# Bayesian Model Calibration

## An Elastic Approach

### 2023-01-24

research.tetonedge.net

## Sandia National Laboratories

• Department of Energy National Lab
• 14,000 Staff across 7 Locations
• Two Statistics Departments
• 30 Full Time Staff, 4 Post-Docs, 10 year round interns

## Outline

• Introduction
• Functional Data Analysis
• Elastic Metric
• Bayesian Model Calibration
• Results
• Simulation
• Sandia Z-Machine

## Introduction

• Question: How can we model functions
• Can we use the functions to classify diseases?
• Can we use them as predictors in a regression model?
• Can we calibrate a computer model?

• One problem occurs when performing these types of analysis is that functional data can contain variability in time (x-direction) and amplitude (y-direction)
• How do we characterize and utilize this variability in the models that are constructed from functional data?

## Functional Data Analysis

• Let $f$ be a real valued-function with the domain $[0,1]$, can be extended to any domain
• Only functions that are absolutely continuous on $[0,1]$ will be considered
• Let $\Gamma$ be the group of all warping functions $\Gamma = \{\gamma:[0,1]\rightarrow[0,1]|\gamma(0)=0,\gamma(1)=1,~\gamma \mbox{ is a diffeo}\}$
• It acts on the function space by composition $(f,\gamma) = f\circ\gamma$
• It is common to use the following objective function for alignment $\min_{\gamma\in\Gamma}\|f_1\circ\gamma-f_2\|$
• Note: It is not a distance function since it is not symmetric.

## Elastic Distance (Fisher-Rao)

Define the Square Root Velocity Function $q:[0,1]\rightarrow\mathbb{R}^1,~q(t)=sign(\dot{f}(t))\sqrt{|\dot{f}(t)|}$ Fisher Rao Distance is $\mathbb{L}^2$ in SRVF space $d_a(f_1,f_2) = \inf_\gamma\|(q_1\circ\gamma)\sqrt{\dot{\gamma}}-q_2\|$ Distance is a proper distance

Can compute distance on warping functions (how much alignment) $d_p(\gamma) = \arccos\left(\int_0^1\sqrt{\dot{\gamma}}\,dt\right)$

## Bayesian Model Calibration

• We wish to calibrate a computer model with parameters $\theta$ to an experiment
• Can compute computer model (simulations) over wide range of $\theta$
• The data is functional in nature and has phase and amplitude variability
• Utilize elastic metrics in a Bayesian Model Calibration Framework

## Elastic Bayesian Model Calibration

• Decompose observation into aligned functions and warping functions $y_i^E(t) = y_i^E(t^*)\circ\gamma_i^E(t)$
• and decompose the simulations $y^M(t,x_j) = y^M(t^*,x_j)\circ\gamma^M(t,x_j)$
• To facilitate modeling, we transform the warping functions into shooting vector space with $v_i^E = \exp_\psi^{-1}\left(\sqrt{\dot{\gamma}_i^E}\right)$ $v^M(x) = \exp_\psi^{-1}\left(\sqrt{\dot{\gamma}^M(x)}\right)$

## Elastic Bayesian Model Calibration

• Calibrate the aligned data and shooting vectors using the following model $y^E(t^*) = y^M(t^*,\theta)+\delta_y(t^*)+\epsilon_y(t^*),~\epsilon_y(t^*)\sim\mathcal{N}(0,\sigma_y^2I)$ $v^E = v^M(\theta) + \delta_v + \epsilon_v,~\epsilon_v\sim\mathcal{N}(0,\sigma_v^2I)$
• Note: The shooting vector will be identity if the data is aligned to the observation (experiment)
• Then if $\theta$ is calibrated correctly the shooting vectors will be identity

## MCMC Sampling

For each experiment the likelihood is a Gaussian likelihood

1. We fit an emulator (Gaussian Process, BASS, MARS) to the simulated data
2. Uniform priors on $\theta$
3. Sample posterior using delayed rejection adaptive Metropolis Hastings
4. Implemented using Impala (LANL) or Dakota (SNL) calibration framework

## Simulation

• Simulation study where each function is parameterized Gaussian pdf
• A set of 100 functions were simulated with $\theta_1,\theta_2$ being drawn from a $U[0,1]$
• Third nuisance parameter $\theta_3$ also drawn from $U[0,1]$

$f_i(t) = \frac{\theta_1}{0.05\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{t-(\sin(2\pi \theta_0^2)/4-\theta_0/10+0.5)}{0.05}\right)^2\right)$

## Calibration

• Trained BASS Emulator on Aligned Functions and Shooting Vectors (using elastic fPCA)
• Calibrated using framework with tempering and adaptive MCMC
• Blue shows draws from posterior distribution at 95% credible interval

## Simulation with Discrepancy

• Repeat same simulation model with with the experiment having a timing discrepancy
• Discrepancy modeling with basis functions in shooting vector space

## Calibration

• Trained BASS Emulator on Aligned Functions and Shooting Vectors (using elastic fPCA)
• Calibrated using framework with tempering and adaptive MCMC
• Blue shows draws from posterior distribution at 95% credible interval

Video

## Calibration of Tantalum

• Calibration equation of state of tantalum generated with pulse magnetic fields
• Estimate the parameters describing the compressibility (relationship between pressure and density) to understand extreme pressures
• Conducted using Sandia Z-machine, a pulse power drive reactor

## Calibration

• We have a good fit between experiment and simulation
• Better residuals over standard calibration method

## Tantalum Parameters

• With the elastic method we have tighter posteriors over previous methods
• Additionally, scientists at Z have confirmed the parameters conform more to physical understanding of the material

## Summary

• Functional metrics provide a global measure of the difference of a function in terms of amplitude and phase
• Integrated elastic functional metrics into Bayesian Model Calibration framework utilizing aligned data and shooting vector representation
• Demonstrated ability on simulated and tantalum equation of state calibration problems
• Future Work
• Additional testing on real world examples

Questions?

jdtuck@sandia.gov