QMC: Transition Workshop - Applying Quasi-Monte Carlo Methods to a Stochastic Eigenvalue Problem - Alexander Gilbert, May 9, 2018

Applying Quasi-Monte Carlo integration to a stochastic
eigenproblem
Alec Gilbert
Joint work with Frances Kuo (UNSW), Ian Sloan (UNSW) and
Ivan Graham (Bath), Rob Scheichl (Bath)
Wednesday May 9, 2018
QMC Transition Workshop, SAMSI, Research Triangle Park, NC, USA
1 / 19

WG VIII: Application of QMC to PDEs with random coefficients
To appear
I. G. Graham, F. Y. Kuo, D. Nuyens, R. Scheichl, and I. H. Sloan, Circulant embedding
with QMC — analysis for elliptic PDE with lognormal coefficients, Numerische
Mathematik. arXiv: 1710.09254
I. G. Graham, F. Y. Kuo, D. Nuyens, R. Scheichl, and I. H. Sloan, Analysis of circulant
embedding methods for sampling stationary random fields, SIAM Journal on Numerical
Analysis. arXiv: 1710.00751
Ongoing/in preparation
A. D. Gilbert, I. G. Graham, F. Y. Kuo, R. Scheichl and I. H. Sloan. Analysis of
Quasi-Monte Carlo methods for elliptic eigenvalue problems with stochastic coefficients.
M. Ganesh, F. Y. Kuo and I. H. Sloan. Quasi-Monte Carlo finite element methods for
stochastic heterogeneous wave propagation models.
2 / 19

The criticality problem for nuclear reactors
−∇ · a(x) ∇u(x) + b(x) u(x) = λ c(x) u(x)
(λ, u(x)) : λ ∈ R measures criticality of reactor
u(x) is the neutron ﬂux at the point x
https://www.youtube.com/watch?v=xIDytUCRtTA
λ1 ≈ 1 =⇒ operating eﬃciently
λ1 > 1 =⇒ not self-sustaining
λ1 < 1 =⇒ supercritical
3 / 19

The criticality problem for nuclear reactors
−∇ · a(x)
diffusion
∇u(x) + b(x)
absorption
u(x) = λ c(x)
fission
u(x)
(λ, u(x)) : λ ∈ R measures criticality of reactor
u(x) is the neutron flux at the point x
λ1 ≈ 1 =⇒ operating efficiently
λ1 > 1 =⇒ not self-sustaining
λ1 < 1 =⇒ supercritical
3 / 19

The UQ criticality problem for nuclear reactors
−∇ · a(x, y)
diffusion
∇u(x, y) + b(x, y)
absorption
u(x, y) = λ(y) c(x)
fission
u(x, y)
(λ(y), u(x, y)) : λ(y) ∈ R measures criticality of reactor
u(x, y) is the neutron flux at the point x
λ1(y) ≈ 1 =⇒ operating efficiently
λ1(y) > 1 =⇒ not self-sustaining
λ1(y) < 1 =⇒ supercritical
3 / 19

The parametrised eigenproblem
−∇ · (a(x, y)∇u(x, y)) + b(x, y)u(x, y) = λ(y)c(x)u(x, y) for x ∈ D ,
u(x, y) = 0 for x ∈ ∂D .
a(x, y) = a0(x) +
∞
j=1
yjaj(x) and b(x, y) = b0(x) +
∞
j=1
yjbj(x)
1. D ⊂ Rd is bounded and convex
2. ∇ are with respect to the x
3. stochastic parameters y ∈ [−1
2, 1
2 ] × [−1
2, 1
2] × · · · := [−1
2, 1
2]N
Quantity of interest:
Ey [λ(y)] =
[− 1
2
, 1
2
]N
λ(y) dy := lim
s→∞ [− 1
2
, 1
2
]s
λ(y1, . . . , ys, 0, 0 . . .) dy1 · · · dys
4 / 19

