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A New Diffusive Representation for Fractional Derivatives, Part II: Convergence Analysis of the Numerical Scheme
Kai Diethelm
TL;DR
The paper addresses the lack of a thorough quantitative convergence analysis for a recently developed method for computing fractional derivatives. It analyzes the diffusive-representation scheme and reports faster-than-algebraic convergence of its quadrature error.
Problem
The previously developed method had numerical results, qualitative convergence analysis, and heuristic support, but lacked a thorough quantitative convergence analysis.
Source: §2 · p. 3Method
Results
The quadrature error converges to zero at a faster-than-algebraic rate.
Source: §2 · p. 4Takeaways & Limitations
The scheme is described as having O(N) computational complexity, O(1) active memory, and freedom to use any discretization.
Source: §1 · pp. 2–3When the input function is less smooth, the convergence order stated in Lemma 3 and Theorem 1 cannot be achieved.
Source: §3.4 · p. 11
Abstract
from arXiv · showhide
Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately evident that the method is fast and memory efficient. Moreover, the method's design is such that good convergence properties may be expected. This paper here starts a systematic investigation of these convergence properties.
1 A Diffusive Representation Based Numerical Scheme for Computing Fractional Derivatives
The section frames the task as approximating a Caputo derivative and describes a diffusive-representation scheme built from quadrature and numerical solution of an initial-value problem. It also identifies computational cost, active memory, and grid flexibility as advantages of the design.
1 A Diffusive Representation Based Numerical Scheme for Computing Fractional Derivatives
- The algorithm approximates the representation’s integral by quadrature and computes its integrand by numerically solving an initial-value problem.It uses Gauss-Laguerre quadrature; the two variants use backward Euler or the trapezoidal method, and this paper focuses on backward Euler.Source: §1 · p. 2
- O(N) computational complexity contrasts with traditional methods’ usual O(N^2) straightforward cost and O(N log N) or O(N log^2 N) costs with more sophisticated implementations.N is the number of time steps.Source: §1 · p. 2
- O(1) active memory contrasts with O(N) for traditional methods and, at best, O(log N) for modified versions.The passage attributes the difference to how the method handles the fractional operators’ inherent memory.Source: §1 · p. 3
- The approach permits any discretization of the solution interval, whereas some traditional schemes require a uniform mesh.Source: §1 · p. 3
2 Convergence Properties of the Numerical Method
The section develops a quantitative convergence analysis by examining quadrature and ODE-solver error. It establishes faster-than-algebraic quadrature convergence and an overall error estimate under stated smoothness, grid, and solver conditions.
2 Convergence Properties of the Numerical Method
- The analysis addresses the lack of a thorough quantitative convergence study in the earlier algorithm paper, which offered qualitative analysis and heuristic arguments.The authors focus on estimating the error expressions under reasonable assumptions on the data.Source: §2 · pp. 3–4
- The quadrature error converges to zero at a faster-than-algebraic rate.Lemma 1 assumes y ∈ C⌈α⌉[a, a + T] and states its bound for all p > 0.Source: §2 · p. 4
- The asymptotic estimates for φD(w, t) hold uniformly over t ∈ [a, a + T].The proof's implied constants can be chosen independently of t.Source: §2 · pp. 4, 7
- For y ∈ C⌈α⌉+1[a, a + T] on a uniform grid, the backward-Euler ODE-solver error admits a bound.The section assumes backward Euler for the initial-value problems.Source: §2 · p. 7
- Under the stated smoothness, uniform-grid, and backward-Euler conditions, Theorem 1 gives an overall error estimate combining quadrature and ODE-solver errors.The total error is analyzed through these two components.Source: §2 · pp. 4, 9
3 Comments and Further Remarks
The remarks examine how quadrature-node placement affects ODE-solver difficulty and error bounds, and discuss quadrature truncation and smoothness as directions for improving convergence. They also outline a possible diffusive-representation approach for fractional integrals.
3.1 The Stiffness of the Differential Equation (2)
- For w ∈ W+, Lipschitz constants can become extremely large when qD is close to 1 or K is large, making the equations stiff and motivating A-stable solvers.For w ∈ W−, the Lipschitz constants are small and the equations do not exhibit stiffness.Source: §3.1 · p. 10
3.2 The Error Bounds for the ODE Solver
- The ODE-solver error bound depends on the largest quadrature node, but may drastically overestimate the true error, so the paper recommends keeping that node small.Source: §3.2 · p. 10
3.3 Choice of the Quadrature Formula
- Truncating the quadrature sum at a properly chosen K* can reduce computational cost, improve ODE-solver approximation quality, and yield a better quadrature convergence rate.The cited proposal omits the largest nodes, whose associated equations have the largest Lipschitz constants.Source: §3.3 · pp. 10–11
3.4 Diffusive Representations for Fractional Integrals
- For fractional initial-value problems with α ∈ (0,1), the smoothness required by Lemma 3 may be rare, and less-smooth functions cannot achieve its and Theorem 1’s convergence order.Source: §3.4 · p. 11
- Rewriting the initial-value problem using the Riemann–Liouville integral and seeking its diffusive representation may yield functions with more favorable smoothness and possibly better ODE-solver convergence.Source: §3.4 · p. 11