An adaptive hierarchical sparse grid
collocation algorithm for the solution of
stochastic differential equations
Nicholas Zabaras and Xiang Ma
Materials Process Design and Control Laboratory
Sibley School of Mechanical and Aerospace Engineering
101 Frank H. T. Rhodes Hall
Cornell University
Ithaca, NY 14853-3801
Email: zabaras@cornell.edu
URL: http://mpdc.mae.cornell.edu/
Symposium on `Numerical Methods for Stochastic Computation and Uncertainty Quantification'
in the 2008 SIAM Annual Meeting, San Diego CA, July 7-11, 2008
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 1
Outline of the presentation
o Stochastic analysis
o Issues with gPC and ME-gPc
o Collocation based approach
o Basics of stochastic collocation
o Issues with existing stochastic sparse grid collocation method
(Smolyak algorithm)
o Adaptive sparse grid collocation method
o Some numerical examples
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 2
Motivation
All physical systems have an inherent associated randomness
SOURCES OF UNCERTAINTIES
Process
•Multiscale material information –
inherently statistical in nature.
Engineering
component
•Uncertainties in process conditions
•Input data
•Model formulation – approximations,
assumptions.
Heterogeneous
random
Microstructural
features
Why uncertainty modeling ?
Assess product and process reliability.
Estimate confidence level in model predictions.
Identify relative sources of randomness.
Control?
Provide robust design solutions.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 3
Problem definition
 Define a complete probability space  , F , P . We are interested to find a
u :  D  such that for P-almost everywhere (a.e.)
stochastic function
the following equation hold:
,
 
L  x, ; u   f  x,   , x  D
B  x, u   g  x  ,
where x   x1 , x2 , , xd  are the coordinates in
operator, and B is a boundary operators.
x  D
d
, L is (linear/nonlinear) differential
 In the most general case, the operator L and B as well as the driving terms f and
g, can be assumed random.
 In general, we require an infinite number of random variables to completely
characterize a stochastic process. This poses a numerical challenge in modeling
uncertainty in physical quantities that have spatio-temporal variations, hence
necessitating the need for a reduced-order representation.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 4
Representing Randomness
Interpreting random variables as functions
1. Interpreting random variables
2. Distribution of the random variable
Random
variable x
E.g. inlet velocity, inlet temperature
MAP
S
Real line
Sample space of
elementary events
Collection of all
possible outcomes
  o 1  0.1x 
Each outcome is
mapped to a
corresponding real
value
A general stochastic process is a random field
with variations along space and time – A
function with domain (Ω, Τ, S)
3. Correlated data
Ex. Presence of impurities, porosity
Usually represented with a correlation function
We specifically concentrate on this.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 5
Representing Randomness
Karhunen-Loèvè expansion
Based on the spectral decomposition of the
covariance kernel of the stochastic process
1. Representation of random process
- Karhunen-Loeve, Polynomial Chaos
expansions
2. Infinite dimensions to finite dimensions
Random
process
- depends on the covariance
Mean
Set of random
variables to
be found
• Need to know covariance
• Converges uniformly to
any second order process
Eigenpairs of
covariance
kernel
Set the number of stochastic dimensions, N
Dependence of variables
Pose the (N+d) dimensional problem
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 6
Karhunen-Loeve expansion

X ( x, t ,  )  X ( x, t )   X i ( x, t )Yi ( )
i 1
ON random variables
Deterministic functions
Stochastic Mean
process function
 Deterministic functions ~ eigen-values, eigenvectors of
the covariance function
 Orthonormal random variables ~ type of stochastic
process
 In practice, we truncate (KL) to first N terms
X ( x, t ,  )  fn( x, t , Y1 ,
CORNELL
U N I V E R S I T Y
, YN )
Materials Process Design and Control Laboratory 7
The finite-dimensional noise assumption
 By using the Doob-Dynkin lemma, the solution of the problem can be described
by the same set of random variables, i.e.
u  x,    u  x, Y1   ,
, YN   
 So the original problem can be restated as: Find the stochastic function u :  D 
such that
L  x, Y; u   f  x, Y  , x  D  
B  x, Y; u   g  x, Y  ,
x  D  
 In this work, we assume that Yi  i 1 are independent random variables with
probability density function i . Let  i be the image of Yi . Then
N
N
  Y    i Yi , Y  
i 1
Y  Y1 ,
is the joint probability density of
, YN  with
support
N
   i 
N
i1
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 8
Uncertainty analysis techniques
 Monte-Carlo : Simple to implement, computationally
expensive
 Perturbation, Neumann expansions : Limited to small
fluctuations, tedious for higher order statistics
 Sensitivity analysis, method of moments : Probabilistic
information is indirect, small fluctuations
 Spectral stochastic uncertainty representation: Basis in
probability and functional analysis, can address second
order stochastic processes, can handle large fluctuations,
derivations are general
 Stochastic collocation: Results in decoupled equations
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 9
Spectral stochastic presentation
 A stochastic process = spatially, temporally varying
random function
X : ( x, t , )
CHOOSE APPROPRIATE
BASIS FOR THE
PROBABILITY SPACE
HYPERGEOMETRIC ASKEY POLYNOMIALS
PIECEWISE POLYNOMIALS (FE TYPE)
GENERALIZED POLYNOMIAL
CHAOS EXPANSION
SUPPORT-SPACE
REPRESENTATION
SPECTRAL DECOMPOSITION
KARHUNEN-LOÈVE
EXPANSION
COLLOCATION, MC (DELTA FUNCTIONS)
SMOLYAK QUADRATURE,
CUBATURE, LH
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 10
Generalized polynomial chaos (gPC)
 Generalized polynomial chaos expansion is used to
represent the stochastic output in terms of the input
X ( x, t ,  )  fn( x, t , Y1 ,
P
, YN )
Stochastic input
Z ( x, t ,  )   Z i ( x, t ) i (Y ( ))
i 0
Stochastic
output
Askey polynomials in input
Deterministic functions
 Askey polynomials ~ type of input stochastic process
 Usually, Hermite, Legendre, Jacobi etc.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 11
Generalized polynomial chaos (gPC)
Advantages:
1) Fast convergence1.
2) Very successful in solving various problems in solid and fluid
mechanics – well understood 2,3.
3) Applicable to most random distributions 4,5.
4) Theoretical basis easily extendable to other phenomena
1.
2.
3.
4.
5.
P Spanos and R Ghanem, Stochastic finite elements: A spectral approach, Springer-Verlag (1991)
D. Xiu and G. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos,
Comput. Methods Appl. Mech. Engrg. 187 (2003) 137--167.
S. Acharjee and N. Zabaras, Uncertainty propagation in finite deformations -- A spectral stochastic
Lagrangian approach, Comput. Methods Appl. Mech. Engrg. 195 (2006) 2289--2312.
D. Xiu and G. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations,
SIAM J. Sci. Comp. 24 (2002) 619-644.
X. Wan and G.E. Karniadakis, Beyond Wiener-Askey expansions: Handling arbitrary PDFs, SIAM J
Sci Comp 28(3) (2006) 455-464.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 12
Generalized polynomial chaos (gPC)
Disadvantages:
1) Coupled nature of the formulation makes implementation nontrivial1.
2) Substantial effort into coding the stochastics: Intrusive
3) As the order of expansion (p) is increased, the number of unknowns
(P) increases factorially with the stochastic dimension (N):
combinatorial explosion2,3.
P 1 
 N  p !
N ! p!
4) Furthermore, the curse of dimensionality makes this approach
particularly difficult for large stochastic dimensions
1.
2.
3.
B. Ganapathysubramaniana and N. Zabaras, Sparse grid collocation schemes for stochastic natural
convection problems, J. Comp Physics, 2007, doi:10.1016/j.jcp.2006.12.014
D Xiu, D Lucor, C.-H. Su, and G Karniadakis, Performance evaluation of generalized polynomial
chaos, P.M.A. Sloot et al. (Eds.): ICCS 2003, LNCS 2660 (2003) 346-354
B. J. Debusschere, H. N. Najm, P. P. Pebray, O. M. Knio, R. G. Ghanem and O. P. Le Maître,
Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes,
SIAM Journal of Scientific Computing 26(2) (2004) 698-719.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 13
Failure of gPC

