u

19 Pages·2016·4.51 MB·English

16 X. ZHU, D. XIU and , Y. Chen
6.1. 2D di↵usion equation with input in layers. Consider the two-dimensional
stochastic elliptic problem
(a(x, !) u(x, !)) = f(x), in D
�r · r (6.2)
u = g(x), on @D,
⇢
with the physical domain D = [ 1, 1]2, with the right-hand-side f = 1, and zero
�
Dirichlet boundary condition. The random di↵usivity field a(x, !) has the covariance
function,
x y 2
2
C(x, y) = exp k � k , x, y [ 1, 1] , (6.3)
� `2 2 �
✓ ◆
where ` is the correlation length. Here we choose a short correlation length, ` = 0.2,
where the global KL expansion results in d = 200 random dimensions. We divide the
physical domain D = [ 1, 1]2 into four rectangular layers with distinct mean values,
�
200, D = [ 1, 1] [1/2, 1],
1
� ⇥
40, D = [ 1, 1] [0, 1/2],
8 2
µ = � ⇥ (6.4)
a
> 8, D = [ 1, 1] [ 1/2, 0],
>> 3
< � ⇥ �
4, D = [ 1, 1] [ 1, 1/2].
4
� ⇥ � �
>
>
>
:
To compute the gPC expansion,we employ the high level (with level 7) sparse grids
stochastic collocation and then approximate the gPC expansion coe�cients using the
Stochaspasrste gircids qcuaodraltulroe ruclea(wtithiolevenl 7) . mThe erefetrehnceoFEdMsso luwtionsiatrehco mputed
over a 64 64 linear elements, which result in negligible spatial discretization errors
⇥
for this case. For domain partitioning, we utilize three sets of subdomains: 4 4,
multi-fidelity models
⇥
8 8, and 16 16 (the number of FEM elements in each subdomain is adjusted in
⇥ ⇥
each case to keep the overall number of elements at 64 64).
⇥
Figure 6.1 shows one realization of the input random field a(x, Z) with distinct
means values in each layer indicated above and the corresponding full finite element
solution (reference solution) of the di↵usion problem in the domain D = [ 1, 1]2.
�
Xueyu Zhu
Scientific Computing and Imaging Institute
University of Utah
[email protected]
with
Utah: Dongbin Xiu, Akil Narayan
Erin m. Linebarger
Fig. 6.1. Left: One realization of the input a(x,Z) with distinct means values in each layer
1
indicated above, Right: Its corresponding full finite element solution (reference solution).
Motivation
Collocation methods are widely used in uncertainty quantification
of practical problems:
‣ Non-intrusive: run different parameter realizations of the
deterministic solver.
‣ Fast convergence can be achieved for smooth problems.
However,
‣ High cost: repetitive runs of deterministic solvers
‣ Extreme case: “I can only afford 10 simulations”
What can we do with this scenario?
2
Motivation
For many problems, there usually exists low fidelity models:
‣ Affordable cost: coarser meshes, simplified physics, coarse-grained
model, …
‣ Low accuracy but resolve some important features
Wishlist:
‣ Efficiency from low fidelity models
‣ Accuracy from high fidelity models
How can we glue models together?
3
Setup
u (x, t, Z) = (u), in D (0, T ] I ,
t Z
L ⇥ ⇥
(u) = 0, on @D [0, T ] I ,
Z
8 B ⇥ ⇥
u = u , in D t = 0 I .
< 0 Z
⇥ { } ⇥
L
u
: low fidelity solution (cheap)
:
H
u : high fidelity solution (expensive)
The goal: build a surrogate of H in a non-instrusive way
u
m
H
v(x, t, z) = c (z)u (x, t, z ), z I
n n n Z
2
n=1
X
z
Q1: How to choose intelligently?
n
Q2: How to compute c without extensive sampling the high-fidelity solver?
n
4
Problems of interest
We seek a solution, , which is constrained
u(x, µ)
Point Selection
through a linear parametrized PDE
The Reduced Basis Approach: A Motivation
(x, µ)u(x, µ) = f(x, µ) x �
M
L � µ R
Key idea: Explore the parameter space by the cheap low-fi model
⇥ D �
The Main Idea: Reduuc(xe ,tµhe) =Bagsi(sx, µ) x ��
�
We might expect a good approximation using a Galerkin approach
‣ Search the space by the low-fi model via greedy algorithm:
using solutions for “well chosen” sampling of parameters as base
Assumption: The solution varies smoothly on a low-
functions.
L L
z = arg max d(u (z), Udimen)sional manifold under parameter variation.
m
m 1
z �
�
2
Where
M
Choosing the samples well,
Candidate set:
we should be able to derive
� = z , . . . , z
1 M good approximations for all
{ }
H
u (z )
u(µ )j parameters
Low-fi approximation space: j uH(µ)
u (z)
L L L
U (�) = u (z ), . . . , u (z ) X
m 1 1 m 1
{ }
� �
✓ Enrich the space by finding the furtherest point away from the
Wednesday, November 2, 11
space spanned by the existed set
✓ The greedy choice is (almost) asymptotically optimal. (Devore, 2013)
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Lifting procedure
The coefficients are possibly inferred from the low-fi model
‣ Construct c as projection coefficients from low-fidelity model:
n
{ }
m
L L L
v(z) = u (z) = c (z)u (z )
UL(�) n n
P
n=1
X
‣ Using the same coefficients, construct the approximation
rule for high-fidelity model
m
H H
v(x, t, z) = c (z)u (z )
n n
n=1
X
•
This is in fact a lifting operator from low-fi space to high-fi space
•
Justification: e.g., linear scaling, coarse/fine mesh
6
Overview of Bi-fidelity algorithm
‣ Run the low-fi models at each point of � to obtain uL(�), U L(�)
�
select the most “important” m points —
‣ Run the high-fi models on those m points to get uH (�)
most expensive part!
‣ Bi-fidelity approximation:
‣ For any given z I , compute low-fi projection coefficients
z
2
m
L L L
v(z) = u (z) = c (z)u (z )
UL(�) n n
P
n=1
X
‣ Apply the same coefficients to u H ( � ) to get the bi-fi approximation
m
H H
v(z) = c (z)u (z )
n n
n=1
X
The approximation quality depends on how well the low-fi model approximates
the functional variation of high-fi model in the parameter space
7
Tri-fidelity senario
‣ Run the low-fi model to select the most “important” m points
‣ Run the medium and high-fidelity models at those m points to get
H M
u (�) u (�)
1
‣ Tri-fidelity approximation:

