3,976 research outputs found
Online interpolation point refinement for reduced order models using a genetic algorithm
A genetic algorithm procedure is demonstrated that refines the selection of
interpolation points of the discrete empirical interpolation method (DEIM) when
used for constructing reduced order models for time dependent and/or
parametrized nonlinear partial differential equations (PDEs) with proper
orthogonal decomposition. The method achieves nearly optimal interpolation
points with only a few generations of the search, making it potentially useful
for {\em online} refinement of the sparse sampling used to construct a
projection of the nonlinear terms. With the genetic algorithm, points are
optimized to jointly minimize reconstruction error and enable dynamic regime
classification. The efficiency of the method is demonstrated on two canonical
nonlinear PDEs: the cubic-quintic Ginzburg-Landau equation and the
Navier-Stokes equation for flow around a cylinder. Using the former model, the
procedure can be compared to the ground-truth optimal interpolation points,
showing that the genetic algorithm quickly achieves nearly optimal performance
and reduced the reconstruction error by nearly an order of magnitude.Comment: 18 pages, 8 figure
An efficient, globally convergent method for optimization under uncertainty using adaptive model reduction and sparse grids
This work introduces a new method to efficiently solve optimization problems
constrained by partial differential equations (PDEs) with uncertain
coefficients. The method leverages two sources of inexactness that trade
accuracy for speed: (1) stochastic collocation based on dimension-adaptive
sparse grids (SGs), which approximates the stochastic objective function with a
limited number of quadrature nodes, and (2) projection-based reduced-order
models (ROMs), which generate efficient approximations to PDE solutions. These
two sources of inexactness lead to inexact objective function and gradient
evaluations, which are managed by a trust-region method that guarantees global
convergence by adaptively refining the sparse grid and reduced-order model
until a proposed error indicator drops below a tolerance specified by
trust-region convergence theory. A key feature of the proposed method is that
the error indicator---which accounts for errors incurred by both the sparse
grid and reduced-order model---must be only an asymptotic error bound, i.e., a
bound that holds up to an arbitrary constant that need not be computed. This
enables the method to be applicable to a wide range of problems, including
those where sharp, computable error bounds are not available; this
distinguishes the proposed method from previous works. Numerical experiments
performed on a model problem from optimal flow control under uncertainty verify
global convergence of the method and demonstrate the method's ability to
outperform previously proposed alternatives.Comment: 27 pages, 6 figures, 1 tabl
Scalable Gaussian Processes with Grid-Structured Eigenfunctions (GP-GRIEF)
We introduce a kernel approximation strategy that enables computation of the
Gaussian process log marginal likelihood and all hyperparameter derivatives in
time. Our GRIEF kernel consists of eigenfunctions found
using a Nystrom approximation from a dense Cartesian product grid of inducing
points. By exploiting algebraic properties of Kronecker and Khatri-Rao tensor
products, computational complexity of the training procedure can be practically
independent of the number of inducing points. This allows us to use arbitrarily
many inducing points to achieve a globally accurate kernel approximation, even
in high-dimensional problems. The fast likelihood evaluation enables type-I or
II Bayesian inference on large-scale datasets. We benchmark our algorithms on
real-world problems with up to two-million training points and
inducing points.Comment: Appears in the proceedings of the International Conference on Machine
Learning (ICML), 201
Reduced-order modeling with artificial neurons for gravitational-wave inference
Gravitational-wave data analysis is rapidly absorbing techniques from deep
learning, with a focus on convolutional networks and related methods that treat
noisy time series as images. We pursue an alternative approach, in which
waveforms are first represented as weighted sums over reduced bases
(reduced-order modeling); we then train artificial neural networks to map
gravitational-wave source parameters into basis coefficients. Statistical
inference proceeds directly in coefficient space, where it is theoretically
straightforward and computationally efficient. The neural networks also provide
analytic waveform derivatives, which are useful for gradient-based sampling
schemes. We demonstrate fast and accurate coefficient interpolation for the
case of a four-dimensional binary-inspiral waveform family, and discuss
promising applications of our framework in parameter estimation.Comment: Published versio
Sparse multiresolution representations with adaptive kernels
Reproducing kernel Hilbert spaces (RKHSs) are key elements of many
non-parametric tools successfully used in signal processing, statistics, and
machine learning. In this work, we aim to address three issues of the classical
RKHS based techniques. First, they require the RKHS to be known a priori, which
is unrealistic in many applications. Furthermore, the choice of RKHS affects
the shape and smoothness of the solution, thus impacting its performance.