Gibbs phenomenon: Due to its global support, it fails to capture the
stochastic discontinuity in the stochastic space.
0.25
0
0.25
0.5
X

0.75
1
9.3E-07
1.4E-06
1.9E-06
2.4E-06
2.9E-06
0.75
Y
0.5
0
-6.4E-08 4.3E-07
1
Y-velocity
0.75
Y
Capture unsteady
equilibrium in
natural convection
X-velocity
9.2E-07 5.7E-06 1.0E-05 1.5E-05 2.0E-05 2.5E-05 3.0E-05 3.4E-05
1
0.5
0.25
0
0
0.25
0.5
0.75
1
X
Therefore, we need some method which can resolve the
discontinuity locally. This motivates the following Multi - element
gPC method.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 14
3.4E-06
Multi-element generalized polynomial chaos (ME-gPC)
 Basic concept:
 Decompose the space of random inputs into small elements.
 In each element of the support space, we generate a new random
variable and apply gPCE.
 The global PDF is also simultaneously discretized. Thus, the global PC
basis is not orthogonal with the local PDF in a random element. We
need to reconstruct the orthogonal basis numerically in each random
element. The only exception is the uniform distribution whose PDF is a
constant.
 We can also decompose the random space adaptively to increase its
efficiency.
1.
X. Wan and G.E. Karniadakis, An adaptive multi-element generalized polynomial chaos method for
stochastic differential equations, SIAM J Sci Comp 209 (2006) 901-928.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 15
Decomposition of the random space
 We define a decomposition B of  as
 k  [ak ,1 , bk ,1 )  [ak ,2 , bk ,2 ) 

B    Mk1  k

 k1  k2  
 [ak ,d , bk ,d ]
where k1 , k2  1, 2 , M
 Based on the decomposition B , we define the following indicator
random variables:
Y k ,
1
k  1, 2 , M
I k  
0 otherwise
 Then a local random vector Yi is defined in each element  i subject to
a conditional PDF
fˆk (Yk | I k  1) 
where
CORNELL
U N I V E R S I T Y
f (Yk )
Pr( I k  1)
J k  Pr( I Bk  1)  0
Materials Process Design and Control Laboratory 16
An adaptive procedure
 Let us assume that the gPC expansion of random field in element k is
Np
uˆk (ξ k )   uˆk ,i k ,i (ξ k )
i 0
 From the orthogonality of gPC, the local variance approximated by PC
with order p is given by
Np
2
 k , p   uˆk2,i  i2
i 1
 The approximate global mean u and variance  2 can be expressed by
N
N
u   uˆk ,0 Pr( I Bk  1),
   [ k2, p  (uˆk ,0  u ) 2 ]Pr( I B  1)
2
k 1
k 1
k
 Let  k denote the error of local variance. We can write the exact global
variance as
N
 2   2    k Pr( I B  1)
k 1
CORNELL
U N I V E R S I T Y
k
Materials Process Design and Control Laboratory 17
An adaptive procedure
 We define the local decay rate of relative error of the gPC approximation
in each element as follows:
k 

2
k, p

 k2, p
2
k , p 1


Np
 i2
uˆ
2
i  N p1 1 k ,i
 k2, p
 Based on  k and the scaled parameter Pr( I B  1) , a random element will
be selected for potential refinement when the following condition is
satisfied

k
k Pr( I B  1)  1,
k
0   1
where  and 1 are prescribed constants.
 Furthermore, in the selected random element k , we use another
threshold parameter 2 to choose the most sensitive random dimension.
We define the sensitivity of each random dimension as
(uˆi , p ) 2  i2, p
ri  N
,
i  1, 2, , d ,
2
2
 j  N 1 uˆ j  j
where we drop the subscript k for clarity and the subscript i , p denotes
the mode consisting only of random dimension Yi with polynomial order
p
p 1
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 18
p
An adaptive procedure
 All random dimensions which satisfy
ri  2  max rj ,
j 1, d
0  2  1, i  1, 2,
, d,
will be split into two equal random elements in the next time step
while all other random dimensions will remain unchanged.
 If Yk corresponds to element [a, b] in the original random space [1,1] , the
ab
ab
element [a, 2 ] and [ 2 , b] will be generated in the next level if the two
criteria are both satisfied.
 Through this adaptive procedure, we can reduce the total element
number while gaining the same accuracy and efficiency.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 19
An Example: K-O problem
So it can successfully resolve the discontinuity in stochastic space.
However, due to its gPC nature in each element and the tree like adaptive
refinement, this method still suffers from the curse of dimensionality.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 20
First summary
 The traditional gPC method fails when there are steep gradients/finite
discontinuities in stochastic space.
 gPC related method suffers from the so called curse of dimensionality
when the input stochastic dimension is large. High CPU cost for large
scale problems.
 Although the h-adaptive ME-gPC can successfully resolve discontinuity in
stochastic space, the number of element grows fast at the order O (2 N ) . So
the method is still a dimension-dependent method.
 Therefore, an adaptive framework utilizing local basis functions that
scales linearly with increasing dimensionality and has the decoupled
nature of the MC method, offers greater promise in efficiently and
accurately representing high-dimensional non-smooth stochastic
functions.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 21
Collocation based framework
The stochastic variable at a point in the domain is an N dimensional
N

function. f :
The gPC method approximates this N dimensional function using a
spectral element discretization (which causes the coupled set of
equations)
Spectral descretization represents the
function as a linear combination of sets of
orthogonal polynomials.
Instead of a spectral discretization, use
other kinds of polynomial approximations
Use simple interpolation!
f:
N

CORNELL
U N I V E R S I T Y
That is, sample the function at some points
and construct a polynomial approximation
Materials Process Design and Control Laboratory 22
Collocation based framework
Need to represent this function
f:
N

