5 research outputs found
Tensor train construction from tensor actions, with application to compression of large high order derivative tensors
We present a method for converting tensors into tensor train format based on
actions of the tensor as a vector-valued multilinear function. Existing methods
for constructing tensor trains require access to "array entries" of the tensor
and are therefore inefficient or computationally prohibitive if the tensor is
accessible only through its action, especially for high order tensors. Our
method permits efficient tensor train compression of large high order
derivative tensors for nonlinear mappings that are implicitly defined through
the solution of a system of equations. Array entries of these derivative
tensors are not directly accessible, but actions of these tensors can be
computed efficiently via a procedure that we discuss. Such tensors are often
amenable to tensor train compression in theory, but until now no efficient
algorithm existed to convert them into tensor train format. We demonstrate our
method by compressing a Hilbert tensor of size , and by forming high order (up to order
derivatives/ order tensors) Taylor series surrogates of the
noise-whitened parameter-to-output map for a stochastic partial differential
equation with boundary output
Projected Wasserstein gradient descent for high-dimensional Bayesian inference
We propose a projected Wasserstein gradient descent method (pWGD) for
high-dimensional Bayesian inference problems. The underlying density function
of a particle system of WGD is approximated by kernel density estimation (KDE),
which faces the long-standing curse of dimensionality. We overcome this
challenge by exploiting the intrinsic low-rank structure in the difference
between the posterior and prior distributions. The parameters are projected
into a low-dimensional subspace to alleviate the approximation error of KDE in
high dimensions. We formulate a projected Wasserstein gradient flow and analyze
its convergence property under mild assumptions. Several numerical experiments
illustrate the accuracy, convergence, and complexity scalability of pWGD with
respect to parameter dimension, sample size, and processor cores
A fast and scalable computational framework for goal-oriented linear Bayesian optimal experimental design: Application to optimal sensor placement
Optimal experimental design (OED) is a principled framework for maximizing
information gained from limited data in inverse problems. Unfortunately,
conventional methods for OED are prohibitive when applied to expensive models
with high-dimensional parameters, as we target here. We develop a fast and
scalable computational framework for goal-oriented OED of large-scale Bayesian
linear inverse problems that finds sensor locations to maximize the expected
information gain (EIG) for a predicted quantity of interest. By employing
low-rank approximations of appropriate operators, an online-offline
decomposition, and a new swapping greedy algorithm, we are able to maximize EIG
at a cost measured in model solutions that is independent of the problem
dimensions. We demonstrate the efficiency, accuracy, and both data- and
parameter-dimension independence of the proposed algorithm for a contaminant
transport inverse problem with infinite-dimensional parameter field
Optimal design of acoustic metamaterial cloaks under uncertainty
In this work, we consider the problem of optimal design of an acoustic cloak
under uncertainty and develop scalable approximation and optimization methods
to solve this problem. The design variable is taken as an infinite-dimensional
spatially-varying field that represents the material property, while an
additive infinite-dimensional random field represents the variability of the
material property or the manufacturing error. Discretization of this optimal
design problem results in high-dimensional design variables and uncertain
parameters. To solve this problem, we develop a computational approach based on
a Taylor approximation and an approximate Newton method for optimization, which
is based on a Hessian derived at the mean of the random field. We show our
approach is scalable with respect to the dimension of both the design variables
and uncertain parameters, in the sense that the necessary number of acoustic
wave propagations is essentially independent of these dimensions, for numerical
experiments with up to one million design variables and half a million
uncertain parameters. We demonstrate that, using our computational approach, an
optimal design of the acoustic cloak that is robust to material uncertainty is
achieved in a tractable manner. The optimal design under uncertainty problem is
posed and solved for the classical circular obstacle surrounded by a
ring-shaped cloaking region, subjected to both a single-direction
single-frequency incident wave and multiple-direction multiple-frequency
incident waves. Finally, we apply the method to a deterministic large-scale
optimal cloaking problem with complex geometry, to demonstrate that the
approximate Newton method's Hessian computation is viable for large, complex
problems
Derivative-Informed Projected Neural Networks for High-Dimensional Parametric Maps Governed by PDEs
Many-query problems, arising from uncertainty quantification, Bayesian
inversion, Bayesian optimal experimental design, and optimization under
uncertainty-require numerous evaluations of a parameter-to-output map. These
evaluations become prohibitive if this parametric map is high-dimensional and
involves expensive solution of partial differential equations (PDEs). To tackle
this challenge, we propose to construct surrogates for high-dimensional
PDE-governed parametric maps in the form of projected neural networks that
parsimoniously capture the geometry and intrinsic low-dimensionality of these
maps. Specifically, we compute Jacobians of these PDE-based maps, and project
the high-dimensional parameters onto a low-dimensional derivative-informed
active subspace; we also project the possibly high-dimensional outputs onto
their principal subspace. This exploits the fact that many high-dimensional
PDE-governed parametric maps can be well-approximated in low-dimensional
parameter and output subspace. We use the projection basis vectors in the
active subspace as well as the principal output subspace to construct the
weights for the first and last layers of the neural network, respectively. This
frees us to train the weights in only the low-dimensional layers of the neural
network. The architecture of the resulting neural network captures to first
order, the low-dimensional structure and geometry of the parametric map. We
demonstrate that the proposed projected neural network achieves greater
generalization accuracy than a full neural network, especially in the limited
training data regime afforded by expensive PDE-based parametric maps. Moreover,
we show that the number of degrees of freedom of the inner layers of the
projected network is independent of the parameter and output dimensions, and
high accuracy can be achieved with weight dimension independent of the
discretization dimension