‣ For any given , compute med-fi projection coefficients
z I
z
2
m
M M M M
v(z) 1 = u 1(z) = c 1(z)u 1(z ) more accurate
UM1(�) n n
P coefficients
n=1
X
‣ Apply the same coefficients to H to get the tri-fi approximation
u (�)
m
H M H
v (z) = c 1(z)u (z )
n n
n=1
X
8
1D Burgers’ equation
u + uu = ⌫u , x ( 1, 1),
� U (0, 0.1)
t x xx
2 �
⇠
u( 1) = 1 + �, u(1) = 1.
⌫ = 0.5
� �
low-fi: 100 points (in physical space), FD mean l2 error
high-fi: 1000 points, FD
ref solution: high-fi solution
0
10
−1
10
−2
10
−3
10
−4
10
−5
10
−6
10
0 2 4 6 8 10 12 14
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1D stochastic elliptic equation
(a(Z, x)u (Z, x)) = 1, (Z, x) I (0, 1)
x x Z
� 2 ⇥
u(Z, 0) = 0, u(Z, 1) = 0.
⇢
d
1
(k)
a(Z, x) = 1 + � cos (2⇡kx)Z , d > 1.
k⇡
k=1
X

d=50
d=20
d=10
−2
10
low-fi: 16 points, Cheb Collocation
high-fi:128 points, Cheb Collocation
−3
10
low fidelity: too coarse to
resolve the feature in the
−4 high-D random space
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