Sample the function at a finite
(i ) M
set of points Θ M  {Y }i1
Use polynomials (Lagrange
polynomials) to get a
approximate
representation
M
I f (Y)   f (Y (i ) )ai (Y (i ) )
i 1
Function value at any point is
simply I f (x )
Spatial domain is approximated using a FE, FD, or FV discretization.
Stochastic function in 2 dimensions
Stochastic domain is approximated using multidimensional interpolating
functions
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 23
From one-dimension to higher dimensions
 Denote the one dimensional interpolation formula as
with the set of support nodes:
 In higher dimension, a simple case is the tensor product formula
For instance, if M=10 dimensions and we use k points in each direction
This quickly becomes impossible to use.
Number of points
Total number of
in each direction, k
sampling points
2
1024
3
59049
4
1.05x106
5
9.76x106
10
1x1010
CORNELL
U N I V E R S I T Y
One idea is only to pick the
most important points from the
tensor product grid.
Materials Process Design and Control Laboratory 24
Conventional sparse grid collocation (CSGC)
In higher dimensions, not all the points are
equally important.
Some points contribute less to the
accuracy of the solution (e.g. points where
the function is very smooth, regions where
the function is linear, etc.). Discard the
points that contribute less: SPARSE GRID
COLLOCATION
1D sampling
Tensor
product
2D sampling
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 25
Conventional sparse grid collocation
 Both piecewise linear as well as polynomial functions, L, can be used to
construct the full tensor product formula above.
 The Smolyak construction starts from this full tensor product
representation and discards some multi-indices to construct the minimal
representation of the function
 Define the differential interpolant, ∆ as the difference between two
interpolating approximations
with U 0  0
 i  U i  U i 1
 The sparse grid interpolation of the function f is given by
Aq , N ( f )   (i1 
iN )( f )
i q
where i  i1   id , is the multi-index representation of the support
nodes. N is the dimension of the function f and q is an integer (q>N).
k = q-N is called the depth of the interpolation. As q is increased more and
more points are sampled.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 26
Conventional sparse grid collocation
 The interpolant can be represented in a recursive manner as
Aq, N ( f )   (i1 
i q
iN )( f )  Aq 1, N ( f )   (i1 
i q
iN )( f )
with
AN 1, N  0
 Can increase the accuracy of the interpolant based on Smolyak’s
construction WITHOUT having to discard previous results.
 Can build on previous interpolants due to the recursive formulation
 To compute the interpolant Aq , N , one only needs the function values at
the sparse grid point given by:
H q,N 
( X i1  ...  X iN )
q  N 1|i| q
One could select the sets X i , in a nested fashion such that X i  X i 1
 Similar to the interpolant representation in a recursive form, the
sparse grid points can be represented as:
H q , M  H q 1, M H q , M
with
CORNELL
U N I V E R S I T Y
H q , M 
|i|  q
( X i1  ...  X iM )
i
i
i 1
and X   X \ X
Materials Process Design and Control Laboratory 27
Choice of collocation points and nodal basis functions
 One of the choice is the Clenshaw-Curtis grid at non-equidistant extrema
of the Chebyshev polynomials. ( D.Xiu etc, 2005, F. Nobile etc, 2006)
 By doing so, the resulting sets are nested. The corresponding univariate
nodal basis functions are Lagrangian characteristic polynomials:
 There are two problems with this grid:
(1) The basis function is of global support, so it will fail when there is
discontinuity in the stochastic space.
(2) The nodes are pre-determined, so it is not suitable for implementation
of adaptivity.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 28
Choice of collocation points and nodal basis functions
 Therefore, we propose to use the Newton-Cotes grid using equidistant
support nodes and use the linear hat function as the univariate nodal
1
basis.
Yji  21i
i
1i
Y ji Y j  2
 It is noted that the resulting grid points are also nested and the grid has
the same number of points as that of the Clenshaw-Curtis grid.
 Comparing with the Lagrangian polynomial, the piecewise linear hat
function has a local support, so it is expected to resolve discontinuities in
the stochastic space.
 The N-dimensional multi-linear basis functions can be constructed using
tensor products:
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 29
Comparison of the two grids
Clenshaw-Curtis grid
CORNELL
U N I V E R S I T Y
Newton-Cotes grid
Materials Process Design and Control Laboratory 30
From nodal basis to multivariate hierarchical basis
 We start from the 1D interpolating formula. By definition, we have
with
and
we can obtain
since
we have
For simplifying the notation, we consecutively number the elements in
and denote the j-th point of X i as Y ji
X i
hierarchical
surplus
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 31
,
Nodal basis vs. hierarchical basis
Nodal basis
CORNELL
U N I V E R S I T Y
Hierarchical basis
Materials Process Design and Control Laboratory 32
Multi-dimensional hierarchical basis
 For the multi-dimensional case, we define a new multi-index set
which leads to the following definition of hierarchical basis
 Now apply the 1D hierarchical interpolation formula to obtain the sparse
grid interpolation formula for the multivariate case in a hierarchical form:
Here we define the hierarchical surplus as:
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 33
Error estimate
Depending on the order of the one-dimensional interpolant, one can come
up with error bounds on the approximating quality of the interpolant1,2.
If piecewise linear 1D interpolating functions are used to construct the
sparse interpolant, the error estimate is given by:
Where N is the number of sampling points,
If 1D polynomial interpolation functions are used, the error bound is:
assuming that f has k continuous derivatives
1.
2.
V. Barthelmann, E. Novak and K. Ritter, High dimensional polynomial interpolation on sparse grids,
Adv. Comput. Math. 12 (2000), 273–288
E. Novak, K. Ritter, R. Schmitt and A. Steinbauer, On an interpolatory method for high dimensional
integration, J. Comp. Appl. Math. 112 (1999), 215–228
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 34
Hierarchical integration
 So now, any function u  can now be approximated by the following
reduced form
 It is just a simple weighted sum of the value of the basis for all collocation
points in the current sparse grid. Therefore, we can easily extract the
useful statistics of the solution from it.
 For example, we can sample independently N times from the uniform
distribution [0,1] to obtain one random vector Y, then we can place this
vector into the above expression to obtain one realization of the solution.
In this way, it is easy to plot realizations of the solution as well as the PDF.
 The mean of the random solution can be evaluated as follows:
where the probability density function   Y  is 1 since we have assumed
uniform random variables on a unit hypercube [0,1]N .
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 35
Hierarchical integration
 The 1D version of the integral in the equation above can be evaluated
analytically:
 Since the random variables are assumed independent of each other, the
value of the multi-dimensional integral is simply the product of the 1D
integrals.
 Denoting
we rewrite the mean as
 Thus, the mean is just an arithmetic sum of the hierarchical surpluses and
the integral weights at each interpolation point.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 36
Hierarchical integration
 To obtain the variance of the solution, we need to first obtain an
approximate expression for u 2
 Then the variance of the solution can be computed as
 Therefore, it is easy to extract the mean and variance analytically, thus
we only have the interpolation error. This is one of the advantages to use
this interpolation based collocation method rather than quadrature based
collocation method.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 37
Adaptive sparse grid collocation (ASGC)
 Let us first revisit the 1D hierarchical interpolation
 For smooth functions, the hierarchical
surpluses tend to zero as the interpolation
level increases.
 On the other hand, for non-smooth function,
steep gradients/finite discontinuities are
indicated by the magnitude of the hierarchical
surplus.
 The bigger the magnitude is, the stronger the
underlying discontinuity is.
 Therefore, the hierarchical surplus is a natural
candidate for error control and implementation
of adaptivity. If the hierarchical surplus is
larger than a pre-defined value (threshold), we
simply add the 2N neighbor points of the
current point.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 38
Adaptive sparse grid collocation
 As we mentioned before, by using the equidistant nodes, it is easy to
refine the grid locally around the non-smooth region.
 It is easy to see this if we consider the 1D equidistant points of the sparse
grid as a tree-like data structure
 Then we can consider the interpolation
level of a grid point Y as the depth of the
tree D(Y). For example, the level of point
0.25 is 3.
 Denote the father of a grid point as F(Y),
where the father of the root 0.5 is itself,
i.e. F(0.5) = 0.5.
 Thus the conventional sparse grid in Ndimensional random space can be
reconsidered as
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 39
Adaptive sparse grid collocation
 Thus, we denote the sons of a grid point Y  Y1 , , YN  by
 From this definition, it is noted that, in general, for
each grid point there are two sons in each
dimension, therefore, for a grid point in a Ndimensional stochastic space, there are 2N sons.
 The sons are also the neighbor points of the father,
which are just the support nodes of the hierarchical
basis functions in the next interpolation level.
 By adding the neighbor points, we actually add the
support nodes from the next interpolation level, i.e.
we perform interpolation from level |i| to level |i|+1.
 Therefore, in this way, we refine the grid locally
while not violating the developments of the
Smolyak algorithm
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 40
Adaptive sparse grid collocation
 To construct the sparse grid, we can either work through the multi-index
or the hierarchical basis. However, in order to adaptively refine the grid,
we need to work through the hierarchical basis. Here we first show the
equivalence of these two ways in a two dimensional domain.
 We always start from the multi-index (1, 1), which means the interpolation
level in first dimension is 1, in second dimension is also 1.
 For level 1, the associated point is only 0.5.
 So the tensor product of the coordinate is
0.5  0.5  (0.5, 0.5)
 Thus, for interpolation level 0 of smolyak
construction, we have
(1,1)
A( N , N )  w(0.5,0.5)a(1,1)
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 41
Adaptive sparse grid collocation
 Recalling, we define the interpolation level k as AN  k , N .
 So, now we come to the next interpolation level, k  1, i  3
For this requirement, the following two multi-indices are satisfied:
(1, 2), (2,1)
Let us look at the index (1,2)
i1  1 The associated set of points is 0.5 (j=1).
i2  2 The associated set of points is 0 (j=1) and 1 (j=2).
So the tensor product is
(0.5)  (0,1)  (0.5, 0), (0.5,1)
Then the associated sum is
(1,2)
1,2
w(0.5,0)a(1,1)
 w(0.5,1)a1,2
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 42
Adaptive sparse grid collocation
 In the same way, for multi-index (2,1), we have the corresponding
collocation points
(0,1)  (0.5)  (0, 0.5), (1, 0.5)
So the associated sum is
(2,1)
2,1
w(0,0.5)a(1,1)
 w(1,0.5)a2,1
i
Recall Aq , N ( f )   ( 1 
iN )( f )
i q
So for Smolyak interpolation level 1, we have

