Tan Matthias Hwai Yong School of Data Science City University of Hong Kong

Transcription

1 Gaussian Process Modeling of Solutions to Partial Differential Equation Models with Information from Boundary and Initial Conditions Tan Matthias Hwai Yong School of Data Science City University of Hong Kong 1

2 Introduction Consider building Gaussian process (GP) emulators for time-consuming finite element simulators that use information from Dirichlet boundary and initial conditions. Finite element method: numerical method used to solve a system of partial differential equations (PDEs) with initial and boundary conditions, which is often called an initial-boundary value problem. It is a deterministic method.

3 Physical modeling: PDE with initial and boundary conditions Heat equation: standard PDE model for heat conduction. Parameters h = convective heat transfer coefficient =ambient temperature = thermal conductivity = volumetric specific heat

4 Physical modeling: PDE with initial and boundary conditions PDE: the heat equation, =, 2h,, [0,]. Dirichlet BCs, =! ", $ ", (0,],, =! &, $ &, (0,], Neumann BC ' ( ",( &, = 0 ), (0,], Robin BC ' ( ",( &, = h, *, (0,], Initial condition,0 =! + $.

5 Finite element method: illustration Approximate PDE solution with a piecewise polynomial, i.e., a polynomial (linear, quadratic) within each element. Use spatial nodes ",, - to generate functional bases. For piecewise linear polynomials, nodes are the vertices of the elements. PDE solution:, = - 02" / 0 1 0, where the 1 0 s are the tent basis functions = 4 1,6 = 7 0,6 8 7.Thus, 0, = / 0. Time nodes ",, :. ; 0 0 = 1,; 0 3 = 0, Output at, :, = - : 02" 32" 0, ; 3. Solve a system of algebraic equations for at time points ",, :, which gives 0, 3,7 = 1,,<,6 = 1,,=.

6 Finite element method: solving the heat equation

7 Computer experiments with varying Dirichlet boundary and initial conditions Uncertainty quantification problem: quantify the effect of uncertainty in the Dirichlet boundary and initial conditions on the PDE solution. Time-consuming PDE emulator. Change the Dirichlet boundary and initial conditions (and other inputs), represented by input vector $, in a computer experiment, use data to approximate relationship between Dirichlet boundary and initial conditions, and the PDE solution? $,. Assume that the Dirichlet boundary! ", $, ", (0,]and initial condition! + $, are cheap-tocompute/ known.

8 Computer experiments with varying Dirichlet boundary and initial conditions Know that PDE solution? satisfies? $,, =! ", $, ", (0,],? $,,0 =! + $,. Propose a GP $,, that gives predictions that are exactly equal to the Dirichlet boundary and initial conditions at the space-time boundaries, $,, =? $,, =! ", $, ", $,,0 =? $,,0 =! + $,. Helps improve prediction accuracy. Proposed GP model is tractable when? $, is observed on a large node set.

9 Example: heat equation ( ",0, = D " B D & ( " =! ", $, ( ",1, = D ) B D * ( " =! &, $ ( ",0, = 30 B 5( ", ( ",1, = 35 5( " ( ",0, = 30 B 0( ", ( ",1, = 30 B 5( "

10 Example Szasz, A., Szasz, N., and Szasz, O. (2010). Hyperthermia results and challenges, InOncothermia: Principles and Practices(pp ). Springer, Dordrecht. PDEs: Pennes eq. Maxwell s eqs. 2D model Monsalvo, J. F., García, M. J., Millwater, H., and Feng, Y. (2017). Sensitivity analysis for radiofrequency induced thermal therapies using the complex finite element method.finite Elements in Analysis and Design, 135, D model

11 Examples in the literature Groundwater flow with random Dirichlet boundary conditions representing random hydraulic heads Shi, L., Yang, J., and Zhang, D. (2009). A stochastic approach to nonlinear unconfined flow subject to multiple random fields.stochastic Environmental Research and Risk Assessment, 23(6), Power transistor model with random Dirichlet boundary conditions representing random potentials at the drains and source pads termaten, E. J. W., Putek, P. A., Günther, M., Pulch, R., Tischendorf, C., Strohm, C.,... and Feng, L. (2016). Nanoelectronic COupledproblems solutions-nanocops: modelling, multirate, model order reduction, uncertainty quantification, fast fault simulation.journal of Mathematics in Industry, 7(1), 2. 11

