USNCCM 2023
Multi-Output Approximate Control
Variate Estimation for Enhanced
Variance Reduction
Thomas Dixon & A. Gorodetsky, University of Michigan
G. Bomarito & J. Warner, NASA Langley Research Center
July 26, 2023
Uncertainty Quantification
● Example system
○ Entry, descent, and landing (EDL)
○ Perseverance Mars rover
● System uncertainties
○ Entry velocity
○ Entry location
● Multiple fidelity models
○ Full physics model
○ Reduced physics models
○ Machine learning
● Multiple outputs
○ Landing location
○ Landing velocity
● Multiple statistics
○ Mean
○ Variance
○ Sobol index
https://mars.nasa.gov/mars2020/timeline/landing/entry-descent-landing/
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Sampling and Estimation
● Process:
○ Sample from the input space
○ Run the samples through the model
○ Estimate statistics using function evaluations
● Problem:
○ Accurate statistics are very expensive
○ Many function evaluations are required for accurate model statistics
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Variance Reduction
● Goal:
○ Reduce the variance of
the estimator
○ Increase the confidence
in an estimation
○ Reduce the cost of
estimation for fixed
confidence
● Monte Carlo Variance
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Multi-Fidelity Estimation
● Use multiple models for
variance reduction
● Cheaper data is available
from multiple sources
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[1] Giles. (2007). “Variance Reduction through Multilevel Monte Carlo Path Calculations”
[2] Peherstorfer, et al. (2016). “Optimal Model Management for Multifidelity Monte Carlo Estimation”
[3] Gorodetsky, et al. (2020). “A generalized approximate control variate framework for multifidelity uncertainty quantification”
[4] Schaden, et al. (2020) “On multilevel best linear unbiased estimators”
Every Method Requires:
Covariance of Estimators
Multi-Output Estimation
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● Multiple outputs of the model
○ Velocity, latitude, longitude
● Multiple statistics of the model
○ Simultaneous mean and variance estimation
● Previous Work:
● Multi-Output MLBLUE
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[5] Croci et al. (2023). “Multi-output multilevel best linear unbiased estimators via
semidefinite programming”
[6] Destouches et al. (2023). “Multivariate extensions of the Multilevel Best Linear
Unbiased Estimator for ensemble-variational data assimilation”
Contributions and Rest of Talk
● Contributions
○ Closed-form expressions for estimator covariances
○ Multi-output approximate control variate (ACV) estimator
● Outline
○ Control variates
■ Estimator covariances
○ Theoretical results
○ Experiments and results
■ EDL example
○ Future work
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Low-fidelity
Many samples
High-fidelity
Few samples
Control Variates
● Using correlated random variable to reduce variance
○ Control variate has known mean
○ Unbiased estimator
● Minimize variance to find optimal weight
● Optimal variance
○ Variance is reduced
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Covariance of Estimators
ACVs
● Control variate with unknown mean
○ Need to estimate the mean of the control variate
○ Control variate estimators take different input samples (z)
● Multiple control variates
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“A generalized approximate control variate framework for
multifidelity uncertainty quantification.” Gorodetsky, et al.
(2020)
(a) MLMC (b) MFMC
(c) ACV Independent
Samples
(a) Multilevel Monte
Carlo (MLMC)
(b) Multifidelity Monte
Carlo (MFMC)
(c) ACV Independent
Samples
Multi-Output ACV (MOACV)
● Creating one estimator for all outputs
○ Leverages correlations between model outputs
○ Uses model outputs across fidelities as control variates for other outputs
● Previously, separate estimators have been created for each model output
● Old estimator equation:
○ Two outputs
○ One low fidelity model
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First model output
Second model output
Unused correlation
High-fidelity
estimator
ACV
estimator
Control variate
weights
Multi-Output ACV (MOACV)
● Creating one estimator for all outputs
○ Leverages correlations between model outputs
○ Uses model outputs across fidelities as control variates for other outputs
● Previously, separate estimators have been created for each model output
● Old estimator equation:
○ Two outputs
○ One low fidelity model
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First model output
Second model output
New weights
High-fidelity
estimator
ACV
estimator
Control variate
weights
Combining Estimators - Multiple Statistics
● Simultaneous statistic estimation
○ Further reduced variance
● Correlations between estimators can also be extracted
● Equation
○ Any number of outputs
■ Mean
■ Variance
○ One low-fidelity model
○ Boxed in green are the new control variate weights
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ACV mean estimator
ACV variance estimator
Types of Estimators
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Estimator Statistics
Quantities of
Interest
Fidelities
MC - Monte Carlo
Single Single Single
ACV
Single Single Multiple
MOACV
Single Multiple Multiple
Combined MOACV
Multiple Multiple Multiple
Theoretical Results
Mean Estimator Covariances
● Mean estimation
● Covariance between mean estimators
● Required covariances
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Theoretical Results
Variance Estimator Covariances
● Variance estimation
● Covariance between variance estimators
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Theoretical Results
Combined Estimator Covariances
● Mean and variance
estimation
● Covariance between
estimators
○ Using previous results
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Theoretical Results
Combined Estimator Covariances
● Mean and variance
estimation
● Covariance between
estimators
○ Using previous results
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Main takeaway:
Closed-form expressions for the
covariance between estimators
EDL Application
● “Multi-Model Monte Carlo Estimators for
Trajectory Simulation” by Warner, et al. (2021)
● 75 uncertain inputs
● 15 Quantities of Interest (QoIs)
● One high-fidelity model
○ Trajectory simulation
● Three low-fidelity models
○ Reduced physics model
○ Coarse time step model
○ Machine learning model
● Mean and variance estimation
○ Optimized sample allocation
○ Minimized weighted sum of variances
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s
Correlation Between Models
● Plotting correlations between high-fidelity and the low-fidelity models
○ Each QoI has a separate correlation between models
● Maximum acceleration has low correlations between models
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QoIs
Types of Estimators
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Estimator Statistics QoIs Fidelities
MC
Single Single Single
ACV
Single Single Multiple
MOACV
Single Multiple Multiple
Combined MOACV
Multiple Multiple Multiple
Results - Mean Estimation
● Variance reduction for each QoI
● Significant reduction in QoI with high correlations
● MC achieved at red line
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Results - Variance Estimation
● Significant variance reduction achieved
compared to traditional approaches
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Future Work
Stochastic Optimization
● Goal: Minimize finite-sum objective function
● Stochastic gradient descent
● Approximate the gradient of the objective
● Variance reduction methods
○ Variance of estimate approaches 0
○ Examples:
■ Stochastic variance reduced gradient (SVRG)
[1]
■ Stochastic average gradient (SAG)
[2]
● Use MOACV to reduce variance further
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[1] Johnson, et al. (2013). “Accelerating stochastic gradient descent using predictive variance reduction”
[2] Roux, et al. (2012). “A stochastic gradient method with an exponential convergence rate for finite training sets”
Conclusion
● Derived closed-form expressions for estimator covariance
● New multi-output estimator outperforms previous methods
● Questions?
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Supplementary Material
