eArtius HMGE Algorithm Applied to Optimization Tasks with 10,000 Design Variables

Boosting Optimization Standards

eArtius HMGE Algorithm Applied to
Optimization Tasks with 10,000
Design Variables
Vladimir Sevastyanov
eArtius, Inc., Irvine, CA 92614, USA
vladimir@eartius.com

©Copyright eArtius Inc 2012 All Rights Reserved October 22, 2012

A number of algebraic models are designed
at eArtius with the following properties:
 2 objective functions
 10,000 design variables
 Only a few hundred design variables are
significant; all others are not significant
 Design space has a predetermined number of
sub-areas (“craters”) where objective functions
really depend on design variables;
 Each such a sub-area has own list of significant
design variables

The models are designed to test scalability of
eArtius optimization algorithms


Hybrid Multi-Gradient Explorer
(HMGE) Optimization Method


How HMGE algorithm works:
 Basically, HMGE works as a Genetic Algorithm;
 Periodically HMGE evaluates gradients, and performs
gradient-based steps, which improves convergence
dramatically
How gradient based steps are performed:
 Evaluate the model on 5-7 points generated in a small sub-
region around current point;
 Recognize the most significant design variables, and build
an approximation for each output variable;
 Estimate gradients based on the approximations;
 Determine a direction of simultaneous improvement for all
objective functions;
 Perform a step in the direction


Hybrid Multi-Gradient Explorer (HMGE)
Optimization Algorithm

Synergy of the features brings
HMGE on unparalleled level of
efficiency and scalability Genetic Algorithm
Framework
HMGE is believed to be the
first global multi-objective
optimization algorithm which
provides:

- Efficiency in finding the Random Mutation Gradient Mutation
global Pareto frontier

- High convergence
typical for gradient-
based methods

- Scalability: Equal
efficiency optimizing
models with dozens, DDRSM – Super Fast Gradient Estimation
hundreds, and even
thousands of design
variables


Dynamically Dimensioned Response Surface
Method (DDRSM) for Gradient Estimation
DDRSM evaluates gradients necessary for any gradient based
optimization algorithms.
Start iteration:
Start iteration: Builds local approximations
Builds local approximations
Determines the most
Determines the most for each response based
for each response based
How DDRSM operates: significant design variables
significant design variables only on the most
only on the most
for each response
for each response significant design variables
significant design variables
variable separately
variable separately

Analytically estimates
Analytically estimates
Performs a gradient
Performs a gradient gradients based on local
gradients based on local
based step
based step approximations
approximations
DDRSM Benefits:
 Equally efficient and accurate for any task dimension
 Requires just 0-7 model evaluations regardless of task dimension
 Fast— it builds a local approximation in 10-30 milliseconds
 Automatic and hidden from users
 Eliminates necessity in global response surface methods
 Eliminates necessity in a sensitivity analysis

Optimization Results for Tasks with
10,000 design Variables


eArtius optimization technology is not
sensitive to the model dimension because
it performs optimization in a sub-space of
the design space related to the most
significant design variables.

All non-significant design variables are
dynamically recognized in runtime, and
simply ignored.

Thus, eArtius algorithms are equally
efficient for low-dimensional and high-
dimensional tasks.

Algebraic Model Description
eArtius has developed an algorithm
which allows to generate algebraic
models with a predetermined number
of design variables, and a
predetermined type of topology.

The functions are similar to Gaussian, but high-dimensional, with 10,000

design variables, and with a given number of “craters”. We tried to

optimize functions with 7, 8, and 10 “craters”.

The following fragments of the algebraic models give an idea about the
functions (both functions require 425KB memory in a text format):
F1 = (-3.309 + (-2300 * exp(-0.03176 * ((X0 + X1 + X2) / 3 + 5))) + (2.655 + (-4200 * exp(-0.03589 * ((X3
+ X4 + X5 + X6 + X7) / 5 + 5))) + …
F2 = (3.02 + (-2100 * exp(-0.02191 * ((X0 + X1 + X2) / 3 + 3))) + (-4.109 + (-3700 * exp(-0.03553 * ((X3 +
X4 + X5) / 3 + -5))) + …


Optimization Results for 7 “Craters”

– 2 objectives to be minimized
– 10,000 design variables with the range [-10, 10]
– 1000 model evaluations
– 53 Pareto optimal solutions



– 97 Pareto Optimal Points


– 54 Pareto Optimal Points

An Engineering Example of an
Optimization Task Solved by HMGE
Optimization Method in ANSYS
Workbench Environment


Multi-Physics Steady State
Thermoelectric Simulation coupled with
Solid Works Shape Optimization and
Transient Radiative Heat Transfer for
Substrate Heat-up

left substrate

right

Design Modeler
(imports geometry parameters
Solid Works from Solid Works, modifies
Workbench+eArtius
model adding symmetry

Complex Multi Physics Problem
Design Modeler Parametric Steady State Thermo- Transient Radiation
Geometry interface with Solid Electric with Surface to with Surface to Surface
Surface Radiation Project folders
Works

Optimization Method
selection (MGE)

WorkBench status Bar
(stop button)

Optimization Messages updates (# of
data points computed)

Optimization Log output


Optimization Parameters
Heater Geometry dimension 1,
from Solid Works

Substrate Temperature
after short term transie
exposure to heater

F1= Tmax-350
F2=Tmin-350
350 =>desired process
temperature we want to
P23=P24 (heating on left reach
Heater Geometry dimension 2, side=heating on right)
from Solid Works
Electrical current runs through 3 separate heating elements creating temperature distribution. Electrical power in each heater equals I*V
and to minimize P18 we need to find optimal ratio of power between center and left/Right elements.


Optimization Results

dT,
Deg. C
dT,
Deg. C

(Tmin-350), Deg. C
Heater Geometry dimension 1

dT,
dT,
Deg. C Want to pick best
Deg. C
Values for
geometry
dimensions
1 and 2

(Tmax-350), Deg. C
©Copyright eArtius Inc 2012 All Rights Reserved Heater Geometry dimension 2 October 22, 2012

Most Essential Result

dT,
Deg. C

Optimal Range of 35
interest found 31

(Tmax-350), Deg. C

These are demo results of overnight –run, so study is not complete. However, we instantly see relationship between key
conflicting variables (P18- maximum temperature difference in substrate) vs F1 – deviation from desired maximum
temperature. The larger F1 the lower maximum temperature during heat-up, that means lower thermal ramp (gradient), lower
power and thus lower temperature difference P18.
It is easy to have low temperature difference if you heat less, it means you loose less heat as well and thermal uniformity is
better. In this problem we need to heat more, thus we are interested in Pareto frontier distribution looking for multiple trade-
offs.


Summary Result

Global computational optimization
of heating module and element
designs to minimize temperature
difference on substrate surface
(DeltaT).

Optimization uses state-of-the art
hybrid genetic-multi-gradient
optimization methodology.

Dimension 2
Dimension 1

Plotting by EXCEL using CSV
export from eArtius

Increase in power
reduces
uniformity Optimal power ratio ~2


eArtius HMGE Algorithm Applied to Optimization Tasks with 10,000 Design Variables

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eArtius HMGE Algorithm Applied to Optimization Tasks with 10,000 Design Variables