198 research outputs found
Linearized Numerical Homogenization Method for Nonlinear Monotone Parabolic Multiscale Problems
Explicit Stabilised Gradient Descent for Faster Strongly Convex Optimisation
This paper introduces the Runge-Kutta Chebyshev descent method (RKCD) for
strongly convex optimisation problems. This new algorithm is based on explicit
stabilised integrators for stiff differential equations, a powerful class of
numerical schemes that avoid the severe step size restriction faced by standard
explicit integrators. For optimising quadratic and strongly convex functions,
this paper proves that RKCD nearly achieves the optimal convergence rate of the
conjugate gradient algorithm, and the suboptimality of RKCD diminishes as the
condition number of the quadratic function worsens. It is established that this
optimal rate is obtained also for a partitioned variant of RKCD applied to
perturbations of quadratic functions. In addition, numerical experiments on
general strongly convex problems show that RKCD outperforms Nesterov's
accelerated gradient descent
Discontinuous Galerkin finite element heterogeneous multiscale method for advection-diffusion problems with multiple scales
A discontinuous Galerkin finite element heterogeneous multiscale method is proposed for advection-diffusion problems with highly oscillatory coefficients. The method is based on a coupling of a discontinuous Galerkin discretization for an effective advection-diffusion problem on a macroscopic mesh, whose a priori unknown data are recovered from micro finite element calculations on sampling domains within each macro element. The computational work involved is independent of the high oscillations in the problem at the smallest scale. The stability of our method (depending on both macro and micro mesh sizes) is established for both diffusion dominated and advection dominated regimes without any assumptions about the type of heterogeneities in the data. Fully discrete a priori error bounds are derived for locally periodic data. Numerical experiments confirm the theoretical error estimates
Multiscale model reduction methods for flow in heterogeneous porous media
In this paper we provide a general framework for model reduction methods applied to fluid flow in porous media. Using reduced basis and numerical homogenization techniques we show that the complexity of the numerical approximation of Stokes flow in heterogeneous media can be drastically reduced. The use of such a computational framework is illustrated at several model problems such as two and three scale porous media
Drift Estimation of Multiscale Diffusions Based on Filtered Data
We study the problem of drift estimation for two-scale continuous time series. We set ourselves in the framework of overdamped Langevin equations, for which a single-scale surrogate homogenized equation exists. In this setting, estimating the drift coefficient of the homogenized equation requires pre-processing of the data, often in the form of subsampling; this is because the two-scale equation and the homogenized single-scale equation are incompatible at small scales, generating mutually singular measures on the path space. We avoid subsampling and work instead with filtered data, found by application of an appropriate kernel function, and compute maximum likelihood estimators based on the filtered process. We show that the estimators we propose are asymptotically unbiased and demonstrate numerically the advantages of our method with respect to subsampling. Finally, we show how our filtered data methodology can be combined with Bayesian techniques and provide a full uncertainty quantification of the inference procedure
Drift Estimation of Multiscale Diffusions Based on Filtered Data
We study the problem of drift estimation for two-scale continuous time
series. We set ourselves in the framework of overdamped Langevin equations, for
which a single-scale surrogate homogenized equation exists. In this setting,
estimating the drift coefficient of the homogenized equation requires
pre-processing of the data, often in the form of subsampling; this is because
the two-scale equation and the homogenized single-scale equation are
incompatible at small scales, generating mutually singular measures on the path
space. We avoid subsampling and work instead with filtered data, found by
application of an appropriate kernel function, and compute maximum likelihood
estimators based on the filtered process. We show that the estimators we
propose are asymptotically unbiased and demonstrate numerically the advantages
of our method with respect to subsampling. Finally, we show how our filtered
data methodology can be combined with Bayesian techniques and provide a full
uncertainty quantification of the inference procedure
Explicit methods for stiff stochastic differential equations
Multiscale differential equations arise in the modeling of many important problems in the science and engineering. Numerical solvers for such problems have been extensively studied in the deterministic case. Here, we discuss numerical methods for (mean-square stable) stiff stochastic differential equations. Standard explicit methods, as for example the Euler-Maruyama method, face severe stepsize restriction when applied to stiff problems. Fully implicit methods are usually not appropriate for stochastic problems and semi-implicit methods (implicit in the deterministic part) involve the solution of possibly large linear systems at each time-step. In this paper, we present a recent generalization of explicit stabilized methods, known as Chebyshev methods, to stochastic problems. These methods have much better (mean-square) stability properties than standard explicit methods. We discuss the construction of this new class of methods and illustrate their performance on various problems involving stochastic ordinary and partial differential equations
Locally periodic unfolding method and two-scale convergence on surfaces of locally periodic microstructures
In this paper we generalize the periodic unfolding method and the notion of
two-scale convergence on surfaces of periodic microstructures to locally
periodic situations. The methods that we introduce allow us to consider a wide
range of non-periodic microstructures, especially to derive macroscopic
equations for problems posed in domains with perforations distributed
non-periodically. Using the methods of locally periodic two-scale convergence
(l-t-s) on oscillating surfaces and the locally periodic (l-p) boundary
unfolding operator, we are able to analyze differential equations defined on
boundaries of non-periodic microstructures and consider non-homogeneous Neumann
conditions on the boundaries of perforations, distributed non-periodically
Rapid covariance-based sampling of linear SPDE approximations in the multilevel Monte Carlo method
The efficient simulation of the mean value of a non-linear functional of the
solution to a linear stochastic partial differential equation (SPDE) with
additive Gaussian noise is considered. A Galerkin finite element method is
employed along with an implicit Euler scheme to arrive at a fully discrete
approximation of the mild solution to the equation. A scheme is presented to
compute the covariance of this approximation, which allows for rapid sampling
in a Monte Carlo method. This is then extended to a multilevel Monte Carlo
method, for which a scheme to compute the cross-covariance between the
approximations at different levels is presented. In contrast to traditional
path-based methods it is not assumed that the Galerkin subspaces at these
levels are nested. The computational complexities of the presented schemes are
compared to traditional methods and simulations confirm that, under suitable
assumptions, the costs of the new schemes are significantly lower.Comment: 18 pages, 5 figures; numerical simulations revised, implementation
section added; To appear in Monte Carlo and Quasi-Monte Carlo Methods -
MCQMC, Rennes, France, July 201
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