scholarly journals Spectral methods for Langevin dynamics and associated error estimates

2018 ◽  
Vol 52 (3) ◽  
pp. 1051-1083 ◽  
Author(s):  
Julien Roussel ◽  
Gabriel Stoltz
Keyword(s):  
Convergence Rates ◽  
Langevin Dynamics ◽  
Galerkin Methods ◽  
Discrete Level ◽  
Rigidity Matrix ◽  
Poisson Equations ◽  
The Poisson Equation ◽  
Convergence Results ◽  
Abstract Setting ◽  
Tensor Basis

We prove the consistency of Galerkin methods to solve Poisson equations where the differential operator under consideration is hypocoercive. We show in particular how the hypocoercive nature of the generator associated with Langevin dynamics can be used at the discrete level to first prove the invertibility of the rigidity matrix, and next provide error bounds on the approximation of the solution of the Poisson equation. We present general convergence results in an abstract setting, as well as explicit convergence rates for a simple example discretized using a tensor basis. Our theoretical findings are illustrated by numerical simulations.

scholarly journals Improved Regularization Method for Backward Cauchy Problems Associated with Continuous Spectrum Operator

2011 ◽  
Vol 2011 ◽  
pp. 1-11
Author(s):  
Salah Djezzar ◽  
Nihed Teniou
Keyword(s):  
Continuous Spectrum ◽  
Original Problem ◽  
Convergence Rates ◽  
Nonlocal Problem ◽  
Cauchy Problems ◽  
Final Time ◽  
Unbounded Linear Operator ◽  
Convergence Results ◽  
Small Parameters ◽  
The Given

We consider in this paper an abstract parabolic backward Cauchy problem associated with an unbounded linear operator in a Hilbert space , where the coefficient operator in the equation is an unbounded self-adjoint positive operator which has a continuous spectrum and the data is given at the final time and a solution for is sought. It is well known that this problem is illposed in the sense that the solution (if it exists) does not depend continuously on the given data. The method of regularization used here consists of perturbing both the equation and the final condition to obtain an approximate nonlocal problem depending on two small parameters. We give some estimates for the solution of the regularized problem, and we also show that the modified problem is stable and its solution is an approximation of the exact solution of the original problem. Finally, some other convergence results including some explicit convergence rates are also provided.

QUASICONFORMAL SOLUTIONS OF POISSON EQUATIONS

2015 ◽  
Vol 92 (3) ◽  
pp. 420-428 ◽  
Author(s):  
PEIJIN LI ◽  
JIAOLONG CHEN ◽  
XIANTAO WANG
Keyword(s):  
Unit Disk ◽  
Poisson Equation ◽  
Lipschitz Continuity ◽  
Poisson Equations ◽  
The Poisson Equation

The main aim of this paper is to establish the Lipschitz continuity of the $(K,K^{\prime })$-quasiconformal solutions of the Poisson equation ${\rm\Delta}w=g$ in the unit disk $\mathbb{D}$.

Quasi-neutral limit for the Euler–Poisson system in the presence of plasma sheaths with spherical symmetry

2016 ◽  
Vol 26 (12) ◽  
pp. 2369-2392 ◽  
Author(s):  
Chang-Yeol Jung ◽  
Bongsuk Kwon ◽  
Masahiro Suzuki
Keyword(s):  
Boundary Layers ◽  
Convergence Rates ◽  
Three Dimensional ◽  
Suitable Condition ◽  
Plasma Sheath ◽  
Limit Behavior ◽  
Dirichlet Boundary Conditions ◽  
Pointwise Estimates ◽  
Poisson Equations ◽  
Sheath Edge

The purpose of this paper is to mathematically investigate the formation of a plasma sheath near the surface of a ball-shaped material immersed in a bulk plasma, and to obtain qualitative information of such a plasma sheath layer. Specifically, we study existence and the quasi-neutral limit behavior of the stationary spherical symmetric solutions for the Euler–Poisson equations in a three-dimensional annular domain. We first propose a suitable condition on the velocity at the sheath edge, referred as to Bohm criterion for the annulus, and under this condition together with the constant Dirichlet boundary conditions for the potential, we show that there exists a unique stationary spherical symmetric solution. Moreover, we study the quasi-neutral limit behavior by establishing [Formula: see text] estimate of the difference of the solutions to the Euler–Poisson equations and its quasi-neutral limiting equations, incorporated with the correctors for the boundary layers. The quasi-neutral limit analysis employing the correctors and their pointwise estimates enables us to obtain detailed asymptotic behaviors including the convergence rates in [Formula: see text] and [Formula: see text] norms as well as the thickness of the boundary layers as a consequence of the pointwise estimates.

