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Stochastic Finite Element Methods (FEMs) for PDEModels with Uncertain Inputs
Catherine Powell
University of Manchester
c.powell@manchester.ac.uk
February 1, 2018
Catherine Powell Stochastic FEMs 1 / 33
Uncertainty Quantification (UQ)
Many physical processes are modelled by PDE(s):
B fluid mechanics
B continuum mechanics
B thermodynamics
B etc...
If we know the geometry, boundary conditions, coefficients, initialconditions etc then we can compute often accurate appproximations usingfinite element methods (FEMs).
In many applications, we don’t know all the inputs. One possibility is torepresent uncertain inputs as random variables.
B We have to solve stochastic/parameter-dependent PDEs.
B We need tailored numerical methods (computer models)
Catherine Powell Stochastic FEMs 2 / 33
Uncertainty Quantification (UQ)
Many physical processes are modelled by PDE(s):
B fluid mechanics
B continuum mechanics
B thermodynamics
B etc...
If we know the geometry, boundary conditions, coefficients, initialconditions etc then we can compute often accurate appproximations usingfinite element methods (FEMs).
In many applications, we don’t know all the inputs. One possibility is torepresent uncertain inputs as random variables.
B We have to solve stochastic/parameter-dependent PDEs.
B We need tailored numerical methods (computer models)
Catherine Powell Stochastic FEMs 2 / 33
Finite Element Approximation
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Q1 finite element subdivision
Catherine Powell Stochastic FEMs 3 / 33
Types of Uncertainty
B Model UncertaintyUncertainty in: form of the model, scales of the physical process, missingphysics etc.
B Parameter/Input UncertaintyUncertainty in: coefficients, material parameters, boundary conditions,initial conditions, geometry etc.
B Numerical UncertaintyUncertainty (error) stemming from choice of discretisation, numericalapproximation etc.
Catherine Powell Stochastic FEMs 4 / 33
Forward UQ
Given a probabilistic description of the uncertain inputs
B mean, covariance
B distribution
B . . .
compute an approximation to a quantity of interest (QoI) related to thesolution u, such as
B E[u], E[f (u)], Var(u)
B P(f (u) > tol)
B . . .
and an estimate of the error/uncertainty in that approximation.
Catherine Powell Stochastic FEMs 5 / 33
1. Navier Stokes
Find u(x) (the velocity) and p(x) (the pressure) such that,
− ν∇2u + u · ∇u +∇p = f in D,
∇ · u = 0 in D,
u = g on ∂DD ,
ν∂u
∂n− np = 0 on ∂DN .
If the viscosity is uncertain, we can model it as a single random variable.For example, a Uniform random variable with
E[ν] = µ, Var[ν] = σ2.
Catherine Powell Stochastic FEMs 6 / 33
Example: Flow over a step
µ = 1/50 and σ = 1/500. Streamlines of the mean flow (top) and meanpressure (bottom) computed with a stochastic (Galerkin) FEM.
−1 0 1 2 3 4 5−1
0
1
−0.3
−0.2
−0.1
0
0.1
Catherine Powell Stochastic FEMs 7 / 33
Example: Flow over a step
Variance of the magnitude of velocity (top) and variance of the pressure(bottom) computed with a stochastic (Galerkin) FEM.
−1 0 1 2 3 4 5 −1
0
12
4
6
8
x 10−4
Catherine Powell Stochastic FEMs 8 / 33
2. Groundwater Flow
A simple model for steady-state fluid flow in a porous medium is:
−a∇p = u in D,
∇ · u = f in D,
p = g on ∂DD ,
u · n = 0 on ∂DN = ∂D\∂DD .
Solution variables: p = p(x) (hydraulic head), u = u(x) (velocity field).
a(x) permeability coefficientf (x) source or sink termsg(x) boundary data
D ⊂ Rd spatial/computational domain
Catherine Powell Stochastic FEMs 9 / 33
WIPP Case Study
Measurements a(xi ) are known at 41 spatial locations xi .
