Multifidelity optimization via pattern search and space mapping
Download
1 / 32

Multifidelity Optimization Via Pattern Search and Space Mapping - PowerPoint PPT Presentation


  • 79 Views
  • Uploaded on

Multifidelity Optimization Via Pattern Search and Space Mapping. Genetha Gray Computational Sciences & Mathematics Research Sandia National Labs, Livermore, CA.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Multifidelity Optimization Via Pattern Search and Space Mapping' - totie


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Multifidelity optimization via pattern search and space mapping

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 Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.


Outline
Outline Mapping

  • Multifidelity Optimization

  • APPSPACK

  • Space Mapping

  • MFO scheme

  • Descriptive Example

  • Groundwater Remediation Example


Multifidelity optimization mfo

Finite Element Mapping

Models of the

Same Component

High Fidelity

800,000 DOF

Low Fidelity

30,000 DOF

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.


Asynchronous parallel pattern search apps
Asynchronous Parallel Pattern Search (APPS) Mapping

  • 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)


Synchronous pattern search
Synchronous Pattern Search Mapping

Inherently (or embarrassingly) parallel,but processor load should be considered.


Processor load balance considerations
Processor Load Balance Considerations Mapping

  • The number of trial points can varyat 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!!


Appspack example

Workers Mapping

Waiting

APPSPACK Example


Appspack example1

Workers Mapping

b

c

d

e

Waiting

APPSPACK Example


Appspack example2

Workers Mapping

c

d

Waiting

APPSPACK Example


Appspack example3

Workers Mapping

f

c

d

g

Waiting

APPSPACK Example


Appspack example4

Workers Mapping

c

d

Waiting

APPSPACK Example


Appspack example5

Workers Mapping

h

c

d

i

Waiting

APPSPACK Example

j,k


Appspack example6

Workers Mapping

i

Waiting

APPSPACK Example


Appspack example7

Workers Mapping

l

m

n

i

Waiting

APPSPACK Example

o


Appspack example8

Workers Mapping

i

Waiting

APPSPACK Example


Appspack example9

Workers Mapping

p

q

r

i

Waiting

APPSPACK Example

s


Appspack example10

Workers Mapping

Waiting

APPSPACK Example

Note: Cannot Prune on Unsuccessful Iteration

s


Appspack example11

Workers Mapping

s

u

t

v

Waiting

APPSPACK Example


Space mapping a conduit between the low and high fidelity model design spaces

x MappingH

xH

high-fi

model

mapped

low-fi model

xL=P(xH)

RL(P(xH))~RH(xH)

such that

RL(P(xH))

RH(xH)

We’re using the mapping

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.


Oracle
Oracle Mapping

  • 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.


The mfo scheme combining appspack and space mapping

Oracle Mapping

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


Mfo algorithm
MFO Algorithm Mapping

  • 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.


A Simple Example Mapping

View of Unmapped Low Fidelity

Design Space

View of High Fidelity Design Space


Mfo results
MFO Results Mapping


When the # response points is 8, there are Mapping

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)



Groundwater remediation via optimization
Groundwater Remediation via Optimization Mapping

  • 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


Optimization

Objective Function Mapping

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

Don’t 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)

Optimization


Mfo numerical test
MFO Numerical Test Mapping

  • 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)


Mfo results1
MFO Results Mapping

Initial cost: $78,586

MODFLOW (mf2k): ~2 seconds

mt3d: ~50 seconds


Mfo results2
MFO Results Mapping


Acknowledgements

MFO development team (Sandia) Mapping

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@sandia.gov

Acknowledgements

  • Software

    • APPSPACK: software.sandia.gov/appspack/version4.0/index.html

    • DAKOTA: http://endo.sandia.gov/DAKOTA


ad