Multiscale Galerkin methods for the nonstationary iterated Tikhonov method with a modified posteriori parameter selection

2018 ◽  
Vol 26 (1) ◽  
pp. 109-120
Author(s):  
Xingjun Luo ◽  
Zhaofu Ouyang ◽  
Chunmei Zeng ◽  
Fanchun Li
Keyword(s):  
Convergence Rates ◽  
Regularization Parameter ◽  
Fredholm Integral Equations ◽  
Galerkin Methods ◽  
A Posteriori ◽  
Parameter Choice ◽  
Compression Technique ◽  
Choice Strategy ◽  
Optimal Convergence Rates ◽  
Parameter Choice Strategy

AbstractIn this paper, we consider a fast multiscale Galerkin method with compression technique for solving Fredholm integral equations of the first kind via the nonstationary iterated Tikhonov regularization. A modified a posteriori regularization parameter choice strategy is established, which leads to optimal convergence rates.

UNIFORM CONVERGENCE RATES OF KERNEL-BASED NONPARAMETRIC ESTIMATORS FOR CONTINUOUS TIME DIFFUSION PROCESSES: A DAMPING FUNCTION APPROACH

2016 ◽  
Vol 33 (4) ◽  
pp. 874-914 ◽  
Author(s):  
Shin Kanaya
Keyword(s):  
Uniform Convergence ◽  
Continuous Time ◽  
Convergence Rates ◽  
Diffusion Processes ◽  
Kernel Functions ◽  
Damping Function ◽  
Nonparametric Estimators ◽  
Empirical Process Theory ◽  
Convergence Results ◽  
The Moment

In this paper, we derive uniform convergence rates of nonparametric estimators for continuous time diffusion processes. In particular, we consider kernel-based estimators of the Nadaraya–Watson type, introducing a new technical device called adamping function. This device allows us to derive sharp uniform rates over an infinite interval with minimal requirements on the processes: The existence of the moment of any order is not required and the boundedness of relevant functions can be significantly relaxed. Restrictions on kernel functions are also minimal: We allow for kernels with discontinuity, unbounded support, and slowly decaying tails. Our proofs proceed by using the covering-number technique from empirical process theory and exploiting the mixing and martingale properties of the processes. We also present new results on the path-continuity property of Brownian motions and diffusion processes over an infinite time horizon. These path-continuity results, which should also be of some independent interest, are used to control discretization biases of the nonparametric estimators. The obtained convergence results are useful for non/semiparametric estimation and testing problems of diffusion processes.

scholarly journals Tight Convergence Rate of Gradient Descent for Eigenvalue Computation

2020 ◽  
Author(s):  
Qinghua Ding ◽  
Kaiwen Zhou ◽  
James Cheng
Keyword(s):  
Fixed Point ◽  
Convergence Rate ◽  
Efficient Algorithm ◽  
Gradient Descent ◽  
Convergence Rates ◽  
Real Data ◽  
Convergence Results ◽  
Fast Convergence Rate ◽  
Quadratic Factor ◽  
Prior Analysis

Riemannian gradient descent (RGD) is a simple, popular and efficient algorithm for leading eigenvector computation [AMS08]. However, the existing analysis of RGD for eigenproblem is still not tight, which is O(log(n/epsilon)/Delta^2) due to [Xu et al., 2018]. In this paper, we show that RGD in fact converges at rate O(log(n/epsilon)/Delta), and give instances to shows the tightness of our result. This improves the best prior analysis by a quadratic factor. Besides, we also give tight convergence analysis of a deterministic variant of Oja's rule due to [Oja, 1982]. We show that it also enjoys fast convergence rate of O(log(n/epsilon)/Delta). Previous papers only gave asymptotic characterizations [Oja, 1982; Oja, 1989; Yi et al., 2005]. Our tools for proving convergence results include an innovative reduction and chaining technique, and a noisy fixed point iteration argument. Besides, we also give empirical justifications of our convergence rates over synthetic and real data.