605 610 615 620
3570
3575
3580
3585
3590
3595
WIPP Site: Sandia Grid and Bore Locations
in thousands of mE
in th
ousa
nds
of m
N
H1H2H3
H4
H5H6
H7
H9
H10
H11
H12
H14
H15H16
H17
H18
DOE1
DOE2
P14
P15P17
P18
WIPP12WIPP13
WIPP18WIPP19WIPP21WIPP22
WIPP25
WIPP26
WIPP27
WIPP28
WIPP30
ERDA9
CB1
ENGLE
USGS1
D268
AEC7
Catherine Powell Stochastic FEMs 10 / 33
Random Field Model
Let a(x) be a random variable at each point in space.
Given a mean µ(x) and covariance C (x, x′) for a(x), we want to makestatistical predications about the pressure p and velocity u.
Catherine Powell Stochastic FEMs 11 / 33
WIPP (Monte Carlo FEM)
Realisations of the path travelled by a particle released into the flow at afixed location (left). Monte Carlo estimate of the expected flow field (right).
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 104
0
0.5
1
1.5
2
2.5
3
x 104
in mE
in m
N
Individial Particle Paths from Unconditioned Simulations
0 5 10 15 200
5
10
15
20
25
30
km
km
Monte Carlo Mean RT0 flux solution
H1H2H3
H4
H5H6
H7
H9
H10
H11
H12
H14
H15H16
H17
H18
DOE1
DOE2
P14
P15P17
P18
WIPP12WIPP13
WIPP18WIPP19WIPP21WIPP22
WIPP25
WIPP26
WIPP27
WIPP28
WIPP30
ERDA9
CB1
ENGLE
USGS1
D268
AEC7
Catherine Powell Stochastic FEMs 12 / 33
Monte Carlo FEM
A popular black-box strategy for performing forward UQ in PDE models is
B choose a statistical model for the uncertain input(s)
B use random field generators to draw samples of the input(s)
B use FEM codes to solve the PDEs for each sample
B average results to obtain estimates of the mean, variance etc.
Commonly cited benefits include:
B allows reuse of existing FEM codes
B ‘trivially’ paralellisable
B no complicated mathematics!
Catherine Powell Stochastic FEMs 13 / 33
What’s wrong with Monte Carlo FEM?
The error in approximating the expected solution is:
Error =
∥∥∥∥∥E [u]− 1
M
M∑i=1
uh(x, yi )
∥∥∥∥∥
and breaks up into FEM discretisation and MC statistical errors:
Error =
∥∥∥∥∥E [u − uh] +
(E [uh]− 1
M
M∑i=1
uh(x, yi )
)∥∥∥∥∥≤ ‖E [u − uh]‖︸ ︷︷ ︸
FEM error
+
∥∥∥∥∥(E [uh]− 1
M
M∑i=1
uh(x, yi )
)∥∥∥∥∥︸ ︷︷ ︸MC sampling error
≈ O(hα) + O(M−1/2).
Catherine Powell Stochastic FEMs 14 / 33
What’s wrong with Monte Carlo FEM?
The error in approximating the expected solution is:
Error =
∥∥∥∥∥E [u]− 1
M
M∑i=1
uh(x, yi )
∥∥∥∥∥and breaks up into FEM discretisation and MC statistical errors:
Error =
∥∥∥∥∥E [u − uh] +
(E [uh]− 1
M
M∑i=1
uh(x, yi )
)∥∥∥∥∥≤ ‖E [u − uh]‖︸ ︷︷ ︸
FEM error
+
∥∥∥∥∥(E [uh]− 1
M
M∑i=1
uh(x, yi )
)∥∥∥∥∥︸ ︷︷ ︸MC sampling error
≈ O(hα) + O(M−1/2).
Catherine Powell Stochastic FEMs 14 / 33
What’s wrong with Monte Carlo FEM?
To obtainError ≤ TOL,
we need to choose
h = O(TOL1/α), M = O(TOL−2)
Suppose an optimal solver for the FEM system is available. The cost toachieve Error ≤ TOL is:
Cost = M × O(h−d)︸ ︷︷ ︸Cost of FEM solve
= O(TOL−2−d/α)
So, even for a toy problem: −∇ · a∇u = f in 2d with linear FEM:
Cost = O(TOL−4).
