19,705 research outputs found
Estimation of the Distribution of Random Parameters in Discrete Time Abstract Parabolic Systems with Unbounded Input and Output: Approximation and Convergence
A finite dimensional abstract approximation and convergence theory is
developed for estimation of the distribution of random parameters in infinite
dimensional discrete time linear systems with dynamics described by regularly
dissipative operators and involving, in general, unbounded input and output
operators. By taking expectations, the system is re-cast as an equivalent
abstract parabolic system in a Gelfand triple of Bochner spaces wherein the
random parameters become new space-like variables. Estimating their
distribution is now analogous to estimating a spatially varying coefficient in
a standard deterministic parabolic system. The estimation problems are
approximated by a sequence of finite dimensional problems. Convergence is
established using a state space-varying version of the Trotter-Kato semigroup
approximation theorem. Numerical results for a number of examples involving the
estimation of exponential families of densities for random parameters in a
diffusion equation with boundary input and output are presented and discussed
Particle-kernel estimation of the filter density in state-space models
Sequential Monte Carlo (SMC) methods, also known as particle filters, are
simulation-based recursive algorithms for the approximation of the a posteriori
probability measures generated by state-space dynamical models. At any given
time , a SMC method produces a set of samples over the state space of the
system of interest (often termed "particles") that is used to build a discrete
and random approximation of the posterior probability distribution of the state
variables, conditional on a sequence of available observations. One potential
application of the methodology is the estimation of the densities associated to
the sequence of a posteriori distributions. While practitioners have rather
freely applied such density approximations in the past, the issue has received
less attention from a theoretical perspective. In this paper, we address the
problem of constructing kernel-based estimates of the posterior probability
density function and its derivatives, and obtain asymptotic convergence results
for the estimation errors. In particular, we find convergence rates for the
approximation errors that hold uniformly on the state space and guarantee that
the error vanishes almost surely as the number of particles in the filter
grows. Based on this uniform convergence result, we first show how to build
continuous measures that converge almost surely (with known rate) toward the
posterior measure and then address a few applications. The latter include
maximum a posteriori estimation of the system state using the approximate
derivatives of the posterior density and the approximation of functionals of
it, for example, Shannon's entropy.
This manuscript is identical to the published paper, including a gap in the
proof of Theorem 4.2. The Theorem itself is correct. We provide an {\em
erratum} at the end of this document with a complete proof and a brief
discussion.Comment: IMPORTANT: This manuscript is identical to the published paper,
including a gap in the proof of Theorem 4.2. The Theorem itself is correct.
We provide an erratum at the end of this document. Published at
http://dx.doi.org/10.3150/13-BEJ545 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Minimax estimation of smooth optimal transport maps
Brenier's theorem is a cornerstone of optimal transport that guarantees the
existence of an optimal transport map between two probability distributions
and over under certain regularity conditions. The main
goal of this work is to establish the minimax estimation rates for such a
transport map from data sampled from and under additional smoothness
assumptions on . To achieve this goal, we develop an estimator based on the
minimization of an empirical version of the semi-dual optimal transport
problem, restricted to truncated wavelet expansions. This estimator is shown to
achieve near minimax optimality using new stability arguments for the semi-dual
and a complementary minimax lower bound. Furthermore, we provide numerical
experiments on synthetic data supporting our theoretical findings and
highlighting the practical benefits of smoothness regularization. These are the
first minimax estimation rates for transport maps in general dimension.Comment: 53 pages, 6 figure
Efficient Optimization of Loops and Limits with Randomized Telescoping Sums
We consider optimization problems in which the objective requires an inner
loop with many steps or is the limit of a sequence of increasingly costly
approximations. Meta-learning, training recurrent neural networks, and
optimization of the solutions to differential equations are all examples of
optimization problems with this character. In such problems, it can be
expensive to compute the objective function value and its gradient, but
truncating the loop or using less accurate approximations can induce biases
that damage the overall solution. We propose randomized telescope (RT) gradient
estimators, which represent the objective as the sum of a telescoping series
and sample linear combinations of terms to provide cheap unbiased gradient
estimates. We identify conditions under which RT estimators achieve
optimization convergence rates independent of the length of the loop or the
required accuracy of the approximation. We also derive a method for tuning RT
estimators online to maximize a lower bound on the expected decrease in loss
per unit of computation. We evaluate our adaptive RT estimators on a range of
applications including meta-optimization of learning rates, variational
inference of ODE parameters, and training an LSTM to model long sequences
Quantification of airfoil geometry-induced aerodynamic uncertainties - comparison of approaches
Uncertainty quantification in aerodynamic simulations calls for efficient
numerical methods since it is computationally expensive, especially for the
uncertainties caused by random geometry variations which involve a large number
of variables. This paper compares five methods, including quasi-Monte Carlo
quadrature, polynomial chaos with coefficients determined by sparse quadrature
and gradient-enhanced version of Kriging, radial basis functions and point
collocation polynomial chaos, in their efficiency in estimating statistics of
aerodynamic performance upon random perturbation to the airfoil geometry which
is parameterized by 9 independent Gaussian variables. The results show that
gradient-enhanced surrogate methods achieve better accuracy than direct
integration methods with the same computational cost
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