A21,2    i1 
i 3

(1,1)
(1,2)
  iN ( f )  w(0.5, 0.5) a(1,1)
 w(0.5, 0) a(1,1)
1,2
(2,1)
2,1
 w(0.5,1) a1,2
 w(0, 0.5) a(1,1)
 w(1, 0.5) a2,1
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 43
Adaptive sparse grid collocation
 So, from this equation, it is noted that we can store the points with the
surplus sequentially. When calculating interpolating value, we only
need to perform a simple sum of each term. We don’t need to search
for the point. Thus, it is very efficient to generate the realizations.
 Here, we give the first three level of multi-index and corresponding nodes.
(1,1)  (0.5, 0.5)
(1, 2)  (0.5, 0), (0.5,1)
(2,1)  (0, 0.5), (1, 0.5)
(2, 2)  (0, 0), (1, 0), (0,1), (1,1)
(1,3)  (0.5, 0.25), (0.5, 0.75)
(3,1)  (0.25, 0.5), (0.75, 0.5)
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 44
Adaptive sparse grid collocation
 In fact, the most important formula is the following
From this equation, in order to calculate the surplus, we need the
interpolation value at just previous level. That is why it requires that we
construct the sparse grid level by level.
 Now, let us revisit the algorithm in a different view. we start from the
hierarchical basis. We still use the two-dimension grid as an example.
Recalling
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 45
Adaptive sparse grid collocation
 As previous given, the first level in a sparse grid is always a point
(0.5,0.5). There is a possibility that the function value at this point is 0 and
thus the refinement terminates immediately. In order to avoid the early
stop for the refinement process, we always refine the first level and keep
a provision on the first few hierarchical surpluses.
 It is noted here, the associated multi-index with this point (0.5, 0.5) is
(1,1),i.e. (0.5, 0.5)  (1,1) and it corresponds to level of interpolation k = 0.
So the associated sum is still
(1,1)
A( N , N )  w(0.5,0.5)a(1,1)
 Now if the surplus is larger than the threshold, we add sons
(neighbors) of this point to the grid.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 46
Adaptive sparse grid collocation
 In one-dimension, the sons of the point 0.5 are 0,1.
According to the definition, the sons of the point (0.5,0.5) are
(0, 0.5), (1, 0.5), (0.5, 0), (0.5,1)
which are just the points in the level 1 of sparse grid. Indeed,
the associated multi-index is
(0, 0.5), (0,1)  (2,1), (0.5, 0), (0.5,1)  (1, 2)
Recalling when we start from multi-index
(1, 2)  (0.5, 0), (0.5,1), (2,1)  (0, 0.5), (1, 0.5)
So actually, we can see they are equivalent.
 Now, let us summarize what we have
k  1: (0, 0.5), (1, 0.5), (0.5, 0), (0.5,1)
So the associated sum is
(1,2)
1,2
(2,1)
2,1
w(0.5,0)a(1,1)
 w(0.5,1)a1,2
 w(0,0.5)a(1,1)
 w(1,0.5)a2,1
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 47
Adaptive sparse grid collocation
 After calculating the hierarchical surplus, now we need to refine.
Here we assume we refine all of the points, i.e. threshold is 0.
 Notice here, all of the son points are actually from next
interpolation level, so we still construct the sparse grid level by level.
 First, in one dimension, the sons of point 0 is 0.25, 1 is 0.75.
 So the sons of each point in this level are:
(0, 0.5)  (0.25, 0.5), (0,1), (0, 0)
(1, 0.5)  (0.75, 0.5), (1,1), (1, 0)
(0.5, 0)  (0.5, 0.25), (1, 0), (0, 0)
(0.5,1)  (0.5, 0.75), (1,1), (0,1)
 Notice here, we have some redundant points. Therefore, it is important
to keep track of the uniqueness of the newly generated (active) points.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 48
Adaptive sparse grid collocation
 After delete the same points, and arrange ,we can have
(0, 0), (1, 0), (0,1), (1,1)  (2, 2)
(0.5, 0.25), (0.5, 0.75)  (1,3)
(0.25, 0.5), (0.75, 0.5)  (3,1)
which are just all of the support nodes in k=2.
 Therefore, we can see by setting the threshold equal zero, conventional
sparse grid is exactly recovered.
 We actually work on point by point instead of the multi-index.
 In this way, we can control the local refinement of the point.
 Here, we equally treat each dimension locally, i.e., we still add the two
neighbor points in all N dimensions. So there are generally 2N son
(neighbor) points.
 If the discontinuity is along one dimension, this method can still detect it.
So in this way, we say this method can also detect the important dimension.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 49
Adaptive sparse grid collocation: implementation
 As shown before, we need to keep track of the uniqueness of the newly
active points.
 To this end, we use the data structure <set> from the standard template
library in C++ to store all the active points and we refer to this as the
active index set.
 Due to the sorted nature of the <set>, the searching and inserting is
always very efficient. Another advantage of using this data structure is
that it is easy for a parallel code implementation. Since we store all of
the active points from the next level in the <set>, we can evaluate the
surplus for each point in parallel, which increases the performance
significantly.
 In addition, when the discontinuity is very strong, the hierarchical
surpluses may decrease very slowly and the algorithm may not stop
until a sufficiently high interpolation level. Therefore, a maximum
interpolation level is always specified as another stopping criterion.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 50
Adaptive sparse grid collocation: Algorithms
 Let   0 be the parameter for the adaptive refinement threshold. We propose the
following iterative algorithm beginning with a coarsest possible sparse grid
, i.e.
with the N-dimensional multi-index i = (1,…,1), which is just the point (0.5,…,0.5).
1)
Set level of Smolyak construction k = 0.
2)
Construct the first level adaptive sparse grid
.
•
Calculate the function value at the point (0.5, …, 0.5).
•
Generate the 2N neighbor points and add them to the active index set.
•
Set k = k + 1;
3)
•
While k  kmax and the active index set is not empty
Copy the points in the active index set to an old index set and clear the active index set.
•
Calculate in parallel the hierarchical surplus of each point in the old index set according
to
Here, we use all of the existing collocation points in the current adaptive sparse grid
This allows us to evaluate the surplus for each point from the old index set in parallel.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 51
Adaptive sparse grid collocation: Algorithms
• For each point in the old index set, if
 Generate 2N neighbor points of the current active point according to the above
equation;
 Add them to the active index set.
• Add the points in the old index set to the existing adaptive sparse grid
adaptive sparse grid becomes
Now the
• Set k = k + 1;
4)
Calculate the mean and variance, the PDF and if needed realizations of the solution.
 In practice, instead of using surplus
of the function value, it is sometimes preferable to
use surplus of the square of the function value as the error indicator. This is because
the hierarchical surplus
is related to the calculation of the variance. In principle, we can
consider it like a local variance. Thus, is more sensitive to the local variation of the
stochastic function than .
 Error estimate of the adaptive interpolant
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 52
NUMERICAL EXAMPLES
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 53
Adaptive sparse grid interpolation
Ability to detect and reconstruct steep gradients
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 54
A dynamic system with discontinuity
 We consider the following governing equation for a particle moving under the
influence of a potential field and friction force:
d2X
dX
dh

