Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1406.1868

High order algorithms for the fractional substantial diffusion equation with truncated Lévy flights

Minghua Chen·Weihua Deng
View original source ↗Read source document ↗Export metadata ↓

Abstract

The equation with the time fractional substantial derivative and space fractional derivative describes the distribution of the functionals of the Lévy flights; and the equation is derived as the macroscopic limit of the continuous time random walk in unbounded domain and the Lévy flights have divergent second order moments. However, in more practical problems, the physical domain is bounded and the involved observables have finite moments. Then the modified equation can be derived by tempering the probability of large jump length of the Lévy flights and the corresponding tempered space fractional derivative is introduced. This paper focuses on providing the high order algorithms for the modified equation, i.e., the equation with the time fractional substantial derivative and space tempered fractional derivative. More concretely, the contributions of this paper are as follows: 1. the detailed numerical stability analysis and error estimates of the schemes with first order accuracy in time and second order in space are given in {\textsl{complex}} space, which is necessary since the inverse Fourier transform needs to be made for getting the distribution of the functionals after solving the equation; 2. we further propose the schemes with high order accuracy in both time and space, and the techniques of treating the issue of keeping the high order accuracy of the schemes for {\textsl{nonhomogeneous}} boundary/initial conditions are introduced; 3. the multigrid methods are effectively used to solve the obtained algebraic equations which still have the Toeplitz structure; 4. we perform extensive numerical experiments, including verifying the high convergence orders, simulating the physical system which needs to numerically make the inverse Fourier transform to the numerical solutions of the equation.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Minghua Chen, Weihua Deng. 2014-08-11. High order algorithms for the fractional substantial diffusion equation with truncated Lévy flights. https://doi.org/10.1137/14097207x

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

How many continuous measurements are needed to learn a vector?

One can recover vectors from $\mathbb{R}^m$ with arbitrary precision, using only $\lceil \log_2(m)\rceil +1$ continuous measurements that are chosen adaptively. This surprising result is explained and discussed, and we present applications to infinite-dimensional approximation problems.

math.NA↗
arXiv2026

IterativeCUR: Large Rank-Adaptive Approximation From a Small Recycled Sketch

The computation of accurate low-rank matrix approximations is central to improving the scalability of various techniques in machine learning, uncertainty quantification, and control. Traditionally, low-rank approximations are constructed using SVD-based approaches such as truncated SVD or Randomized SVD. Although these SVD approaches---especially Randomized SVD---have proven to be very computationally efficient, other low-rank approximation methods can offer even greater performance. One such approach is the CUR decomposition, which forms a low-rank approximation using direct row and column subsets of a matrix. Because CUR uses direct matrix subsets, it is also often better able to preserve native matrix structures like sparsity or non-negativity than SVD-based approaches and can facilitate data interpretation in many contexts. This paper introduces IterativeCUR, which draws on previous work in randomized numerical linear algebra to build a new algorithm that is highly competitive compared to prior work. IterativeCUR is adaptive in the sense that it takes as an input parameter the desired tolerance $ε$ and outputs (with arbitrarily high probability) an approximation of error bounded by $ε$, rather than requiring an a priori guess of the numerical rank. IterativeCUR typically runs significantly faster than both existing CUR algorithms and techniques such as Randomized SVD. Its asymptotic complexity is $\mathcal{O}(mn + (m+n)r^2)$ for an $m\times n$ matrix of output rank $r$. IterativeCUR relies on a single small sketch from the matrix that is successively downdated as the algorithm proceeds. We demonstrate through extensive experiments that IterativeCUR achieves up to $4\times$ speed-up over state-of-the-art pivoting-on-sketch approaches with no loss of accuracy, and up to $40\times$ speed-up over rank-adaptive randomized SVD approaches.

math.NA↗
arXiv2026

Multigrid with Linear Storage Complexity

As the discretization error for the solution of a partial differential equation (PDE) decreases, the precision required to store the corresponding coefficients naturally increases. Storing the solution's finite element coefficients explicitly requires $\mathcal O(n \log n)$ bits of storage, where $n$ is the number of degrees of freedom (DoFs). This paper presents a full multigrid method to compute the solution in a compressed format that reduces the storage complexity of the solution and intermediate vectors to $\mathcal O(n)$ bits. This reduction allows a matrix-free implementation to solve elliptic PDEs with an overall linear space complexity. For problems limited by the memory capacity of current supercomputers, we expect a memory footprint reduction of about an order of magnitude compared to state-of-the-art mixed-precision methods. We demonstrate the applicability of our algorithm by solving two model problems. Depending on the PDE and polynomial degree, but irrespective of the problem size, the solution vector on the finest grid requires between 4 and 12 bits per DoF, and the residual and correction require 3 to 6 bits each. Additional data is stored on the coarse grids with modestly increasing bit widths toward coarser grids.

math.NA↗