THESIS
2019

**Abstract**
Gaussian Process(GP) is a popular technique for simulation metamodeling due to
its flexibility and analytical tractability. Its computational bottleneck is the inversion
of a covariance matrix, which takes O(n

^{3}) time in general and becomes prohibitive for
large n, where n is the number of design points. Moreover, the covariance matrix is
often ill-conditioned for large n, and thus the inversion is prone to numerical instability,
resulting in erroneous parameter estimation and prediction. These two numerical issues
preclude the use of stochastic kriging at a large scale. We presents a novel approach to
address them. We construct a class of covariance functions, called Markovian covariance
functions (MCFs), which have two properties: (i) the associated covariance matrices can
be...[

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Gaussian Process(GP) is a popular technique for simulation metamodeling due to
its flexibility and analytical tractability. Its computational bottleneck is the inversion
of a covariance matrix, which takes O(n

^{3}) time in general and becomes prohibitive for
large n, where n is the number of design points. Moreover, the covariance matrix is
often ill-conditioned for large n, and thus the inversion is prone to numerical instability,
resulting in erroneous parameter estimation and prediction. These two numerical issues
preclude the use of stochastic kriging at a large scale. We presents a novel approach to
address them. We construct a class of covariance functions, called Markovian covariance
functions (MCFs), which have two properties: (i) the associated covariance matrices can
be inverted analytically, and (ii) the inverse matrices are sparse. With the use of MCFs,
the inversion-related computational time is reduced to O(n

^{2}) in general, and can be
further reduced by orders of magnitude with additional assumptions on the simulation
errors and design points. The analytical invertibility also enhance the numerical stability
dramatically. The key in our approach is that we identify a general functional form of
covariance functions that can induce sparsity in the corresponding inverse matrices. We
also establish a connection between MCFs and linear ordinary differential equations. Such
a connection provides a flexible, principled approach to constructing a wide class of MCFs.

One example for this principled approach is that we can incorporate boundary information
to our GPs. Boundary information (i.e., the behavior of the phenomena along input
boundaries) is oftentimes known beforehand, either from governing physics or scientific
knowledge. While recent work suggests that incorporating boundary information can
improve GP emulation, such methods do not allow for amenable theoretical analysis of
this improvement. To this end, we propose a new boundary constrained Gaussian Process
(BoundGP) model for incorporating boundary information, and show that it has improved
convergence rates over existing GP models under sparse grid sampling. BoundGP enjoys an
L

^{p} convergence rate of O(n

^{−1}) and a probabilistic convergence rate of O

_{p}(n

^{−1}[log n]

^{2d−3/2}),
where d is the dimension of the input space. These rates also suggest that BoundGP is
more resistant to the well-known “curse-of-dimensionality” in nonparametric regression.
Our proofs make use of a novel connection between GP modeling and finite element
modeling.

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