f


dt
dx
dt 2
X t  0  x0 , dX / dt t  0  v0
If we set the potential field as h  x   35/ 8 x4  15/ 4 x2, then the differential equation
has two stable fixed points x   15 / 35 and an unstable fixed point at x  0 .
 The stochastic version of this problem with random initial position in the interval
can be expressed as
d2X
dX
35 3 15

f


X  X,
2
dt
2
2
dt
with stochastic initial condition
X  t  0, Y   X 0  X  Y ,
dX
|t  0  0
dt
where X t, Y  denotes the response of the stochastic system, X 0   x1  x2  / 2,
 1,1.
X | x1  x2 | / 2 and Y is uniformly distributed random variable over [1,1] .
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 55
A dynamic system with discontinuity
 If we choose X 0  0.05, X  0.2 and a relatively large friction coefficient f  2, a
steady –state solution is achieved in a short time. The analytical steady-state
solution is given by
 X  t  , Y    15 / 35, Y  0.25

 X  t  , Y    15 / 35, Y  0.25
 This results in the following statistical moments
  X  t  , Y   
1
15 / 35
4
Var  X  t  , Y   
45
112
 The analytic PDF is two dirac mass with unequal strength.
 In the following computations, the time integration of the ODE is performed using
a fourth-order Runge-Kutta scheme and a time step t  0.001 was used.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 56
Failure of gPC expansion
4
1
3.5
0.5
3
2.5
X
PDF
0
-0.5
2
1.5
1
-1
0.5
-1.5
-1
-0.5
0
Y
0.5
0
-2
1
-1.5
-1
Y
-0.5
0
0.5
p  10
3.5
1
3
0.5
2.5
0
X
PDF
2
1.5
-0.5
1
-1
0.5
-1.5
-1
-0.5
0
Y
0.5
1
0
-2.5
-2
-1.5
-1
Y
-0.5
0
p  20
Evolution of X (t , Y ) for 0  t  10
CORNELL
U N I V E R S I T Y
Solution at t  10
PDF at t  10
Materials Process Design and Control Laboratory 57
0.5
0.3
0.43
0.25
0.42
0.2
0.41
0.15
0.4
Var (X)
Mean (X)
Failure of gPC expansion
0.1
0.05
0.38
gPC
Exact
0
-0.05
0.39
5
10
Order
15
gPC
Exact
0.37
20
0.36
5
10
Order
15
20
Mean and variances exhibit oscillations with the increase of expansion
order
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 58
Failure of gPC expansion
1
0.5
X
Steady-state solution at t  25
0
p=5
p = 10
p = 15
p = 20
Exact
-0.5
-1
-1
-0.5
0
Y
0.5
1
 Therefore, the gPC expansion fails to converge the exact solutions with
increasing expansion orders. It fails to capture the discontinuity in the
stochastic space due to its global support, which is the well known “Gibbs
phenomenon” when applying global spectral decomposition to problems with
discontinuity in the random space.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 59
Failure of Lagrange polynomial interpolation
9
0.6
8
0.4
7
0.2
6
PDF
10
X
1
0.8
0
5
-0.2
4
-0.4
3
-0.6
2
-0.8
1
-1
-1
-0.5
0
Y
0.5
0
-1
1
-0.5
0
Y
0.5
1
0
Y
0.5
1
k 5
0.8
35
0.6
30
0.4
25
0.2
PDF
X
20
0
15
-0.2
10
-0.4
5
-0.6
-0.8
-1
-0.5
0
Y
k  10
Evolution of X (t , Y ) for 0  t  10
Solution at t  10
0.5
1
0
-1
-0.5
PDF at t  10
Much better than gPC, but still exhibit oscillation near discontinuity point.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 60
Failure of Lagrange polynomial interpolation
0.5
0.17
Lagrange Polynomial interpolation
Exact
0.166
0.3
0.164
0.2
0.1
0.162
2
4
6
Level
8
10
0.16
10
12
0.5
11
Level
12
13
0.403
Lagrange Polynomial interpolation
Exact
0.45
0.4
Lagrange Polynomial interpolation
Exact
0.402
Var (X)
Var (X)
Lagrange Polynomial interpolation
Exact
0.168
Mean (X)
Mean (X)
0.4
0.35
0.3
0.401
0.25
0.2
2
4
CORNELL
U N I V E R S I T Y
6
Level
8
10
12
0.4
10
11
Level
12
13
Materials Process Design and Control Laboratory 61
Failure of Lagrange polynomial interpolation
1
X
0.5
Steady-state solution at t  25
0
k=3
k=5
k=7
k=9
Exact
-0.5
-1
-1
-0.5
0
Y
0.5
1
 Similarly, the CSGC method with Lagrange polynomials also fails to capture
the discontinuity. However, better predictions of the low-order moments are
obtained than the gPC method. Though the results indeed converge, the
converged values are not accurate.
 Near the discontinuity point of the solution, there are still several wriggles,
which will not disappear even for higher interpolation levels.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 62
Sparse grid collocation with linear basis function
k 5
k  10
Evolution of X (t , Y ) for 0  t  10
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 63
Sparse grid collocation with linear basis function
60
1
50
0.5
PDF
X
40
0
k=3
k=5
k=7
k=9
Exact
-0.5
-1
-1
-0.75
-0.5
-0.25
0
Y
0.25
0.5
Solution at t  25
0.75
30
20
10
1
0
-0.8
-0.6
-0.4
-0.2
0
Y
0.2
0.4
0.6
0.8
PDF at t  25
 Thus, it is correctly predicting the discontinuity behavior using linear basis
function. And it is also noted that there are no oscillations in the solutions
shown here.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 64
Adaptivity
-1
10
20
10-2
-3
15
-4
10
-5
10
Level
Absolute error of variance
10
10-6
-7
10
CSGC
-2
ASGC:  = 10
10
5
10-8
-9
10
10-10
1
10
2
10
3
10
#Points
4
10
5
10
0
-1
-0.75
-0.5
-0.25
0
Y
0.25
0.5
0.75
1
 For CSGC method, a maximum interpolation level of 15 is used and 32769
points are need to reach accuracy of 1105. For the ASGC method, a
maximum interpolation level of 30 is used. Only 115 number of points are
required to achieve accuracy of 31010.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 65
Highly discontinuous solution
 Now we consider a more difficult problem. We set f  0.05, X 0  1 and X  0.1 .
All other conditions remain as before. The reduction of the friction coefficient,
together with higher initial energy will result in the oscillation of the particle for
several cycles between each of the two stable equilibrium points.
1.5
1
1
0.5
0
X
X
0.5
0
-0.5
-0.5
-1
-1.5
-1
-0.5
0
Y
0.5
1
-1
-1
-0.5
0
Y
0.5
1
Failure of gPC (left) and polynomial interpolation (right) at t = 250s. The
failure of Lagrange polynomial interpolation is much more obvious.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 66
Highly discontinuous solution
k=4
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-1
-0.5
0.5
1
-0.8
-1
k=8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
-0.2
-0.4
-0.4
-0.6
-0.6
-0.5
0
Y
CORNELL
U N I V E R S I T Y
0.5
1
0
Y
0.5
1
0.5
1
k = 10
Results using
the CSGC
method with
linear basis
functions.
0
-0.2
-0.8
-1
-0.5
0.8
X
X
0
Y
k=6
0.8
X
X
0.8
-0.8
-1
-0.5
0
Y
Materials Process Design and Control Laboratory 67
The Kraichnan-Orszag three-mode problem
 The governing equations are (X.Wan, G. Karniadakis, 2005),
dx1
 x2 x3
dt
dx2
 x1 x3
dt
dx3
 2 x1 x2
dt
subject to stochastic initial conditions:
x1 (0)  x1 (0;  ), x2 (0)  x2 (0;  ), x3 (0)  x3 (0; )
 There is a discontinuity when the random initial condition approaches
x1  1 and x2  1 (x1=x2) . So let us first look at the discontinuity in the
deterministic solution.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 68
The Kraichnan-Orszag three-mode problem
1.5
x1(0) = 1, x2(0) = 0.95, x3(0) = 1
x1(0) = 1, x2(0) = 0.97, x3(0) = 1
x1(0) = 1, x2(0) = 0.99, x3(0) = 1
x1(0) = 1, x2(0) = 1.00, x3(0) = 1
1
x2
0.5
0
-0.5
-1
-1.5
0
5
10
15
20
25
30
Time
Singularity at plane x2  1
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 69
The Kraichnan-Orszag three-mode problem
 For computational convenience and clarity we first perform the following
transformation
 2