12 Examples in the literature Burgers equation with initial conditions generated from a Gaussian process Cho, H., Venturi, D., and Karniadakis, G. E. (2014). Statistical analysis and simulation of random shocks in stochastic Burgers equation.proc. R. Soc. A,470(2171),

13 GP modeling of functional outputs $ =vector of inputs to simulator Output is a function? $, (solution to the PDE) defined on a space-time domain E. Observations on? $, at the points in the set M = ",, G are used as data. Abusing notation, can refer to a spatial coordinate (for time-independent problems), or a space-time coordinate (for time-dependent problems). M is the set of nodes or Cartesian product of nodes and time grid. Experiment design: H = $ ",,$ I K.

14 Standard separable GP model for functional outputs Stationary and separable GP prior L $,. Prior covariance function: cov L $,,L $, = Q R & S ", T " S & $,$ T &. Q R & is the variance, S " and S & are stationary correlation functions with parameters T " and T &. Prior mean function: U L $, = /. Experiment data: V = L $ ", ",,L $ ", G,,L $ I, ",,L $ I, G Distribution of V(used to derive likelihood): V~Y /Z IG,Q & R [ & [ " W

15 Standard separable GP model for functional outputs Computationally efficient formulas for the posterior mean and variance functions, and the MLEs can be obtained using Kronecker product matrix algebra formulas: [ ]" = [ & ]" [ " ]" and [ = [ & G [ " I However, because Y can be very large, computing [ " ]" can be very time consuming, which makes use of the standard separable GP model time consuming.

16 KL-GP model (KL=Karhunen Loève) We shall consider a GP model given by the collection of random variables W ^ $ $, ",,@ $, G :$ K only interested to predict PDE solution at M = ",, G E instead of all E. Justification: The simulator output is? $, = 02"? $, To match with simulator, our model should be $, = $, Assumption: ` = cov ^ $ and a = U ^ $ do not depend on $. Denote the eigendecomposition of `by ` = bcb W. bis an Y Y matrix with the 7thcolumn given by the 7thorthonormal W eigenvector b 0 = ; 0 ",,; 0 G, and c = diag h",,h G is the diagonal matrix of eigenvalues. G

17 KL-GP model Model: ^ $ = a B bc "/& c ]"/& b W ^ $ a = a B bc "/& j $. a = U ^ $ = k ",,k G W. j $ = c ]"/& b W ^ $ a = l " $,,l G $ W is a vector of independent standard normal random variables for any fixed $. l ",,l G are independent GP s defined on Kand l 0 has correlation function denoted by R 0, n 0, where n 0 is a vector of parameters. Covariance function: cov ^ $,^ $ = G W 02" h 0 R 0 $,$ n 0 b 0 b 0. cov ^ $,^ $ is separable with respect to $and if and only if all R 0 s are the same.

18 KL-GP model Estimation method Notation: ^0 $ 0, ",,@ $ 0, G W, ^ = ^",,^I. ap = ^q = ^ Z/r. I `s = t = " I ^0 W 02" ^q ^0 ^q = " I ^ u ZZ W /r ^ W. Eigendecompositionof sample covariance matrix: t = bscsbs W estimates of eigenvalues and eigenvectors. l 0 $ 3 = h 0 ]"/& ^3 a W b 0. js 0 = lv 0 $ ",,lv 0 $ I W = h v 0 ]"/& ^ ^qz W W bs 0, if hv 0 > 0. hv 0 = 0, lv 0 $ 3 is arbitrary.

19 KL-GP model Fit independent GP models to the data lv 0 $ ",,lv 0 $ I,7 = 1,,Y. Interpolation $ 3, y = kz y B G 02" hv 0 lv 0 $ 3 ;{ 0 y. Orthonormal eigenvectors (the b 0 s) are unique only up to a change in sign if the eigenvalues (the h 0 ) are unique (no duplicity). Changing the sign of b 0 and bs 0 leads to a change in sign of l 0 and lv 0 but no change in model predictions. Leads to essentially the same PCA approach used by Higdon et al. (2008) and Fricker et al. (2013), but these authors did not give a statistical model for the procedure.