Catherine Powell Stochastic FEMs 15 / 33
What’s wrong with Monte Carlo FEM?
To obtainError ≤ TOL,
we need to choose
h = O(TOL1/α), M = O(TOL−2)
Suppose an optimal solver for the FEM system is available. The cost toachieve Error ≤ TOL is:
Cost = M × O(h−d)︸ ︷︷ ︸Cost of FEM solve
= O(TOL−2−d/α)
So, even for a toy problem: −∇ · a∇u = f in 2d with linear FEM:
Cost = O(TOL−4).
Catherine Powell Stochastic FEMs 15 / 33
What’s wrong with Monte Carlo FEM?
To obtainError ≤ TOL,
we need to choose
h = O(TOL1/α), M = O(TOL−2)
Suppose an optimal solver for the FEM system is available. The cost toachieve Error ≤ TOL is:
Cost = M × O(h−d)︸ ︷︷ ︸Cost of FEM solve
= O(TOL−2−d/α)
So, even for a toy problem: −∇ · a∇u = f in 2d with linear FEM:
Cost = O(TOL−4).
Catherine Powell Stochastic FEMs 15 / 33
Alternative Stochastic FEMs
There are many alternative schemes that can exploit existing FEM codes.
B Sampling methods- Multifidelity & Multilevel Monte Carlo methods.- Stochastic collocation FEMs (SC-FEMs)- Reduced basis FEMs (RB-FEMs)
B Not sampling methods- Stochastic Galerkin FEMs (SG-FEMs)
SC-FEMs and SG-FEMs are based on polynomial approximation.
RB-FEMs can be combined with any sampling method.
Catherine Powell Stochastic FEMs 16 / 33
Stochastic Galerkin FEM
We compute an approximation of the form
u(x, y) ≈N∑j=1
uj(x)ψj(y)
where {ψ1, . . . , ψN} are polynomials in the parameters y (Polynomial
Chaos) by solving one linear system,
Au = f
of dimension NNh × NNh.
B Nh is the size of the FEM problem
B N depends on no. of parameters and chosen polynomial degree
Catherine Powell Stochastic FEMs 17 / 33
Example: Flow over a step
PC component 0 norm=9.9995e−01
−1 0 1 2 3 4 5−1
0
1PC component 1 norm=2.5326e−02
−1 0 1 2 3 4 5−1
0
1
PC component 2 norm=2.0076e−03
−1 0 1 2 3 4 5−1
0
1PC component 3 norm=1.8752e−04
−1 0 1 2 3 4 5−1
0
1
PC component 4 norm=1.5520e−05
−1 0 1 2 3 4 5−1
0
1PC component 5 norm=1.5102e−06
−1 0 1 2 3 4 5−1
0
1
Catherine Powell Stochastic FEMs 18 / 33
Estimating the vorticity
Components of the vorticity along the bottom channel wall, E[Re] ≈ 296.
0 5 10 15−1
−0.5
0
0.5
1
x
PC component 0
0 5 10 15−0.2
00.20.40.6
x
PC component 1
0 5 10 15−0.4
−0.2
0
0.2
0.4
x
PC component 2
0 5 10 15−0.2
0
0.2
x
PC component 3
0 5 10 15−0.04−0.02
00.020.040.06
x
PC component 4
0 5 10 15−0.02
0
0.02
x
PC component 5
Catherine Powell Stochastic FEMs 19 / 33
Standard deviation of the vorticity
Here, just one parameter (viscosity), ν = µ+ σy . Quantity of interest:
Q(ν(y)) :=
∫ 7
3
ω(x, y)ds
The standard deviation of Q increases as we increase σ and we need morepolynomials to accurately compute the QoI.
