Transcript On a Trust Region Inexact Newton Interior

Hybrid methods for solving large-scale
parameter estimation problems
Carlos A. Quintero1 Miguel Argáez1 Hector Klie2
Leticia Velázquez1 Mary Wheeler2
1
Department of Mathematical Sciences
University of Texas at El Paso
2
ICES-The Center for Subsurface Modeling
The University of Texas at Austin
13th GAMM - IMACS International Symposium on Scientific
Computing, Computer Arithmetic, and Verified Numerical
Computations SCAN'2008
El Paso, Texas, USA
October 2 , 2008
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Outline
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Introduction
Problem Formulation
 Parameter Estimation
Optimization Framework
 Global Search
 Surrogate Model
 Local Search
Numerical Results
Conclusion and Future Work
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Goal
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

We present a hybrid optimization approach for solving
global optimization problems, in particular automated
parameter estimation models.
The hybrid approach is based on the coupling of the
Simultaneous Perturbation Stochastic Approximation
(SPSA) and a Newton-Krylov Interior-Point method
(NKIP) via a surrogate model.
We implemented the hybrid approach on several test
cases.
3
Problem Formulation
We consider the global optimization problem in
the form:
minimize f  x 
f : R  R (1)
n
where the global solution x* is such that
f x *  f x for all x  R
n
We are interested in problem (1) that have many
local minima.
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Types of local minima
local
minima
global
minimum
x
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Hybrid Optimization Framework
• Global Method: SPSA
• Surrogate Model
• Local Method: NKIP
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Global Method: SPSA
Stochastic steepest descent direction algorithm
• (James Spall, 1998)
•Advantage
• SPSA gives region(s) where the function value is
low, and this allows to conjecture in which region(s) is
the global solution.
• Uses only two objective function evaluations at each
iteration to obtain the update of the parameters.
Disadvantages
•Slow method
•Do not take
constraints
into
account
equality/inequality
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Global Search: SPSA
SPSA performs random simultaneous perturbations
of all model parameters to generate a descent
direction at each iteration
This process may be performed by starting with
different initial guesses (multistart).
Multistart increases the chances for finding a global
solution, and yields to find a vast sampling of the
parameter space.
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SPSA: Pseudocode
function[x]  SP SA(x, n, k max , a, A, c, ,  )
1 : for k  1 : k max
a
c
and
c

k
(k  A)
k
3 : Set therandomperturbation vector k  2(round(rand(n;1)) - 1)
2 : Set a k 
4 : Simultaneous P erturbation x  x  c k  and x-  x - c k ;
5 : Functionevaluation y   f(x ) and y -  f(x- );
6 : Set step x  (y - y - )./(2ck )
7:
Updatex  x - a k x
8 : Check upper and lower constraints on x
end
Note: A  (10,k max/10),  1.101,  0.602
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Observations
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SPSA gives region(s) where the function value
is low, and this allows to conjecture in which
region(s) is the global solution.
This give us a motivation to apply a local
method in the region(s) found by SPSA.
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Local Method
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Advantages
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
to
add
Disadvantage
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Fast Method: Newton Type Methods
Interior-Point
Methods
allow
equality/inequality constraints
Needs first/second order information
Solution

Construct a Surrogate Model using the SPSA
function values inside the conjecture region(s)
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Surrogate Model
A surrogate model f s ( xk ) is created by using an
interpolation
method
with
the
data,
( xk , f ( xk )), k  1,..., p, provided by SPSA.
This can be performed in different ways, e.g.,
radial basis functions, kriging, regression analysis,
or using artificial neural networks.
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Why Surrogate Models?
Most real problems require thousands or millions
of objective and constraint function evaluations,
and often the associated high cost and time
requirements render this infeasible.
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Radial Basis Function (RBF)
RBF is typically parameterized by two sets of
parameters: the center c which defines its position,
and shape r that determines its width or form.
An RBF interpolation algorithm
characterizes the uncertainty parameters:
(Orr,1996)
cij , rij , wj , i  1,...,n, j  1,..,m.
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Surrogate Model
Our goal is to optimize the surrogate function
m
f s ( x)   w j h j ( x),
j 1
Where the radial basis functions can be defined as:
n
h j ( x)  1  
i 1
( xi  cij ) 2
rij
2
or h j ( x)  e