2

2
 y1  
 y    2
 2  2
 y3  
 0


2
2
2
0
0
x 
 1
0   x2 
 x 
1  3 


(X.Wan, G. Karniadakis, 2005)
 The resulting governing equations become:
dy1
 y1 y3 ,
dt
dy2
  y2 y3 ,
dt
dy3
  y12  y22
dt
 The discontinuity then occurs at the planes y1  0 and y2  0 .
 Thus, we first study the stochastic response subject to the following
random 1D initial input:
y1 (0;  )  1, y2 (0;  )  0.1 Y (0;  ), y3 (0;  )  0
where Y U (1,1) . Since the random initial data y2 (0;  ) can cross the plane
y2 know
0 that gPC will fail for this case.
we
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 70
The Kraichnan-Orszag three-mode problem
2
y1(0) = 0.1, y2(0) = 1.0, y3(0) = 1.0
y1(0) = 0.05, y2(0) = 1.0, y3(0) = 1.0
y1(0) = 0.02, y2(0) = 1.0, y3(0) = 1.0
y1(0) = 0.0, y2(0) = 1.0, y3(0) = 1.0
1.5
y1
1
0.5
0
-0.5
0
5
10
15
20
25
30
Time
Singularity at plane y1  0
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 71
One-dimensional random input: variance
0.8
0.25
MC-SOBOL: 1,000,000
ASGC:  = 10-1
ASGC:  = 10-2
ASGC:  = 10-3
gPC: p = 30
0.6
0.15
Var( y1 )
0.1
Variance of y2
Variance of y1
0.2
MC-SOBOL: 1,000,000
ASGC:  = 10-1
ASGC:  = 10-2
-3
ASGC:  = 10
gPC: p = 30
0.4
Var( y2 )
0.2
0.05
0
0
5
10
15
Time
20
25
MC-SOBOL: 1,000,000
ASGC:  = 10-1
ASGC:  = 10-2
-3
ASGC:  = 10
gPC: p = 30
0.8
Variance of y3
0
5
10
15
Time
20
25
30
Var( y3 )
0.6
 gPC fails after t = 6s while the ASGC
method converges even with a large
threshold. The solid line is the result
from MC-SOBOL sequence with 106
iterations.
0.4
0.2
0
0
30
0
5
10
CORNELL
U N I V E R S I T Y
15
Time
20
25
30
Materials Process Design and Control Laboratory 72
One-dimensional random input: adaptive sparse grid
15
Adaptive
sparse grid
with
Level
10
  102
5
0
-1
-0.5
0
Y
0.5
1
 From the adaptive sparse grid, it is seen that most of the refinement
after level 8 occurs around the discontinuity point Y = 0.0, which is
consistent with previous discussion.
 The refinement stops at level 16, which corresponds to 425 number of
points, while the conventional sparse grid requires 65537 points.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 73
One-dimensional random input
The ‘exact’
solution is taken
as the results
given by MCSOBOL
The error level
is defined as
max Var( yi )  Var( yi , MC ) t 30
i 1,2,3
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 74
One-dimensional random input: long term behavior
0.2
1
0.15
MC-SOBOL: 1,000,000
ASGC:  = 10-2
-3
ASGC:  = 10
0.9
MC-SOBOL: 1,000,000
ASGC:  = 10-2
-3
ASGC:  = 10
0.8
Mean of y1
Variance of y1
0.7
0.1
0.6
0.5
0.4
0.3
0.05
0.2
0.1
0
0
20
40
Time
60
80
100
0
0.7
MC-SOBOL: 1,000,000
ASGC:  = 10-2
-3
ASGC:  = 10
Time
60
80
100
MC-SOBOL: 1,000,000
-2
ASGC:  = 10
ASGC:  = 10-3
0.6
0.5
0.5
Variance of y3
Variance of y2
40
0.7
0.6
0.4
0.3
0.4
0.3
0.2
0.2
0.1
0.1
0
20
0
20
40
CORNELL
U N I V E R S I T Y
Time
60
80
100
0
0
20
40
Time
60
80
100
Materials Process Design and Control Laboratory 75
One-dimensional random input: Realizations
 These realizations are reconstructed using hierarchical surpluses. It is
seen that at earlier time, the discontinuity has not yet been developed,
which explains the reason that the gPC is accurate at earlier times.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 76
Two-dimensional random input
 Next, we study the K-O problem with 2D input:
y1 (0;  )  1, y2 (0;  )  0.1Y1 (0;  ), y3 (0; )  Y2 (0; )
 Now, instead of a point, the discontinuity region becomes a line.
0.25
MC-SOBOL: 1,000,000
-1
ASGC:  = 10
ASGC:  = 10-2
ASGC:  = 10-3
gPC: p = 10
1
MC-SOBOL: 1,000,000
-1
ASGC:  = 10
-2
ASGC:  = 10
ASGC:  = 10-3
gPC: p = 10
0.8
Variance of y2
Variance of y1
0.2
0.15
0.1
0.05
0
0.6
0.4
0.2
0
2
CORNELL
U N I V E R S I T Y
4
Time
6
8
10
0
0
2
4
Time
6
8
10
Materials Process Design and Control Laboratory 77
Two-dimensional random input
1
MC-SOBOL: 1,000,000
ASGC:  = 10-1
-2
ASGC:  = 10
ASGC:  = 10-3
gPC: p = 10
0.8
0.5
Y2
Variance of y3
0.6
0
0.4
-0.5
0.2
0
0
2
4
Time
6
8
10
-1
-1
-0.5
0
Y1
0.5
1
 It can be seen that even though the gPC fails at a larger time, the
adaptive collocation method converges to the reference solution given
by MC-SOBOL with 106 iterations.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 78
Two-dimensional random input
The error level
is defined as
max Var( yi )  Var( yi , MC ) t 10
i 1,2,3
The number of
points of the
conventional
grid with
interpolation 20
