IMSPE_AL: IMSPE optimal design point search

In MuFiMeshGP: Multi-Fidelity Emulator for Computer Experiments with Tunable Fidelity Levels

IMSPE optimal design point search

Description

Search for the best next design point according to the IMSPE criterion given a current MuFiMeshGP model fit.

Usage

IMSPE_AL(
  object,
  t.min,
  t.max,
  cost.func,
  cost.new = 0,
  gr = FALSE,
  gr_cost.func = NULL,
  DesCand = NULL,
  Wijs = NULL,
  Hijs = NULL,
  control = list(multi.start.n = 20, maxit = 20, DesStart = NULL, seed = NULL, ncores =
    1)
)

Arguments

object	Current MuFiMeshGP model fit.
t.min, t.max	Lower and upper bounds on the fidelity space for the search.
cost.func	Function that maps the tunable parameter t to the corresponding cost running a simulation at that fidelity level. For example, function(t) 1/t^2.
cost.new	(optional) Cost of running a new simulation at a new fidelity level, scalar.
gr	whether the gradient should be used in the optimization of the IMSPE. (Not recommended due to numerical errors)
gr_cost.func	If grad is TRUE, the user needs to specify the gradient of the cost function, as a function.
DesCand	Design candidates to evaluate from.
Wijs, Hijs	(optional) Matrices from previous IMSPE search to obtain faster computation through matrix decomposition.
control	list of arguments udes for the optimization.

Value

a list with:

• x: the optimal input parameter, a vector.

• t: the optimal tuning parameter, a scalar.

• value: the IMSPE reduction at the optimal design location.

• new: whether the optimal tuning parameter defines a new fidelity level.

• id: the index of the optimal design location if DesCand is used.

Examples

# Example code

f <- function(x, t){
  x <- c(x)
  return(exp(-1.4*x)*cos(3.5*pi*x)+sin(40*x)/10*t^2)
}

set.seed(1)
X <- matrix(runif(15,0,1), ncol = 1)
tt <- runif(15,0.5,2)

Y <- f(c(X), tt)

fit.mufimeshgp <- MuFiMeshGP(X, tt, Y)

xx <- matrix(seq(0,1,0.01), ncol = 1)
ftrue <- f(xx, 0)

# predict
pred.mufimeshgp <- predict(fit.mufimeshgp, xx, rep(0,101))

mu <- pred.mufimeshgp$mean
s <- pred.mufimeshgp$sd
lower <- mu + qnorm(0.025)*s
upper <- mu + qnorm(0.975)*s

# plot

oldpar <- par(mfrow = c(1,2))
plot(xx, ftrue, "l", ylim = c(-1,1.3), ylab = "y", xlab = "x")
lines(c(xx), mu, col = "blue")
lines(c(xx), lower, col = "blue", lty = 2)
lines(c(xx), upper, col = "blue", lty = 2)
points(c(X), Y, col = "red")

### RMSE ###
print(sqrt(mean((ftrue - mu))^2))

best <- IMSPE_AL(fit.mufimeshgp, 0.5, 2, function(t) return(1 / t^2))
new.Y <- f(best$x, best$t)
fit.mufimeshgp <- update(fit.mufimeshgp, best$x, best$t, new.Y)

pred.mufimeshgp <- predict(fit.mufimeshgp, xx, rep(0, 101))
mu <- pred.mufimeshgp$mean
s <- pred.mufimeshgp$sd
lower <- mu + qnorm(0.025)*s
upper <- mu + qnorm(0.975)*s

plot(xx, ftrue, "l", ylim = c(-1,1.3), ylab = "y", xlab = "x")
lines(c(xx), mu, col = "blue")
lines(c(xx), lower, col = "blue", lty = 2)
lines(c(xx), upper, col = "blue", lty = 2)
points(c(X), Y, col = "red")
points(c(best$x), new.Y, col = "green")
par(oldpar)

### RMSE ###
print(sqrt(mean((ftrue - mu))^2))

MuFiMeshGP documentation built on Sept. 1, 2025, 5:09 p.m.