N = 2 N = 3 N = 4 N = 5 N = 6σ = µ/10 0.6553 0.6492 0.6491 * *σ = 2µ/10 1.2639 1.2150 1.2139 1.2138 *σ = 3µ/10 1.7752 1.6197 1.6180 1.6167 1.6162σ = 4µ/10 2.1379 1.8133 1.8320 1.8287 1.8291
Catherine Powell Stochastic FEMs 20 / 33
Reduced Basis FEM
If using a sampling method, for each y of interest, if we use standard FEM(high fidelity methods),
u(x, y) ≈ uh(x, y)
we have to solve a sparse linear system of Nh equations
A(y)︸︷︷︸Nh×Nh
u(y) = f.
If our UQ method (e.g. Monte Carlo) uses M samples, and we have anoptimal iterative solver for each system, then the cost is
M︸︷︷︸Live with this!
× O(Nh)︸ ︷︷ ︸Try to reduce this!
Catherine Powell Stochastic FEMs 21 / 33
Reduced Basis FEM
If using a sampling method, for each y of interest, if we use standard FEM(high fidelity methods),
u(x, y) ≈ uh(x, y)
we have to solve a sparse linear system of Nh equations
A(y)︸︷︷︸Nh×Nh
u(y) = f.
If our UQ method (e.g. Monte Carlo) uses M samples, and we have anoptimal iterative solver for each system, then the cost is
M︸︷︷︸Live with this!
× O(Nh)︸ ︷︷ ︸Try to reduce this!
Catherine Powell Stochastic FEMs 21 / 33
Basic Idea: Reduced Problem
Instead, for each y of interest we now seek to approximate:
uh(x, y) ≈ uR(x, y)
where uR(x, y) belongs to a lower-dimensional space.
B Run the expensive FEM code NR times with specially chosen sets ofinput parameters yi .
B Collect the solution vectors (snapshots) into a matrix
V = [u(y1),u(y2), · · · ,u(yNR)].
B Make the columns orthonormal to produce a new matrix Q
V → Q = [q1,q2, · · · ,qNR].
which has size Nh × NR (tall and skinny).
Catherine Powell Stochastic FEMs 22 / 33
Basic Idea: Reduced Problem
Instead, for each y of interest we now seek to approximate:
uh(x, y) ≈ uR(x, y)
where uR(x, y) belongs to a lower-dimensional space.
B Run the expensive FEM code NR times with specially chosen sets ofinput parameters yi .
B Collect the solution vectors (snapshots) into a matrix
V = [u(y1),u(y2), · · · ,u(yNR)].
B Make the columns orthonormal to produce a new matrix Q
V → Q = [q1,q2, · · · ,qNR].
which has size Nh × NR (tall and skinny).
Catherine Powell Stochastic FEMs 22 / 33
Standard Snapshot Basis
For each new y of interest, approximate u(y) ≈ QuR(y), where uR(y) isfound by solving the reduced system
(Q>A(y)Q
)︸ ︷︷ ︸NR×NR
uR(y) = Q>f
with uR(y) ∈ RNR .
The reduced problem can be used as a surrogate.
Catherine Powell Stochastic FEMs 23 / 33
Test problem 1: Groundwater Flow
5
4
3
2
1
0.2
0.4
0.6
0.8
High-Fidelity Mixed FEM: Nh = 217, 464.
Catherine Powell Stochastic FEMs 24 / 33
Timings: System Solves
Average time in seconds to assemble and solve the high fidelity FEMsystems (T) and reduced basis FEM systems (t):
ε1 NR T t1e-3 36 2.13e0 5.57e-41e-4 59 2.14e0 1.06e-31e-5 84 2.14e0 1.75e-31e-6 108 2.08e0 5.03e-3
Catherine Powell Stochastic FEMs 25 / 33
High Fidelity vs Reduced Basis
Total time to compute mean and variance of FEM solution.
ε = 10−5 RB-FEM Standard FEMM NR offline online high fidelity
351 64 510.9 0.4 746.01471 84 795.0 2.6 3058.05503 96 1,002.1 14.3 11,523.5
Note: In this problem, we have only 5 parameters and a nice structure. Whenoffline times are taken into account, 10 times faster than standard FEM.
Catherine Powell Stochastic FEMs 26 / 33
Test Problem 2
Now we have 12 parameters, and a more complicated PDE structure.