scholarly journals Lagrange–Galerkin methods for the incompressible Navier-Stokes equations: a review

2016 ◽  
Vol 7 (3) ◽  
pp. 26-55
Author(s):  
Rodolfo Bermejo ◽  
Laura Saavedra
Keyword(s):  
Stokes Equations ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Past Century ◽  
Galerkin Methods ◽  
Numerical Experience ◽  
Navier Stokes Equations ◽  
The Past ◽  
Convergence Results ◽  
High Reynolds

Abstract We review in this paper the development of Lagrange-Galerkin (LG) methods to integrate the incompressible Navier-Stokes equations (NSEs) for engineering applications. These methods were introduced in the computational fluid dynamics community in the early eighties of the past century, and at that time they were considered good methods for both their theoretical stability properties and the way of dealing with the nonlinear terms of the equations; however, the numerical experience gained with the application of LG methods to different problems has identified drawbacks of them, such as the calculation of specific integrals that arise in their formulation and the calculation of the ow trajectories, which somehow have hampered the applicability of LG methods. In this paper, we focus on these issues and summarize the convergence results of LG methods; furthermore, we shall briefly introduce a new stabilized LG method suitable for high Reynolds numbers.

scholarly journals A Finite Time Analysis of Temporal Difference Learning with Linear Function Approximation

2021 ◽  
Author(s):  
Jalaj Bhandari ◽  
Daniel Russo ◽  
Raghav Singal
Keyword(s):  
Linear Function ◽  
Finite Time ◽  
Function Approximation ◽  
Gradient Descent ◽  
Convergence Rates ◽  
Temporal Difference ◽  
Temporal Difference Learning ◽  
Convergence Results ◽  
Linear Function Approximation ◽  
Markov Reward

Temporal difference learning (TD) is a simple iterative algorithm widely used for policy evaluation in Markov reward processes. Bhandari et al. prove finite time convergence rates for TD learning with linear function approximation. The analysis follows using a key insight that establishes rigorous connections between TD updates and those of online gradient descent. In a model where observations are corrupted by i.i.d. noise, convergence results for TD follow by essentially mirroring the analysis for online gradient descent. Using an information-theoretic technique, the authors also provide results for the case when TD is applied to a single Markovian data stream where the algorithm’s updates can be severely biased. Their analysis seamlessly extends to the study of TD learning with eligibility traces and Q-learning for high-dimensional optimal stopping problems.

Discrete maximum principle for interior penalty discontinuous Galerkin methods

2013 ◽  
Vol 11 (4) ◽  
Author(s):  
Tamás Horváth ◽  
Miklós Mincsovics
Keyword(s):  
Maximum Principle ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Mesh Size ◽  
Discrete Maximum Principle ◽  
Galerkin Methods ◽  
Penalty Parameter ◽  
Discrete Level ◽  
Interior Penalty ◽  
Theoretical Results

AbstractA class of linear elliptic operators has an important qualitative property, the so-called maximum principle. In this paper we investigate how this property can be preserved on the discrete level when an interior penalty discontinuous Galerkin method is applied for the discretization of a 1D elliptic operator. We give mesh conditions for the symmetric and for the incomplete method that establish some connection between the mesh size and the penalty parameter. We then investigate the sharpness of these conditions. The theoretical results are illustrated with numerical examples.

scholarly journals A finite element method for exterior interface problems

1980 ◽  
Vol 3 (2) ◽  
pp. 311-350 ◽  
Author(s):  
R. C. Maccamy ◽  
S. P. Marin
Keyword(s):  
Variational Problem ◽  
Convergence Rates ◽  
Interface Problem ◽  
Existence Of A Solution ◽  
Local Boundary ◽  
Convergence Results ◽  
Local Boundary Condition ◽  
Infinite Region ◽  
Non Local ◽  
Inner Region

A procedure is given for the approximate solution of a class of two-dimensional diffraction problems. Here the usual inner boundary conditions are replaced by an inner region together with interface conditions. The interface problem is treated by a variational procedure into which the infinite region behavior is incorporated by the use of a non-local boundary condition over an auxiliary curve. The variational problem is formulated and existence of a solution established. Then a corresponding approximate variational problem is given and optimal convergence results established. Numerical results are presented which confirm the convergence rates.

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