2
 n ( xi cij ) 
 

r


ij
 i1



that are the multiquadric and the Gaussian basis
functions, respectively
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Hybrid Optimization Scheme (1)
Explore Parameter
space
Multistart
(x0)1
(x0)2
.
.
.
Global Search
Via SPSA
.
.
.
(x0)k
Target Region
Filtering
+
Sampling
Surrogate
Model
m
f s ( x)   w j h j ( x),
j 1
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Test Case 2
Rastrigin’s Problem:
n
min f ( x1 , x2 ,..., xn )  10n   ( xi2  10cos(2 xi )),
i 1
s.t.  5.12  xi  5.12
for i  1, 2,..., n
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Test Case 2: Rastringin
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Test Case 2: Rastringin
x*=(0,0) global solution
f(x*)=0
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Test Case 2: Restringin
Sampling Data from SPSA
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Test Case 2: Restringin
x*=(0,0) global solution
f(x*)=0
Sampling Data from SPSA
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Restringin: Surrogate Model
We plot the original model function and the surrogate function:
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Restringin: Surrogate Model
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Local Search: NKIP
NKIP is a globalized and derivative dependent
optimization method based on the global strategy
introduced by Miguel Argaéz and Richard Tapia in
2002.
This method calculates the directions using the
conjugate gradient algorithm, and a linesearch is
implemented to guarantee a sufficient decrease of the
objective function.
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Local Search: NKIP
We consider the optimization problem in the form:
minimize f S ( x)
subject to a  x  b
where a and b are determined by the sampled
points given by SPSA.
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Hybrid Optimization Scheme (2)
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Hybrid Optimization Scheme
Convergence history for minimizing Rastringin’s function (n=50).
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Parameter Estimation Problem
The objective function can be written as a nonlinear
least squares problem:
minimize || G(x) - d T ||
2
where G(x) : R n  R m is a nonlinear function
and dT  Rm is a set of data observations
n
m
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Application
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Our goal is to estimate the permeability based
on sensor pressure data.
Despite having full information of the pressure
field, the inverse problem is highly ill-posed
and has multiple local minima.
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Large Scale Two-Phase Flow Problem
3D permeability field
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Large Scale Two-Phase Flow Problem
Figure 2. The true and initial permeability and pressure fields.
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High Performance Computing
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We run the simulator IPARS (Integrated Parallel
Accurate Reservoir Simulator) on a Linux-based
multicore network of workstations.
Our goal is to estimate a permeability field that involves
2000 parameters. The field is parameterized by SVD
reducing the parameter space to 20 (each parameter
represents a scale resolution level).
We choose several initials points by adding a percentage
of noise to the true singular values: 5%, 10% and 15%
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Numerical Results
Noise
Initial starting points by SPSA
Range of f(x0)
SPSA Iterations
Range of fspsa(x)
Best fspsa(x)
Data points used in the surrogate model
NKIP Iterations
fnkip(x)
% Gain by Hybrid Approach
5%
10%
15%
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20
20
8.56 to 10.3
18.72 to 53.9
29.46 to 120.56
3645
2248
1647
0.568 to 2.5
0.858 to 6.785
1.541 to 15.89
0.568
0.858
1.541
156
81
57
94
67
110
0.478
0.747
1.465
16
13
5
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Conclusions
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We combine SPSA and NKIP strategies into a Hybrid
Scheme that exploit the best of these two approaches
for a given problem in order to achieve maximum
efficiency and robustness.
The hybrid approach finds a good estimate of the global
solution for all the test cases.
As the noise level was increased, the hybrid approach
presented more difficulties in reproducing the true
permeability field. However, the estimation is very
accurate in all cases.
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Future Work
Improve a parallel version of the multi-start technique
Parallelize the conjugate gradient of NKIP
Add equality constraints to the minimization of the
surrogate model
Add more Global Optimization Techniques (Simulated
Annealing, Evolutionary Algorithms)
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Thank you very much for your
attention
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