Approximation strategy
1. Dimension truncation:
a(x, y) ≈ a0(x) +
s
j=1
yjaj(x) and b(x, y) ≈ b0(x) +
s
j=1
yjbj(x) ,
which gives
λ(y) ≈ λ(y1, y2, . . . , ys, 0, 0 . . .) := λs(ys)
2. Finite element approximation:
λs(ys) ≈ λs,h(ys)
3. Quasi-Monte Carlo integration:
Ey [λs,h(y)] ≈
1
N
N−1
k=0
λs,h(tk) := QN (λs,h)
5 / 19

The stochastic coeﬃcients & properties of the eigenvalues
a(x, y) = a0(x) +
∞
j=1
∞
j=1
yjbj(x)
Assume: For all x ∈ D, y ∈ [−1
2 , 1
2]N
0 < amin ≤ a(x, y), b(x, y), c(x) and
∞
j=1
aj L∞ + bj L∞ < ∞
6 / 19

The stochastic coeﬃcients & properties of the eigenvalues
a(x, y) = a0(x) +
∞
j=1
∞
j=1
yjbj(x)
Assume: For all x ∈ D, y ∈ [−1
2 , 1
2]N
0 < amin ≤ a(x, y), b(x, y), c(x) and
∞
j=1
aj L∞ + bj L∞ < ∞
=⇒ Self-adjoint, uniformly elliptic EVP =⇒ facts:
1. 0 < λ1(y) ≤ λ2(y) ≤ . . .
2. λk(y) → ∞ as k → ∞
3. λ2(y) − λ1(y) ≥ ρ > 0 for all y
4. Simple eigenpairs λ(y), u(x, y) are analytic in y [Andreev, Schwab 2012]
5. λ1(y) ≤ C independently of y
6. minimal eigenvalue: λ(y) := λ1(y).
6 / 19

Why is simplicity important?
λ1(y) is simple for all y
λ2(y)
λ1(y)
7 / 19

λ2(y)
λ1(y)
λ1(y)
7 / 19

λ2(y)
λ1(y)
λ1(y)
×
7 / 19

Quasi-Monte Carlo integration
N-point randomly shifted lattice rule:
[− 1
2
, 1
2
]s
f(y) dy ≈ QN,∆f =
1
N
N−1
k=0
f (tk) .
The quadrature points:
tk =
kz
N
+ ∆ − 1
2
,
z ∈ Ns the generating vector,
uniformly distributed random shift
∆ ∈ [0, 1]s.
CBC algorithm eﬃciently constructs good z.
1
2
1
2
Figure: 2D lattice rule with N = 55,
z = (1, 34).
8 / 19

The “standard” QMC setting
Assumes f ∈ Ws,γ =: s-dimensional weighted space [Sloan, Woźniakowski 1998] with:
- weights: γ := {γv > 0 : v ⊆ {1, . . . , s}}.
- (unanchored) weighted norm
f 2
s,γ =
v⊆{1,...,s}
1
γv [0,1]|v| [0,1]s−|v|
∂|v|
∂yv
f(y) dy−v
2
dyv .
CBC error bound
For f ∈ Ws,γ, N prime and z constructed using CBC
E∆


[− 1
2
, 1
2
]s
f(y) dy − QN,∆(f)
2

 ≤ Cδ,γN−1+δ
f s,γ
9 / 19

The truncation + FE + QMC algorithm
Given s ∈ N, h > 0, N ∈ N:
1. Construct the s-dimensional generating vector z using the CBC algorithm.
2. Generate the random shift ∆ ∈ [0, 1]s.
3. For k = 0, 1, . . . , N − 1:
i. Construct the shifted lattice point
tk =
kz
N
+ ∆ − 1
2
and the coefficients a(x, (tk, 0, 0, . . .)) and b(x, (tk, 0, 0, . . .)).
ii. Find the minimal eigenvalue of the FE matrix eigenproblem:
A ((tk, 0, 0, . . .); us,h, vh) = λs,h(tk)M(us,h, vh) for all vh ∈ Vh.
4. The final estimate is
QN,∆(λs,h) =
1
N
N−1
k=0
λs,h(tk) .
5. In practice, we repeat Steps 2–4 for a small number of shifts and take the final
estimate to be the average. 10 / 19