ε = 10−5 RB-FEM Standard FEMM NR offline online high fidelity
2,649 52 1548.9 8.8 3,182.517,265 53 9,327.1 57.6 20,746.193,489 53 53,685.8 300.0 116,049.5
When offline times are taken into account, only twice as fast.
Catherine Powell Stochastic FEMs 27 / 33
Summary
B Applied mathematicians are mainly concerned with the accuracy ofoutputs of computer models with respect to the true solution of thecontinuous physical model.
B Standard Monte Carlo is infeasible for computer models consisting ofFEM discretizations for PDE models.
B Stochastic Galerkin FEM (polyomial chaos type methods) may beappropriate if there are a modest no. of input parameters and the truesolution is well behaved enough.
B Building surrogate FEM models using reduced basis ideas might offersome savings if time can be invested in the offline stage.
Catherine Powell Stochastic FEMs 28 / 33
Standard Reduced Basis Methods
� A. Quarteroni, A. Manzoni, F. Negri, Reduced basis methods forPDEs: an introduction, Springer (2015)
� J. Hesthaven, G. Rozza B. Stamm, Certified Reduced Basis Methodsfor Parameterized PDEs, Springer (2015).
� M. Barrault, Y. Maday, N. Nguyen, A.T. Patera, An empiricalinterpolation method: application to efficient reduced basis discretizationof PDEs, Comptes Rendus Mathematique 339(9), 2004.
� G. Rozza, K. Veroy, On the stability of the reduced basis method forStokes equations in parametrized domains, Computer methods inapplied mechanics and engineering, 196(7), 2007.
Catherine Powell Stochastic FEMs 29 / 33
Stochastic Collocation
� I. Babuska, F. Nobile, R. Tempone, A Stochastic Collocation Methodfor Elliptic Partial Differential Equations with Random Input Data ,SIAM. Journal on Numerical Analysis, 45(3), 2007.
� F. Nobile, R. Tempone, C. Webster , A Sparse Grid StochasticCollocation Method for PDEs with Random Input Data, SIAM. Journalon Numerical Analysis, 46(2), 2008.
� D. Xiu, J. Hesthaven, High-Order Collocation Methods for DifferentialEquations with Random Inputs, SIAM. Journal on ScientificComputing, 27(3), 2005.
Catherine Powell Stochastic FEMs 30 / 33
RBMs + Stochastic Collocation
� H.C. Elman & Q. Liao, Reduced Basis Collocation Methods for PDEswith Random Coefficients, SIAM/ASA Journal on UQ, 1(1), 2013.
� P. Chen, A. Quarteroni & G. Rozza, Comparison Between ReducedBasis and Stochastic Collocation Methods for Elliptic Problems, Journalof Scientific Computing, 59(1), 2014.
� C. Newsum & C.E. Powell, Efficient reduced basis methods for saddlepoint problems with applications in groundwater flow. (2016). MIMSEPrint: 2016.60.
Catherine Powell Stochastic FEMs 31 / 33
Offline vs Online
RB methods decompose the computational work into two stages.
� Offline: The reduced basis matrix Q is constructed, and any matrices &vectors that are too expensive to compute online.
Warning: Offline costs may be very expensive!
� Online: All the reduced systems are solved,(QTA(yi )Q
)uR(yi ) = f, i = 1, . . . ,N,
and quantities of interest computed.
Key Philosophy: Costs should be independent of Nh.
Catherine Powell Stochastic FEMs 32 / 33
Standard Greedy Method
Training set Θtrain ⊂ Γ, error tolerance ε, error estimator ∆R(y).Initialise Q = [u(y0)] with some y0 ∈ Γ.while maxy∈Θtrain
∆R(y) > ε doFind yk = arg maxy∈Θtrain
∆R(y)Solve high-fidelity problem for u(yk)Set Q = [Q,u(yk)] & Orthonormalise
end
If the error estimator satisfies
‖uh(y)− uR(y)‖Vh≤ ∆R(y)
then we say the method is certified.
Catherine Powell Stochastic FEMs 33 / 33
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