is 12582913.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 79
Two-dimensional random input: Long term behavior
-3
3
4
ASGC:  = 10-1
3
ASGC:  = 10-2
ASGC:  = 10-3
2
1
0
1
-3
ASGC:  = 10-1
ASGC:  = 10-2
ASGC:  = 10-3
0
-1
-1
-2
0
x 10
2
Error in variance
Error in mean
5
x 10
10
20
30
Time
40
50
-2
0
10
20
30
40
50
Time
Error in the mean (left) and the variance (right) of y1 with different
thresholds for the 2D K-O problem in [0, 50].
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 80
Three-dimensional random input
 Next, we study the K-O problem with 3D input:
y1 (0;  )  Y1 (0;  ), y2 (0;  )  Y2 (0;  ), y3 (0;  )  Y3 (0; )
 This problem is much more difficult than any of the other problems
examined previously. This is due to the strong discontinuity ( the
discontinuity region now consists of the planes Y1 = 0 and Y2 = 0) and the
moderately-high dimension.
 It can be verified from comparison with the result obtained by MC-SOBOL
that unlike the previous results, here 2 x 106 iterations are needed to
correctly resolve the discontinuity.
 Due to the symmetry of y1 and y2 and the corresponding random input,
the variances of y1 and y2 are the same.
 Finally, in order to show the accuracy of the current implementation of the
h-adaptive ME-gPC, we provide the results for this problem.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 81
Three-dimensional random input:ASGC
0.36
0.39
0.34
MC-SOBOL: 2,000,000
ASGC:  = 10-1
ASGC:  = 10-2
ASGC:  = 10-3
0.38
MC-SOBOL: 2,000,000
ASGC:  = 10-1
ASGC:  = 10-2
ASGC:  = 10-3
0.32
Variance of y3
Variance of y1
0.37
0.36
0.35
0.28
0.26
0.34
0.33
0.3
0.24
0.22
0
1
2
3
4
5
Time
6
7
8
9
10
0
1
2
3
4
5
Time
6
7
8
9
Evolution of the variance of y1=y2 (left) and y3 (right) for the 3D K-O
problem using ASGC
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 82
10
Three-dimensional random input: ME-gPC
0.39
0.36
MC-SOBOL: 2,000,000
ME-gPC: p = 3
ME-gPC: p = 4
ME-gPC: p = 5
ME-gPC: p = 6
0.32
Variance of y3
0.37
Variance of y1
0.34
MC-SOBOL: 2,000,000
ME-gPC: p = 3
ME-gPC: p = 4
ME-gPC: p = 5
ME-gPC: p = 6
0.38
0.36
0.3
0.28
0.35
0.26
0.34
0.33
0.24
0
1
2
3
4
5
Time
6
7
8
9
10
0.22
0
1
2
3
4
5
Time
6
7
8
9
Evolution of the variance of y1=y2 (left) and y3 (right) for the 3D K-O
problem using ME-gPC with 1  104 and  2  103
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 83
10
Three-dimensional random input
Due to the strong
discontinuity in this
problem, it took
much longer time
for both methods
to arrive at the
same accuracy.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 84
Stochastic elliptic problem
 In this example, we compare the convergence rate of the CSGC and
ASGC methods through a stochastic elliptic problem in two spatial
dimensions.
 As shown in previous examples, the ASGC method can accurately
capture the non-smooth region in the stochastic space. Therefore, when
the non-smooth region is along a particular dimension (i.e. one dimension
is more important than others), the ASGC method is expected to identify it
and resolve it.
 In this example, we demonstrate the ability of the ASGC method to detect
important dimensions when each dimension weigh unequally.
 This is similar to the dimension-adaptive method, especially in high
stochastic dimension problems.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 85
Stochastic elliptic problem
 Here, we adopt the model problem
   aN ,  u ,    f N ,  in D  
u  ,    0
on  D  
with the physical domain D  x   x, y   0,1  . To avoid introducing errors of
any significance from the domain discretization, we take a deterministic
smooth load f N , x, y   cos  x  sin  y  with homogeneous boundary conditions.
2
 The deterministic problem is solved using the finite element method with
900 bilinear quadrilateral elements. Furthermore, in order to eliminate the
errors associated with a numerical K-L expansion solver and to keep the
random diffusivity strictly positive, we construct the random diffusion
coefficient aN , x  with 1D spatial dependence as
  L   n 18  L
log  aN , x   0.5  1   
cos  2 x  n  1  Yn  
 e
2 
n 1 
where Yn   , n  1, N are independent uniformly distributed random variables
in the interval  3, 3 
N
CORNELL
U N I V E R S I T Y
1/ 2
2
2 2
Materials Process Design and Control Laboratory 86
Stochastic elliptic problem
 This expansion is similar to a K-L expansion of a 1D random field with
stationary covariance
   x1  x2 2 

