32,214 research outputs found
On Adaptive Multiple-Shooting Method for Stochastic Multi-Point Boundary Value Problems
This paper presents an adaptive multiple-shooting method to solve stochastic
multi-point boundary value problems. The heuristic to choose the shooting
points is based on separating the effects of drift and diffusion terms and
comparing the corresponding solution components with a pre-specified initial
approximation. Having obtained the mesh points, we solve the underlying
stochastic differential equation on each shooting interval with a first-order
strongly-convergent stochastic Runge-Kutta method. We illustrate the
effectiveness of this approach on 1-dimentional and 2-dimentional test problems
and compare our results with other non-adaptive alternative techniques proposed
in the literature.Comment: 18 Pages, 2 figure
Adaptive Weak Approximation of Diffusions with Jumps
This work develops Monte Carlo Euler adaptive time stepping methods for the
weak approximation problem of jump diffusion driven stochastic differential
equations. The main result is the derivation of a new expansion for the
omputational error, with computable leading order term in a posteriori form,
based on stochastic flows and discrete dual backward problems which extends the
results in [STZ]. These expansions lead to efficient and accurate computation
of error estimates. Adaptive algorithms for either stochastic time steps or
quasi-deterministic time steps are described. Numerical examples show the
performance of the proposed error approximation and of the described adaptive
time-stepping methods.Comment: 27 page
Lower Error Bounds for Strong Approximation of Scalar SDEs with non-Lipschitzian Coefficients
We study pathwise approximation of scalar stochastic differential equations
at a single time point or globally in time by means of methods that are based
on finitely many observations of the driving Brownian motion. We prove lower
error bounds in terms of the average number of evaluations of the driving
Brownian motion that hold for every such method under rather mild assumptions
on the coefficients of the equation. The underlying simple idea of our analysis
is as follows: the lower error bounds known for equations with coefficients
that have sufficient regularity globally in space should still apply in the
case of coefficients that have this regularity in space only locally, in a
small neighborhood of the initial value. Our results apply to a huge variety of
equations with coefficients that are not globally Lipschitz continuous in space
including Cox-Ingersoll-Ross processes, equations with superlinearly growing
coefficients, and equations with discontinuous coefficients. In many of these
cases the resulting lower error bounds even turn out to be sharp
Towards Automatic Global Error Control: Computable Weak Error Expansion for the Tau-Leap Method
This work develops novel error expansions with computable leading order terms
for the global weak error in the tau-leap discretization of pure jump processes
arising in kinetic Monte Carlo models. Accurate computable a posteriori error
approximations are the basis for adaptive algorithms; a fundamental tool for
numerical simulation of both deterministic and stochastic dynamical systems.
These pure jump processes are simulated either by the tau-leap method, or by
exact simulation, also referred to as dynamic Monte Carlo, the Gillespie
algorithm or the Stochastic simulation algorithm. Two types of estimates are
presented: an a priori estimate for the relative error that gives a comparison
between the work for the two methods depending on the propensity regime, and an
a posteriori estimate with computable leading order term
A Separation-based Approach to Data-based Control for Large-Scale Partially Observed Systems
This paper studies the partially observed stochastic optimal control problem
for systems with state dynamics governed by partial differential equations
(PDEs) that leads to an extremely large problem. First, an open-loop
deterministic trajectory optimization problem is solved using a black-box
simulation model of the dynamical system. Next, a Linear Quadratic Gaussian
(LQG) controller is designed for the nominal trajectory-dependent linearized
system which is identified using input-output experimental data consisting of
the impulse responses of the optimized nominal system. A computational
nonlinear heat example is used to illustrate the performance of the proposed
approach.Comment: arXiv admin note: text overlap with arXiv:1705.09761,
arXiv:1707.0309
The optimal free knot spline approximation of stochastic differential equations with additive noise
In this paper we analyse the pathwise approximation of stochastic
differential equations by polynomial splines with free knots. The pathwise
distance between the solution and its approximation is measured globally on the
unit interval in the -norm, and we study the expectation of this
distance. For equations with additive noise we obtain sharp lower and upper
bounds for the minimal error in the class of arbitrary spline approximation
methods, which use free knots. The optimal order is achieved by an
approximation method , which combines an Euler scheme on
a coarse grid with an optimal spline approximation of the Brownian motion
with free knots.Comment: arXiv admin note: text overlap with arXiv:1306.445
Stochastic Gradient Descent as Approximate Bayesian Inference
Stochastic Gradient Descent with a constant learning rate (constant SGD)
simulates a Markov chain with a stationary distribution. With this perspective,
we derive several new results. (1) We show that constant SGD can be used as an
approximate Bayesian posterior inference algorithm. Specifically, we show how
to adjust the tuning parameters of constant SGD to best match the stationary
distribution to a posterior, minimizing the Kullback-Leibler divergence between
these two distributions. (2) We demonstrate that constant SGD gives rise to a
new variational EM algorithm that optimizes hyperparameters in complex
probabilistic models. (3) We also propose SGD with momentum for sampling and
show how to adjust the damping coefficient accordingly. (4) We analyze MCMC
algorithms. For Langevin Dynamics and Stochastic Gradient Fisher Scoring, we
quantify the approximation errors due to finite learning rates. Finally (5), we
use the stochastic process perspective to give a short proof of why Polyak
averaging is optimal. Based on this idea, we propose a scalable approximate
MCMC algorithm, the Averaged Stochastic Gradient Sampler.Comment: 35 pages, published version (JMLR 2017
Numerical low-rank approximation of matrix differential equations
The efficient numerical integration of large-scale matrix differential
equations is a topical problem in numerical analysis and of great importance in
many applications. Standard numerical methods applied to such problems require
an unduly amount of computing time and memory, in general. Based on a dynamical
low-rank approximation of the solution, a new splitting integrator is proposed
for a quite general class of stiff matrix differential equations. This class
comprises differential Lyapunov and differential Riccati equations that arise
from spatial discretizations of partial differential equations. The proposed
integrator handles stiffness in an efficient way, and it preserves the symmetry
and positive semidefiniteness of solutions of differential Lyapunov equations.
Numerical examples that illustrate the benefits of this new method are given.
In particular, numerical results for the efficient simulation of the weather
phenomenon El Ni\~no are presented
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
Model Error in Data Assimilation
This chapter provides various perspective on an important challenge in data
assimilation: model error. While the overall goal is to understand the
implication of model error of any type in data assimilation, we emphasize on
the effect of model error from unresolved scales. In particular, connection to
related subjects under different names in applied mathematics, such as the
Mori-Zwanzig formalism and the averaging method, were discussed with the hope
that the existing methods can be more accessible and eventually be used
appropriately. We will classify existing methods into two groups: the
statistical methods for those who directly estimate the low-order model error
statistics; and the stochastic parameterizations for those who implicitly
estimate all statistics by imposing stochastic models beyond the traditional
unbiased white noise Gaussian processes. We will provide theory to justify why
stochastic parameterization, as one of the main theme in this book, is an
adequate tool for mitigating model error in data assimilation. Finally, we will
also discuss challenges in lifting this approach in general applications and
provide an alternative nonparametric approach.Comment: This note is prepared for a chapter in "Nonlinear and Stochastic
Climate Dynamics. Eds. C.L.E. Franzke and T.J. O'Kane, Cambridge University
Pres
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