Vladimir Fonoberov (Aimdyn Inc, USA) Igor Mezic (UC Santa Barbara, USA)

Computer models of physical and biological systems Matching model outputs to known data Global sensitivity GoSUM software Calibration of model of human cardiovascular system Conclusion 2

Complex models: dynamical systems 100s of parameters 10s of outputs 3

Commonly, the cost function to be minimized is L 2 norm of the difference between model outputs and real data For time-dependent outputs, both spectral and time domains can be matched, resulting in a better fit static modeldynamic model 4

5 Global measure of importance of model parameters for the outputs of interest NOT local sensitivity, e.g. partial derivatives around the nominal value Global sensitivity allows us to find parameters which are important over the entire range of interest of model parameters Very computationally intensive for very large models

6 Derivative-based, e.g. First-order sensitivity index Total effect sensitivity index

7 Direct evaluation of sensitivity integrals requires the model executable to be available and has complexity of N samp  Dimension model evaluations An analytical model representation can be created by “learning” the model (when dimension or model runtime is very large, this is the only way) We employ robust support vector regression (SVR) algorithms to learn the model The learning is based on maximizing predictive power of the analytical model representation

8 Analytical model representation where is k th input parameter sample When the number of parameters D is large and the number of samples N is comparable to or less than D, the analytical model representation may become over-smoothed This is because all parameters are considered equally important => model reduction is needed

9 Removing irrelevant parameters from the model allows us to Simplify the model Find more accurate analytical model representation and perform predictions Identify important parameters and speed-up uncertainty quantification, optimization, etc.

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11 Each of the D input parameters has a prescribed distribution (uniform, Gaussian, exponential, discrete, etc.) Depending on available time, one estimates the number of times N the model executable can be run GoSUM generates exactly N (D-dimensional) points that accurately sample the joint probability distribution of input parameters using Latinized CVT (up to a few thousand samples) and DSample (any number of samples) 300 samples MC versus Latinized CVT 5,000 samples MC versus DSample

12 Supported model executable formats Windows executable MATLAB / Octave function Simulink model No executable (have model outputs) For every input sample, GoSUM evaluates the Model Executable in parallel (using a specified number of CPU cores) Model evaluations can be stopped by user

Model of a medium office building: 3 floors, 50,000 square feet, 15 zones 941 parameters, 10% uncertainty, uniform and exponential distributions 16 outputs: yearly energy consumption for different systems 5,000 samples Our recent publications: B. Eisenhower, Z. O’Neill, S. Narayanan, V. A. Fonoberov, and I. Mezic, "A Methodology for Meta-Model Based Optimization in Building Energy Models," Energy and Buildings 47, pp , 2012 B. Eisenhower, Z. O'Neill, V. A. Fonoberov, and I. Mezic, "Uncertainty and Sensitivity Decomposition of Building Energy Models," Journal of Building Performance Simulation 5, pp ,

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15 GoSUM leans models using kernel-based support vector machine methods With regularization of noisy data SVR (L2) Epsilon-SVR (L1) Nu-SVR (L1) Without noisy data regularization RS (Response Surface) The learning is based on maximizing predictive power of analytical model representation

16 Global derivative sensitivity (L2) Global derivative sensitivity (L1) Average derivative Global variance sensitivity ANOVA decomposition based on analytical model representation

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19 The problem is to minimize (or maximize) a function of model parameters and model outputs f(par 1, par 2, …, out 1, out 2, …) in a given range of parameters a i  par i  b i, i = 1, …, D and subject to arbitrary equality and/or inequality constrains 1000s of dimensions global maximum

Global stability option in GoSUM will perform global optimization of f(par 1, par 2, …, out 1, out 2, …, unc 1, unc 2, …) in a given range of parameters a i  par i  b i, i = 1, …, D subject to arbitrary equality and/or inequality constrains and in the presence of uncertain parameters unc 1, unc 2, … with known distributions 20 requirement level operation point out of specification 1000s of dimensions functional regime

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23 When input samples are imported, arbitrary correlated distributions are automatically recognized Correlations are supported for all distributions types: continuous, discrete, categorical All correlations are regularized to reveal uncorrelated parameters Samples can be generated from arbitrary data-defined correlated distributions Optimization volume can be constrained by data boundaries

Respi- ration Vuev Vusv Vrv Vlv Emaxrv Emaxlv Heart Period xTsxTv B1, B2, 6P tasks 300 seconds each 24 about 20 states and 100 parameters

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27 Define cost function to be minimized: Use GoSUM to analyze global sensitivity of cost function terms and identify the most important parameters Use GoSUM to minimize the cost function with respect to the most important parameters Use GoSUM to study sensitivity of the solution Repeat the steps above for all subjects of interest HPL - relative difference (error) between power spectral density (PSD) of the calculated and experimental heart periods for low frequencies; HPH - heart period relative PSD error for high frequencies; HPV - heart period relative PSD error for very low frequencies; HP - relative error in mean heart period; SAP - relative error in mean systolic blood pressure; DAP - relative error in mean diastolic blood pressure; SAPT - systolic arterial pressure relative PSD error; DAPT - diastolic arterial pressure relative PSD error

Statistics over 24 subjects 28

29 Model personalization is very important

30 A method for fast global sensitivity analysis of arbitrary high- dimensional models is developed A complex model of human cardiovascular system is studied and calibrated for individual subjects For the considered model, global-sensitivity based calibration required optimization over only 25% of model parameters All algorithms are implemented in our software GoSUM: software documentation and online tutorials are available at We would like to thank our sponsors and customers: Hamilton Sundstrand, Ford, Boeing, AFOSR, NIH, and DARPA