Multifidelity Optimization Via Pattern Search and Space Mapping

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Multifidelity Optimization Via Pattern Search and
Space Mapping

• Genetha Gray
• Computational Sciences Mathematics Research
• Sandia National Labs, Livermore, CA

Sandia is a multiprogram laboratory operated by
Sandia Corporation, a Lockheed Martin Company,
for the United States Department of Energys
National Nuclear Security Administration under
contract DE-AC04-94AL85000.
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Outline

• Multifidelity Optimization
• APPSPACK
• Space Mapping
• MFO scheme
• Descriptive Example
• Groundwater Remediation Example

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Multifidelity Optimization (MFO)

• The low fidelity model retains many of the
  properties of the high fidelity model but is
  simplified in some way
• Decreased physical resolution
• Decreased FE mesh resolution
• Simplified physics
• MFO optimizes an inexpensive, low fidelity model
  while making periodic corrections using the
  expensive, high fidelity model.
• Works well when low-fidelity trends match
  high-fidelity trends.

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Asynchronous Parallel Pattern Search (APPS)

• Direct method ? no derivatives required
• Pattern of search directions drives search and
  determines new trial points for evaluation
• Objective function can be an entirely separate
  program
• Achieves parallelism by assigning function
  evaluations to different processors
• Freely available software under the GNU public
  license (APPSPACK)

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Synchronous Pattern Search
Inherently (or embarrassingly) parallel,but
processor load should be considered.
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Processor Load Balance Considerations

• The number of trial points can vary at each
  iteration.
• Cached function values
• Search patterns change
• Constraints (infeasible trial points are not
  evaluated)
• Evaluation times can vary for each trial point.
• Different processor characteristics
• Effect of input on function
• Function evaluation faults
• MFO different function models with different
  evaluation times!!

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APPSPACK Example
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APPSPACK Example
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APPSPACK Example
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APPSPACK Example
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APPSPACK Example
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APPSPACK Example
j,k
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APPSPACK Example
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APPSPACK Example
o
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APPSPACK Example
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APPSPACK Example
s
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APPSPACK Example
Note Cannot Prune on Unsuccessful Iteration
s
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APPSPACK Example
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Space Mapping A Conduit Between the Low and
High Fidelity Model Design Spaces

• x design variables
• R - response
• P - mapping

xL
?
P(xH)
low-fi model
RL(xL)

• Space mapping is a technique that maps the
  design space of a low fidelity model to the
  design space of high fidelity model such that
  both models result in approximately the same
  response.
• The parameters within xH need not match
  the parameters within xL

Developed by John Bandler, et. al.
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Oracle

• An oracle predicts points at which a decrease in
  the objective function might be observed.
• Analytically, an oracle can choose points by any
  finite process.
• Oracle points are used in addition to the points
  defined by the search pattern.
• The MFO scheme employs an oracle framework to do
  a space mapping so that APPSPACK convergence is
  not adversely affected.
• Future work may include investigating any
  convergence improvement.

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The MFO Scheme Combining APPSPACK and Space
Mapping
Outer Loop
Inner Loop
multiple xH,f(xH)
Space Mapping Via Nonlinear Least
Squares Calculation
High Fidelity Mode Optimization via APPSPACK
a,b,g
Low Fidelity Model Optimization a (xH) b g
xHtrial
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MFO Algorithm

• Start the Outer Loop (APPSPACK)
• Evaluate N high fidelity response points
• Produce xH, fH(xH) pairs
• Start the Inner Loop
• Take data pairs from APPSPACK
• Run LS optimization
• At each iteration, evaluate N low fidelity
  responses
• At conclusion, obtain a, b, g for space map
  a(xH)b g
• Optimize low fidelity model within space mapped
  high fidelity space. In other words, minimize
  fL(a(xH)b g) with respect to xH to obtain xH.
• Return xH to APPSPACK to determine if a new best
  point has been found.

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A Simple Example
View of Unmapped Low Fidelity Design Space
View of High Fidelity Design Space
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MFO Results
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When the response points is 8, there are two
calls to inner loop.
Approximate Inner Loop Call Locations within
Hi-Fi Model
(-0.8,-1.2)
2
1
(-0.76,2.0)
1

• The numbered white boxes show approximately where
  the inner loop was called
• The point in red brackets is where APPSPACK is
  before the inner loop call
• The point in green was found by the inner loop

(-0.56,1.6)
2
(-0.61,1.25)
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Groundwater Remediation
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Groundwater Remediation via Optimization

• Optimization techniques can aid the design
  process to result in lower clean up costs.
• Use Hydraulic Capture (HC) models to alter the
  groundwater flow direction and control plume
  migration
• Transport Based Concentration Control (TBCC)
• Computationally expensive
• Well defined plume boundary
• ? MFO high fidelity model
• Flow Based Hydraulic Control (FBHC)
• Orders of magnitude faster
• Constraints require calibration
• ? MFO low fidelity model

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Optimization

• Objective Function
• J(u) installation costs operation costs
• Evaluation requires results of a simulation
• Design Variables
• Number of wells
• Well pumping rates
• Well locations
• Constraints
• Well capacity
• Net pumping rate
• Dont flood or dry out land
• No useless wells

• Implementation
• Derivatives are unavailable
• Simulators
• MODFLOW used for flow equation (USGS)
• MT3D used for transport equation (EPA)

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MFO Numerical Test

• Test the MFO method on the HC problem included in
  the community problems set. (Mayer, Kelley,
  Miller)
• The FBHC formulation has been shown to be
  sufficient for this simple domain. (Fowler,
  Kelley, Kees, Miller)
• Other approaches are needed for heterogeneous
  more realistic domains. (Ahlfeld, Page, Pinder)

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MFO Results
Initial cost 78,586
MODFLOW (mf2k) 2 seconds mt3d 50 seconds
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MFO Results
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Acknowledgements

• MFO development team (Sandia)
• Joe Castro (PI), Electrical Microsystem
  Modeling, NM
• Tony Giunta, Validation Uncertainty
  Quantification Processes, NM
• Patty Hough, CSMR, CA

• Groundwater Application
• Katie Fowler, Clarkson University
• Questions??
• Genetha Gray
• gagray_at_sandia.gov

• Software
• APPSPACK software.sandia.gov/appspack/version4.0/
  index.html
• DAKOTA http//endo.sandia.gov/DAKOTA