1,999 research outputs found
Multigrid waveform relaxation for the time-fractional heat equation
In this work, we propose an efficient and robust multigrid method for solving
the time-fractional heat equation. Due to the nonlocal property of fractional
differential operators, numerical methods usually generate systems of equations
for which the coefficient matrix is dense. Therefore, the design of efficient
solvers for the numerical simulation of these problems is a difficult task. We
develop a parallel-in-time multigrid algorithm based on the waveform relaxation
approach, whose application to time-fractional problems seems very natural due
to the fact that the fractional derivative at each spatial point depends on the
values of the function at this point at all earlier times. Exploiting the
Toeplitz-like structure of the coefficient matrix, the proposed multigrid
waveform relaxation method has a computational cost of
operations, where is the number of time steps and is the number of
spatial grid points. A semi-algebraic mode analysis is also developed to
theoretically confirm the good results obtained. Several numerical experiments,
including examples with non-smooth solutions and a nonlinear problem with
applications in porous media, are presented
Online dictionary learning from big data using accelerated stochastic approximation algorithms
Applications involving large-scale dictionary learning tasks motivate well online optimization algorithms for generally non-convex and non-smooth problems. In this big data context, the present paper develops an online learning framework by jointly leveraging the stochastic approximation paradigm with first-order acceleration schemes. The generally non-convex objective evaluated online at the resultant iterates enjoys quadratic rate of convergence. The generality
of the novel approach is demonstrated in two online learning applications: (i) Online linear regression using the total least-squares approach; and, (ii) a semi-supervised dictionary learning approach
to network-wide link load tracking and imputation of real data with missing entries. In both cases, numerical tests highlight the potential of the proposed online framework for big data network analytics
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