20 KL-GP model The mean square convergence rates of `sand ap are both r ]" under increasing domain asymptotics. Small sample performance in estimating aand `can be poor. However, due to interpolation property, prediction performance of KL-GP model is good even when sample size is small. Computationally efficient for large Ybecause we only need to compute the eigendecompositionof tonce and fit a small number of independent GP models with rdata points. For standard GP model, need to compute the Y Y matrix [ " ]" many times when optimizing the likelihood.

21 Proposed model Dirichlet boundary and initial conditions are defined on a subset of the space-time boundary. Let the boundary nodes be denoted by B = ~ ",,~. Then, the Dirichlet boundary and initial conditions can be written compactly as? $,, = $,, $ K,, B M. Note that Kis an uncountable set cannot update standard GP model with this information.

22 Inverse distance interpolator of boundary and initial conditions Use an inverse distance interpolator of Dirichlet boundary and initial conditions $,, = & 02" $,~ 0 / ƒ,,~ 0 / 02" 1/ & ƒ,,~ 0, where ƒ is a distance metric with parameter h: & ƒ,,, = & 02" ( 0 ( 0ˆ B h &. Select h by minimizing sum of squared difference between the interpolator and the experiment data, i.e., values of? $,, observed on H M.

23 Proposed model (Modified KL-GP) Model the difference between the data and this interpolator with the KL- GP $,, =? $,, $,,. Proposed model $,, B $,,. $,, B $,, =? $,, = $,, $ K,, B, i.e., it satisfies the Dirichlet boundary and initial conditions. Standard GP/ KL-GP models do not. Use of $,, is reasonable even though the data points for this interpolator is only at parts of the boundary (large portions of the spacetime domain without interpolation points) because $,, is a convex combination of the values $,~ ",, $,~ won t give unreasonable values. ( ",0, = 30 B 5( ", ( ",1, = 35 5( "

24 Example: Electromagnetic heating of a tissue containing tumor Heating of a tissue containing tumor subjected to an electromagnetic field, which has applications in cancer treatment via hyperthermia (Thiebaut and Lemonnier, 2002; Lv et al., 2005), can be modeled via the Pennesequation and the potential equation. Objective: find conditions under which the temperature in the tumor exceeds 42 and the temperature in the healthy tissue does not exceed 42. PennesEq: 2h B Q /2 1 & & B U = 0 Potential Eq: Š 1 = 0 We shall consider only steady state, time-independent case.

25 Example: Electromagnetic heating of a tissue containing tumor D " is the frequency of the electromagnetic wave. Q D " and Š D " are the conductivity and dielectric permittivity, which are functions of D ". Cole-Cole models: Q Œ Ž = 1/ 3.31 B ( )/ 1 B D " / )*, Q š = 1/ 1.94 B ( )/ 1 B D " / ). For healthy tissue: Q D " = real Q Œ Ž For healthy tissue: Š D " = imag Q Œ Ž. / 2ŸD " Similarly for tumor.

26 Example: Electromagnetic heating of a tissue containing tumor Spatial domain and mesh Tumor represented by red square Electrodes represented by two thick black lines. Rest of the domain are healthy tissue. D & =potential difference of electrodes D ) =temperature at top boundary, D * =temperature at bottom boundary D,D =thermal conductivity of healthy tissue and tumor D,D =value of 2h (product of perfusion rate and volumetric specific heat of blood) for healthy tissue and tumor Ranges of variables in experiment: D 1 D 2 D 3 D 4 D 5 D 6 D 7 D 8 Minimum Maximum

27 Example: Electromagnetic heating of a tissue containing tumor Inverse distance interpolator $, of Dirichlet boundary condition depends on D ) and D * but not on other D 0 s. Left figure plots $, when D ) = 36 and D * = 28.

28 Example: Electromagnetic heating of a tissue containing tumor Proposed model= Modified KL-GP

29 Matthias Tan Thank you!