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Register a new userA discretization of Caputo derivatives with application to time fractional SDEs and gradient flows
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Lei Li
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We consider a discretization of Caputo derivatives resulted from deconvolving a scheme for the corresponding Volterra integral. Properties of this discretization, including signs of the coefficients, comparison principles, and stability of the corresponding implicit schemes, are proved by its linkage to Volterra integrals with completely monotone kernels. We then apply the backward scheme corresponding to this discretization to two time fractional dissipative problems, and these implicit schemes are helpful for the analysis of the corresponding problems. In particular, we show that the overdamped generalized Langevin equation with fractional noise has a unique limiting measure for strongly convex potentials and establish the convergence of numerical solutions to the strong solutions of time fractional gradient flows. The proposed scheme and schemes derived using the same philosophy can be useful for many other applications as well.
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In this paper, we investigate fast algorithms to approximate the Caputo derivative $^C_0D_t^alpha u(t)$ when $alpha$ is small. We focus on two fast algorithms, i.e. FIR and FIDR, both relying on the sum-of-exponential approximation to reduce the cost of evaluating the history part. FIR is the numerical scheme originally proposed in [16], and FIDR is an alternative scheme we propose in this work, and the latter shows superiority when $alpha$ is small. With quantitative estimates, we prove that given a certain error threshold, the computational cost of evaluating the history part of the Caputo derivative can be decreased as $alpha$ gets small. Hence, only minimal cost for the fast evaluation is required in the small $alpha$ regime, which matches prevailing protocols in engineering practice. We also present a stability and error analysis of FIDR for solving linear fractional diffusion equations. Finally, we carry out systematic numerical studies for the performances of both FIR and FIDR schemes, where we explore the trade-off between accuracy and efficiency when $alpha$ is small.
Time fractional PDEs have been used in many applications for modeling and simulations. Many of these applications are multiscale and contain high contrast variations in the media properties. It requires very small time step size to perform detailed computations. On the other hand, in the presence of small spatial grids, very small time step size is required for explicit methods. Explicit methods have many advantages as we discuss in the paper. In this paper, we propose a partial explicit method for time fractional PDEs. The approach solves the forward problem on a coarse computational grid, which is much larger than spatial heterogeneities, and requires only a few degrees of freedom to be treated implicitly. Via the construction of appropriate spaces and careful stability analysis, we can show that the time step can be chosen not to depend on the contrast or scale as the coarse mesh size. Thus, one can use larger time step size in an explicit approach. We present stability theory for our proposed method and our numerical results confirm the stability findings and demonstrate the performance of the approach.
We develop the theory of fractional gradient flows: an evolution aimed at the minimization of a convex, l.s.c.~energy, with memory effects. This memory is characterized by the fact that the negative of the (sub)gradient of the energy equals the so-called Caputo derivative of the state. We introduce the notion of energy solutions, for which we provide existence, uniqueness and certain regularizing effects. We also consider Lipschitz perturbations of this energy. For these problems we provide an a posteriori error estimate and show its reliability. This estimate depends only on the problem data, and imposes no constraints between consecutive time-steps. On the basis of this estimate we provide an a priori error analysis that makes no assumptions on the smoothness of the solution.
We consider the multidimensional space-fractional diffusion equations with spatially varying diffusivity and fractional order. Significant computational challenges are encountered when solving these equations due both to the kernel singularity in the fractional integral operator and to the resulting dense discretized operators, which quickly become prohibitively expensive to handle because of their memory and arithmetic complexities. In this work, we present a singularity-aware discretization scheme that regularizes the singular integrals through a singularity subtraction technique adapted to the spatial variability of diffusivity and fractional order. This regularization strategy is conveniently formulated as a sparse matrix correction that is added to the dense operator, and is applicable to different formulations of fractional diffusion equations. We also present a block low rank representation to handle the dense matrix representations, by exploiting the ability to approximate blocks of the resulting formally dense matrix by low rank factorizations. A Cholesky factorization solver operates directly on this representation using the low rank blocks as its atomic computational tiles, and achieves high performance on multicore hardware. Numerical results show that the singularity treatment is robust, substantially reduces discretization errors, and attains the first-order convergence rate allowed by the regularity of the solutions. They also show that considerable savings are obtained in storage ($O(N^{1.5})$) and computational cost ($O(N^2)$) compared to dense factorizations. This translates to orders-of-magnitude savings in memory and time on multi-dimensional problems, and shows that the proposed methods offer practical tools for tackling large nonlocal fractional diffusion simulations.
We present a new fractional Taylor formula for singular functions whose Caputo fractional derivatives are of bounded variation. It bridges and ``interpolates the usual Taylor formulas with two consecutive integer orders. This enables us to obtain an analogous formula for the Legendre expansion coefficient of this type of singular functions, and further derive the optimal (weighted) $L^infty$-estimates and $L^2$-estimates of the Legendre polynomial approximations. This set of results can enrich the existing theory for $p$ and $hp$ methods for singular problems, and answer some open questions posed in some recent literature.
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