10−2
10−1
100
univariate grid size
Abs. err. for energy, CH4, pseudo
O(h2)
O(h3)
200 400 600 800 10000
5
10
15
20
25
30
univariate grid sizem
inu
tes
time per SCF iteration
The total energy is calculated by
EHF = 2
NorbX
i=1
λi −
NorbX
i=1
“eJi − eKi
”
with eJi = (ϕi , VH ϕi )L2 and eKi = (ϕi ,Vexϕi )L2 (i = 1, ..., Norb ) the Coulomb and exchange integrals,
computed with respect to the orbitals ϕi .
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solution3D nonlinear EVP solver by TS methods
Iterations of the nonlinear 3D EVP solver
Left: all electron case of H2O.Right: the pseudopotential case of CH3OH .
5 10 1510
−6
10−4
10−2
100
iterations
residual in HF EVP, all electron case H2O
5 10 15 20 2510
−5
10−4
10−3
10−2
10−1
100
CH3OH, residual
n=64
n=128
n=256
n=512
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations
Overview Tensor-structured (TS) methods in Rd Accurate TS computation of integral operators in R
d , d ≥ 3 TS numerical solution3D nonlinear EVP solver by TS methods
Thank you for attention
The talk is based on
1 H.-J. Flad, V. Khoromskaia, B. Khoromskij.Solution of the Hartree-Fock Equation by the Tensor-Structured Methods.
Preprint MPI MIS 44/2009.
2 V. Khoromskaia.Computation of the Hartree-Fock Exchange by the Tensor-structured Methods.
Preprint MPI MIS 25/2009
3 T. Blesgen, V. Gavini and V. Khoromskaia.Approximation of the Electron Density of Aluminium Clusters in Tensor-product Format,
Preprint MPI MIS 66/2009
4 V. Khoromskaia and B. N. Khoromskij.Multigrid Accelerated Tensor Approximation of Function Related Multi-dimensional Arrays.
SIAM Journal on Scientific Computing, vol. 31, No. 4, pp. 3002-3026, 2009.
5 B. N. Khoromskij, V. Khoromskaia, S.R. Chinnamsetty and H.-J. Flad.Tensor Decomposition in Electronic Structure Calculations on 3D Cartesian Grids.
Journal of Computational Physics, 228(2009), pp. 5749-5762, 2009.
6 B. N. Khoromskij and V. Khoromskaia.Low Rank Tucker-Type Tensor Approximation to Classical Potentials.
Central European Journal of Mathematics v.5, N.3, 2007, pp.523-550.
http://personal-homepages.mis.mpg.de/vekh/
Venera Khoromskaia GAMM 2010, Karlsruhe, March 22-26, 2010.TS methods in electronic structure calculations