QMC root-mean-square error
Theorem
Assume
∞
j=1
aj L∞ + bj L∞
p
< ∞ for p ∈ (0, 1) ,
then λs(y) ∈ Ws,γ and the root-mean-square error of the CBC lattice rule
approximation satisﬁes
E∆ |Ey [λs(y)] − QN,∆(λs(y))|2
≤ CαN−α
where
α =
1 − δ for all δ > 0 if p ∈ (0, 2
3] ,
1
p − 1
2 if p ∈ (2
3, 1) .
12 / 19

Key points for Theorem
1. Start with CBC error bound
E∆ |E[λs] − QN,∆(λs)|2
≤ Cδ,γN−1+δ
λs s,γ
2. To bound the norm we use our bounds on derivatives
∂|v|
∂yv
λs(y) ≤ C(|v|!)1+ǫ
j∈v
1
λ2(y) − λ1(y)
aj L∞ + bj L∞
which gives a constant that depends on sum of aj L∞ , bj L∞ , eigenvalue gap
λ2 − λ1 > ρ and the weights γv.
3. The weights γ are chosen to minimise the error bound.
4. Determine values of δ for which the error bound is independent of s.
13 / 19

Numerical results
−∇ · (a(x, y)u(x, y)) = λ(y)u(x, y) for x = (x1, x2) ∈ (0, 1)2
with
a0 = 2, aj(x1, x2) =
1
1 + (jπ)q
sin(jπx1) sin((j − 1)πx2)
- q ≥ 4/3 determines the summability for convergence rates:
aj L∞ =
1
1 + (πj)q
<
1
jq
=⇒ p ∈ (1/q, 1)
- use a CBC generated lattice rule with weights depending on q
- N = 101, 199, 499, 997, 1999, 4001, 8009, 16001
- Fix s = 100, h = 1/256 and r = 8 random shifts.
14 / 19

QMC convergence: p ≈ 1/2
10
3
10
4
10
5
Number of quadrature points: N
10
-9
10
-8
10-7
10-6
10
-5
10
-4
Std.Error
QMC vs. MC: q = 2 (with LS fit)
MC
MC Least-Squares fit
Exp. MC rate: 1/ N
QMC
QMC Least-Squares fit
Exp. QMC rate: 1/N
Figure: N-convergence for h = 1/256.
15 / 19

QMC convergence II: p ≈ 3/4 (restricted convergence)
10
3
10
4
10
5
Number of quadrature points: N
10-9
10
-8
10
-7
10
-6
10
-5
10
-4
Std.Error
QMC vs. MC: q = 4/3 (with LS fit)
MC
MC Least-Squares fit
Exp. MC rate: 1/ N
QMC
QMC Least-Squares fit
Exp. QMC rate: 1/N 5/6
Figure: N-convergence for h = 1/256.
16 / 19

Summary
1. Designed a combined QMC + FE method to approximate the minimal eigenvalue.
2. Obtained an a priori bound on the total error which gives a good convergence rate
in N and is independent of the stochastic dimension. For almost all p it is the
same convergence rate as the source problem.
3. Similar results hold for linear functionals of the eigenfunction u1.
Future work...
1. log-normal coeﬃcients:
a(x, y) = exp a0(x) +
j≥1
yjaj(x) yj ∼ N(0, σ2
).
How to handle λ2(y) − λ1(y)?
2. Study neutron transport equation (harder integro-diﬀerential equation)
3. higher-order QMC rules
17 / 19

QMC: Transition Workshop - Applying Quasi-Monte Carlo Methods to a Stochastic Eigenvalue Problem - Alexander Gilbert, May 9, 2018

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QMC: Transition Workshop - Applying Quasi-Monte Carlo Methods to a Stochastic Eigenvalue Problem - Alexander Gilbert, May 9, 2018