mcGP package

Chih-Li Sung November 10, 2023

This R package allows the emulation using a mesh-clustered Gaussian process (mcGP) model for partial differential equation (PDE) systems. More details can be seen in Sung, Wang, Ding, and Wang (2024).

You can install the package using install_github function as follows:

library(devtools)
install_github("ChihLi/mcGP")

A few examples are given below. This is a PDE system on an L-shaped membrane based on Poisson’s equation. The data is provided in this package, which can be loaded by data(poisson_dat). More details about the dataset can be found by help(poisson_dat).

load data

We see that X is the input data which is one dimensional and has five observations. S is the mesh specification, which is two dimensional and has 401 mesh nodes. Y is the multiple outputs on the 401 mesh nodes, so it has five columns (i.e., sample size of X) and has 401 rows.

library(mcGP)

# example: Poisson equation simulations 
# (data was generated by finite element methods via MATLAB)

data(poisson_dat)   
attach(poisson_dat)
print(dim(X)) # input: sample size is 5; one input variable

## [1] 5 1

print(dim(S)) # mesh node coordinates: 2-dimensional mesh; 401 mesh nodes

## [1] 401   2

print(dim(Y)) # outputs at these mesh nodes: sample size is 5; 401 outputs at mesh nodes

## [1] 401   5

Visualize two training examples

Plot the output data of $x=-0.8$ and $x=0.8$.

# visualize two training examples
library(ggplot2)
pde.plot <- function(u, nodes, ...){
  df <- data.frame(S, u)
  colnames(df) <- c("s1", "s2", "z")
  ggplot(df, aes(x=s1, y=s2, alpha=z)) + 
    geom_point(color="#69b3a2",size=5) + theme_bw()
}

pde.plot(Y[,1], S) # when x=-0.8

pde.plot(Y[,5], S) # when x=0.8

Model fitting

Perform model fitting by mcGP function.

# model fitting
set.seed(123) # for reproducibility
fit <- mcGP(X,Y,S)

Prediction

Perform predictions by predict.mcGP function. We make predictions on the test dataset X.test, which is $x=-0.25$.

# making predictions
pred <- predict(fit, Y, xnew = X.test)

Visualize the variational distribution

Visualize the variational distribution $q(Z)$.

# qZ
pde.plot(fit$q_Z[,1], S)

pde.plot(fit$q_Z[,2], S)

pde.plot(fit$q_Z[,4], S)

pde.plot(fit$q_Z[,5], S)

pde.plot(fit$q_Z[,6], S)

pde.plot(fit$q_Z[,7], S)

Visualize predictions

# prediction mean
pde.plot(pred$mean, S)

# prediction variance
pde.plot(pred$sig2, S)

Parallel computing

Parallel computing can be done by parallel=TRUE. You could specify the number of cores by the argument n.cores. More details can be found by help(mcGP).

#model fitting (with parallel computing)
fit2 <- mcGP(X,Y,S, parallel=TRUE)

print(fit$time.elapsed)  # without parallel computing

##    user  system elapsed 
##   9.506   0.052   9.638

print(fit2$time.elapsed) # with parallel computing

##    user  system elapsed 
##   7.946   0.075  10.137