log  aN  , x   0.5   x1 , x2   exp 
2



L

 Small values of the correlation L correspond to a slow decay, i.e. each
stochastic dimension weighs almost equally. On the other hand, large
values of L results in fast decay rates, i.e., the first several stochastic
dimensions corresponds to large eigenvalues weigh relatively more.
 By using this expansion, it is assumed that we are given an analytic
stochastic input. Thus, there is no truncation error. This is different from
the discretization of a random filed using the K-L expansion, where for
different correlation lengths we keep different terms accordingly.
 In this example, we fix N and change L to adjust the importance of each
stochastic dimension. In this way, we investigate the effects of L on the
ability of the ASGC method to detect the important dimensions.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 87
Stochastic elliptic problem: N = 5
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 88
Stochastic elliptic problem: N = 5
 We estimate the L2  D  approximation error for the mean and variance.
Specifically, to estimate the computation error in the q-th level, we fix N
and compare the results at two consecutive levels, e.g. the error for the
mean is
E  Aq , N  u N   Aq 1, N  u N  
 The previous figures shows results for different correlation lengths at N=5.
 For small L, the convergence rates for the CSGC and ASGC are nearly
the same. On the other hand, for large L, the ASGC method requires
much less number of collocation points than the CSGC for the same
accuracy.
 This is because more points are placed only along the important
dimensions which are associated with large eigenvalues.
 Next, we study some higher-dimensional cases. Due to the rapid increase
in the number of collocation points, we focus on moderate to higher
correlation lengths so that the ASGC is effective.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 89
Stochastic elliptic problem: higher-dimensions
#points:4193
Comparison:
CSGC: 3.5991x1011
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 90
Stochastic elliptic problem: higher-dimensions
 In order to further verify our results, we compare the mean and the
variance when N = 75 using the AGSC method with   108 with the ‘exact’
solution obtained by MC-SOBOL simulation with 106 iterations.
E  Aq , N  u N    E  uMC 
E  uMC 
Relative error of variance
Relative error of mean
0.0015
0.0004
0.0003
0.001
0.0002
0.0005
0.0001
0
0
0
0.2
0.2
y
0.6
0.6
0.4
0.4
x
y
0.8
0.8
0.2
0.2
0.4
0.4
0
0
0
0.6
0.6
1 1
Computational time: ASGC : 0.5 hour
CORNELL
U N I V E R S I T Y
0.8
0.8
1 1
x
MC: 2 hour
Materials Process Design and Control Laboratory 91
Application to Rayleigh-Bénard instability
Cold wall  = -0.5
Insulated
0.75
Y
Insulated
1
0.5
0.25
0
0
0.25
0.5
0.75
hot wall 
1
X
 Filled with air (Pr = 0.7). Ra = 2500, which is larger than the critical
Rayleigh number, so that convection can be initialized by varying the hot
wall temperature. Thus, the hot wall temperature is assumed to be a
random variable.
 Prior stochastic simulation, several deterministic computations were
performed in order to find out the range where the critical point lies in.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 92
Rayleigh-Bénard instability: deterministic case
0
10
Evolution of the
average kinetic
energy for different
hot wall temperatures
h = 0.30
h = 0.40
h = 0.50
h = 0.55
h = 0.60
h = 0.70
[K]1/2
10-1
-2
10
10-3
20
40
Time
60
80
100
 As we know, below the critical temperature, the flow vanishes, which
corresponds to a decaying curve. On the other hand, the growing curve
represents the existence of heat convection. Therefore, the critical
temperature lies in the range [0.5,0.55].
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 93
Rayleigh-Bénard instability: deterministic formulation
 Steady state Nusselt number which denotes the rate of heat transfer:
1
Nu 
h  c

0 y
1
y 0
dx
Nu  1  Conduction
Nu  1  Convection
 This again verifies the critical hot wall temperature lies in the range
[0.5,0.55]. However, the exact critical value is not known to us. So, now
we try to capture this unstable equilibrium using the ASGC method.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 94
Rayleigh-Bénard instability: stochastic formulation
 We assume the following stochastic boundary condition for the hot wall
temperature:
h  0.4  0.3Y
where Y is a uniform random variable in the interval [0,1]. Following the
discussion before, both conductive and convective modes occur in this
range.
12
0.15
10
0.1
Nu
Level
8
0.05
6
4
0
2
0.4
0.45
0.5
0.55
h
0.6
 Nu  Nu  1
CORNELL
U N I V E R S I T Y
0.65
0.7
0.4
0.45
0.5
0.55
h
0.6
0.65
0.7
0.541
Materials Process Design and Control Laboratory 95
Rayleigh-Bénard instability: stochastic formulation
3
0.08
0.06
v
u
2
0.04
u
v
1
0.02
0
0
0.4
0.45
0.5
0.55
h
0.6
0.65
0.7
0.4
0.25
0.45
0.5
0.55
h
0.6
0.65
0.7
0.541
0.2


0.15
0.1
0.05
0
-0.05
0.4
0.45
0.5
CORNELL
U N I V E R S I T Y
0.55
h
0.6
0.65
0.7
nonlinear:
convection
Solution of the
variables versus hotwall temperature at
point (0.1,0.5)
Linear:
conduction
Materials Process Design and Control Laboratory 96
Rayleigh-Bénard instability: stochastic formulation
Max = 0.436984
Max = 0.436984
0.35
0.25
0.15
0.05
-0.05
-0.15
-0.25
-0.35
-0.45
0.35
0.25
0.15
0.05
-0.05
-0.15
-0.25
-0.35
-0.45
 Prediction of the temperature when h  0.436984 using ASGC (left) and the
solution of the deterministic problem using the same  h (right).
 Fluid flow vanishes, and the contour distribution of the temperature is
characterized by parallel horizontal lines which is a typical distribution for
heat conduction.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 97
Rayleigh-Bénard instability: stochastic formulation
Max = 4.33161
Max = 0.667891
Max = 4.31858
4
3
2
1
0
-1
-2
-3
-4
Max = 4.33384
Max = 4.3208
4
3
2
1
0
-1
-2
-3
-4
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
4
3
2
1
0
-1
-2
-3
-4
Max = 0.667891
4
3
2
1
0
-1
-2
-3
-4
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
 Prediction of the temperature when h  0.66789 using ASGC (top) and the
solution of the deterministic problem using the same  h (bottom).
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory 98
Rayleigh-Bénard instability: Mean
Max = 1.73709
Max = 1.73123
1.6
1.2
0.8
0.4
0
-0.4
-0.8
-1.2
-1.6
1.6
1.2
0.8
0.4
0
-0.4
-0.8
-1.2
-1.6
Max = 1.73128
Max = 1.73715
1.6
1.2
0.8
0.4
0
-0.4
-0.8
-1.2
-1.6
ASGC (top)
CORNELL
U N I V E R S I T Y
Max = 0.55
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
Max = 0.549918
1.6
1.2
0.8
0.4
0
-0.4
-0.8
-1.2
-1.6
MC-SOBOL (bottom)
Materials Process Design and Control Laboratory 99
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
Rayleigh-Bénard instability: Variance
Max = 3.28535
Max = 3.30617
3
2.6
2.2
1.8
1.4
1
0.6
0.2
3
2.6
2.2
1.8
1.4
1
0.6
0.2
ASGC (top)
U N I V E R S I T Y
3
2.6
2.2
1.8
1.4
1
0.6
0.2
Max = 3.27903
Max = 3.29984
CORNELL
Max = 0.0129218
0.012
0.011
0.01
0.009
0.008
0.007
0.006
0.005
0.004
0.003
0.002
0.001
Max = 0.0128783
3
2.6
2.2
1.8
1.4
1
0.6
0.2
MC-SOBOL (bottom)
Materials Process Design and Control Laboratory100
0.012
0.011
0.01
0.009
0.008
0.007
0.006
0.005
0.004
0.003
0.002
0.001
Conclusion
 An adaptive hierarchical sparse grid collocation method is developed
based on the error control of local hierarchical surpluses.
 Besides the detection of important and unimportant dimensions as
dimension-adaptive methods can do, additional singularity and local
behavior of the solution can also be revealed.
 By employing adaptivity, great computational saving is also achieved
than the conventional sparse grid method and traditional gPC related
method.
 Solution statistics is easily extracted with a higher accuracy due to the
analyticity of the linear basis function used.
 The current computational code is a highly efficient parallel program
and has a wider application in various uncertainty quantification area.
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory101