Adaptive Response Surface Methods ARSM for Design Optimization

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Adaptive Response Surface Methods (ARSM) for
Design Optimization
CAD CG National Research Center Zhejiang
University August 19, 2002

• Gary Wang, Assistant Prof.
• U. of Manitoba, Canada

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Computation-intensive Design Problems

• At least one function requires a
  computation-intensive analysis for function
  evaluation this function can be either the
  objective function or a constraint function.
• The gradient information or any other properties
  of the computation-intensive function is not
  available, or computing those properties is too
  costly or unreliable
• A goal is to minimize the total number of
  computation-intensive function evaluations and
  maximizing the amount of parallel computation.
• The simulations and analyses are deterministic

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An Example
Each simulation takes 36-160 hrs for Ford Motor
Inc.
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Westland Helicopters
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Virtual Prototyping-based Design
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Design for Mafg./Production
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The Challenge

• Design with computation-intensive processes
• Global optimum with constraints?
• Balance

Global Optimum
Compu-tation cost
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Outline

• Review of the Response Surface Method (RSM)
• Essence of the Adaptive Response Surface Method
  (ARSM)
• ARSM Inherited Samples
• Fuzzy clustering based space reduction method

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Conventional Optimization

• Problems
• Sequential
• Time (Iteration)x(160 hrs, e.g.)
• Only one optimum, which may not be acceptable
• What if the process got stuck?

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Response Surface Method

• RSM, metamodeling, approximation-based design,
  etc.
• RS metamodel, surrogate

• Idea
• Treat f(x) unknown when given an input x0, it
  gives an output f(x0)
• Sampling regression ? Optimization

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Response Surface Models

• A response surface model derives its name from
  being a model of a response that looks like a
  surface in 2-D
• General form of a response surface
• where
• f(x) can be linear, linearinteractions, or a
  full second-order polynomial function of x
• bi are parameters used to fit the model using
  least squares regression bi (XX)-1XY

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Response Surface Method

• Benefits
• Parallel computation
• Quick, cheap (1160 hr)
• No software integration, easy coding
• Identify relative importance of parameters (i.e.,
  gain knowledge of the surface)

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Response Surface Method
What are the problems?
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Approaches in Literature

• Sampling, e.g. DOE, D-optimality, Latin
  Hypercube, OA, Hammersley
• Approximation Models, e.g., Polynomial, Kriging,
  RBF, ANN, MARS,
• Design Space Reduction
• Dimensionality
• Size

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Our Previous Work

• Adaptive Response Surface Method (ARSM)
• (Engineering Optimization, Vol. 33, No. 6, 2001,
  pp. 707-734)
• ARSM with Inherited Latin Hypercube Design Points
  
• (Journal of Mechanical Design, Transactions of
  the ASME, in print)

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Overview of the ARSM

• Assume f is unknown
• Gradually reduce the search space
• The minimum of the second fitted function is
  close to the real optimum at about x2.13

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Algorithm of the ARSM
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Identification of the Reduced Space
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Identification of the Reduced Design Space
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Difficulties of the ARSM

• Central Composite Designs (CCD) 2n 2n 1
• A new set of CCDs in a new design space
• Inherit the optimum from the last iteration?

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Central Composite Design (CCD) vs. Latin
Hypercube Design (LHD)
CCD for n2 M2n 2n 1
LHD for n2 M arbitrary
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Latin Hypercube Designs

• Space filling property ideal for computer
  experiments
• Uniform random sampling
• Number of points controllable (n1)(n2)/2
• Potential point inheritance

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LHD Inheritance
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LHD Inheritance
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Review of the Difficulties of the Previous ARSM

• Central Composite Designs (CCD) 2n 2n 1
• A new set of CCDs in a new design space
• Inherit the optimum from the last iteration?

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Test Problems

• Goldstein and Price function (GP), n 2.
• Six-hump camel-back function (SC), n 2.
• Branin function (BR), n 2.
• Generalized polynomial function (GF), n 2.
• Rastrigin function (RS), n 2.
• Geometric container function (Constrained) (GC),
  n 3.
• Hartman function (HN), n 6.

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Test of ARSM
Goldstein and Price Function
Goldstein and Price Function
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Test Results
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Design Results
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Advantages of the ARSM

• Numerical models are black-boxes to the design
  integrator no API coding
• A design can be simultaneously analyzed from
  various perspectives with different models,
  either commercial or home-developed models
  shorter analysis time
• The significance of each design variable and
  their interrelationship is obvious engineering
  insight
• The total number of function evaluations
  (numerical analyses) is minimum low cost and
  quick optimization
• Global design optimum can be obtained better
  product quality

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Summary

• RSM is suitable for computation-intensive
  problems
• ARSM is a better approach than RSM in terms of
  efficiency and accuracy
• LHD is better than CCD for the ARSM
• A lot more interesting issues to be solved