Second, RKHSs are ill-equipped to deal with heterogeneous degrees of
smoothness, i.e., with functions that are smooth in some parts of their domain
but vary rapidly in others. Finally, the computational complexity of evaluating
the solution of these methods grows with the number of data points, rendering
these techniques infeasible for many applications. Though kernel learning,
local kernel adaptation, and sparsity have been used to address these issues,
many of these approaches are computationally intensive or forgo optimality
guarantees. We tackle these problems by leveraging a novel integral
representation of functions in RKHSs that allows for arbitrary centers and
different kernels at each center. To address the complexity issues, we then
write the function estimation problem as a sparse functional program that
explicitly minimizes the support of the representation leading to low
complexity solutions. Despite their non-convexity and infinite dimensionality,
we show these problems can be solved exactly and efficiently by leveraging
duality, and we illustrate this new approach in simulated and real data
A Tale of Three Probabilistic Families: Discriminative, Descriptive and Generative Models
The pattern theory of Grenander is a mathematical framework where patterns
are represented by probability models on random variables of algebraic
structures. In this paper, we review three families of probability models,
namely, the discriminative models, the descriptive models, and the generative
models. A discriminative model is in the form of a classifier. It specifies the
conditional probability of the class label given the input signal. A
descriptive model specifies the probability distribution of the signal, based
on an energy function defined on the signal. A generative model assumes that
the signal is generated by some latent variables via a transformation. We shall
review these models within a common framework and explore their connections. We
shall also review the recent developments that take advantage of the high
approximation capacities of deep neural networks
Classification of Spatio-Temporal Data via Asynchronous Sparse Sampling: Application to Flow Around a Cylinder
We present a novel method for the classification and reconstruction of time
dependent, high-dimensional data using sparse measurements, and apply it to the
flow around a cylinder. Assuming the data lies near a low dimensional manifold
(low-rank dynamics) in space and has periodic time dependency with a sparse
number of Fourier modes, we employ compressive sensing for accurately
classifying the dynamical regime. We further show that we can reconstruct the
full spatio-temporal behavior with these limited measurements, extending
previous results of compressive sensing that apply for only a single snapshot
of data. The method can be used for building improved reduced-order models and
designing sampling/measurement strategies that leverage time asynchrony.Comment: 19 pages, 4 figure
A Goal-Oriented Adaptive Discrete Empirical Interpolation Method
In this study we propose a-posteriori error estimation results to approximate
the precision loss in quantities of interests computed using reduced order
models. To generate the surrogate models we employ Proper Orthogonal
Decomposition and Discrete Empirical Interpolation Method. First order
expansions of the components of the quantity of interest obtained as the
product between the components gradient and model residuals are summed up to
generate the error estimation result. Efficient versions are derived for
explicit and implicit Euler schemes and require only one reduced forward and
adjoint models and high-fidelity model residuals estimation. Then we derive an
adaptive DEIM algorithm to enhance the accuracy of these quantities of
interests. The adaptive DEIM algorithm uses dual weighted residuals singular
vectors in combination with the non-linear term basis. Both the a-posteriori
error estimation results and the adaptive DEIM algorithm were assessed using
the 1D-Burgers and Shallow Water Equation models and the numerical experiments
shows very good agreement with the theoretical results.Comment: 34 pages, 19 figure
Environment Identification in Flight using Sparse Approximation of Wing Strain
This paper addresses the problem of identifying different flow environments
from sparse data collected by wing strain sensors. Insects regularly perform
this feat using a sparse ensemble of noisy strain sensors on their wing. First,
we obtain strain data from numerical simulation of a Manduca sexta hawkmoth
wing undergoing different flow environments. Our data-driven method learns
low-dimensional strain features originating from different aerodynamic
environments using proper orthogonal decomposition (POD) modes in the frequency
domain, and leverages sparse approximation to classify a set of strain
frequency signatures using a dictionary of POD modes. This bio-inspired machine
learning architecture for dictionary learning and sparse classification permits
fewer costly physical strain sensors while being simultaneously robust to
sensor noise. A measurement selection algorithm identifies frequencies that
best discriminate the different aerodynamic environments in low-rank POD
feature space. In this manner, sparse and noisy wing strain data can be
exploited to robustly identify different aerodynamic environments encountered
in flight, providing insight into the stereotyped placement of neurons that act
as strain sensors on a Manduca sexta hawkmoth wing.Comment: 23 pages, 12 figures. Revised version. Accepted for publication in
Journal of Fluids and Structure
Transported snapshot model order reduction approach for parametric, steady-state fluid flows containing parameter dependent shocks
A new model order reduction approach is proposed for parametric steady-state
nonlinear fluid flows characterized by shocks and discontinuities whose spatial
locations and orientations are strongly parameter dependent. In this method,
solutions in the predictive regime are approximated using a linear
superposition of parameter dependent basis. The sought after parametric
reduced-basis are obtained by transporting the snapshots in a spatially and
parametrically dependent transport field. Key to the proposed approach is the
observation that the transport fields are typically smooth and continuous,
despite the solution themselves not being so. As a result, the transport fields
can be accurately expressed using a low-order polynomial expansion. Similar to
traditional projection-based model order reduction approaches, the proposed
method is formulated mathematically as a residual minimization problem for the
generalized coordinates. The proposed approach is also integrated with
well-known hyper-reduction strategies to obtain significant computational
speed-ups. The method is successfully applied to the reduction of a parametric
1-D flow in a converging-diverging nozzle, a parametric 2-D supersonic flow
over a forward facing step and a parametric 2-D jet diffusion flame in a
combustor
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