Source-linked AI summary
On stable numerical differentiation
N. S. Hoang, A. G. Ramm
TL;DR
Stable differentiation is difficult because small errors in function data can produce large derivative errors. The paper compares direct regularized Volterra solving with a DSM-based approach; DSM competes favorably with VR, while boundary effects and parameter choice constrain use.
Problem
Small errors in noisy function data can lead to large derivative errors, making stable differentiation an ill-posed problem.
Source: §1 · p. 1Method
The paper presents direct solution of a regularized Volterra equation and a DSM-based approach for solving its discretized form.
Source: §1 · p. 1, §2.1 · p. 2, §2.2 · p. 3Results
DSM competes favorably with VR in accuracy and computation time, while first-method accuracy varies with noise level and boundary behavior.
Source: §3.1 · p. 5, §3.2 · p. 6, §4 · p. 7Takeaways & Limitations
For interval data, the paper advises collecting data on a larger interval and differentiating at points away from the boundary.
Source: §3.2 · p. 6No a posteriori choice of α(δ) is known for the first method.
Source: §4 · p. 7
Abstract
from arXiv · showhide
Based on a regularized Volterra equation, two different approaches for numerical differentiation are considered. The first approach consists of solving a regularized Volterra equation while the second approach is based on solving a disretized version of the regularized Volterra equation. Numerical experiments show that these methods are efficient and compete favorably with the variational regularization method for stable calculating the derivatives of noisy functions.
1 Introduction
Stable differentiation of noisy data is ill-posed, and the paper proposes a regularized Volterra-equation approach to address it.
1 Introduction
- Small errors in noisy function data can produce large derivative errors, making stable differentiation ill-posed.The paper frames the task as recovering the derivative of a smooth function from data whose error is bounded by δ.Source: §1 · p. 1
- The paper proposes stable differentiation by solving a regularized Volterra equation, alongside the commonly used Variational Regularization method.The introduction points to related work on the proposed approach.Source: §1 · p. 1
- The approach used in this paper and relies on the operator A having a nonnegative quadratic form in real Hilbert space L2(0, a).Here a is stated to be a positive constant.Source: §1 · p. 1
2 Methods
The methods section presents direct solution of a regularized equation and a DSM-based iterative approach for discretized equations.
2 Methods
- Two approaches solve the regularized equation: direct solution, or a DSM iterative scheme applied to its discretized version.The DSM scheme is based on an iterative method from.Source: §2 · p. 2
2.1 First method
- The first method uses an a priori parameter choice α(δ) = δ^k/c, with 0 < k < 1, then computes an integral to obtain a stable-differentiation scheme.The passage notes that no algorithm is known for choosing optimal k and c, and that runtime is very small compared with VR and DSM.Source: §2.1 · p. 2
2.2 An iterative scheme of DSM for solving discretizations of the regularized Volterra equation
- The DSM formulation applies to a closed, densely defined linear operator in a Hilbert space, with T = A∗A and a positive nonincreasing a(t) tending to zero.An iterative scheme stops at the first time its discrepancy-principle-type condition is satisfied; the initial regularization parameter is selected by iteration.Source: §2.2 · p. 3
3 Numerical experiments
Numerical experiments compare the first method with DSM and VR for first derivatives, and DSM with VR for second derivatives. Relative accuracy depends on noise level, endpoint behavior, and the test case.
3.1 Computing the first derivatives of a noisy function
- The experiments use trapezoidal quadrature; higher-order interpolation or collocation may not improve accuracy with noisy data and can yield worse results.The paper reports larger condition numbers and low-accuracy solutions with higher-order collocation methods.Source: §3.1 · p. 4
- For first derivatives, the first method has larger errors than VR and DSM at δ = 0.02 but smaller errors at δ = 0.002; away from endpoints, DSM and VR are more accurate.The first method's regularized solution is zero at both endpoints, unlike the true derivative in Figure 1.Source: §3.1 · p. 5
- When the true first derivative is zero at both endpoints, DSM and VR give more accurate results than the first method.This is the reported outcome for the second noisy first-derivative example.Source: §3.1 · p. 5
3.2 Computing the second derivatives of a noisy function
- DSM is more accurate than VR and takes about the same or less computation time; both methods perform better in case 2 than case 1.Table 1 reports results for cases 1 and 2.Source: Table 1 · p. 6, §3.2 · p. 6
- The regularized scheme forces zero endpoint derivatives, so the proposed method should use data from a larger interval and evaluate points away from its boundaries.The paper notes this boundary issue for computing derivatives of noisy functions on an interval.Source: §3.2 · p. 6
4 Concluding remarks
The paper presents two approaches to stable differentiation of noisy functions; the DSM approach competes favorably with variational regularization in computation time and accuracy.
- The first approach avoids matrix inversion and linear-system solves and runs quickly, but has no known a posteriori choice of α(δ).Source: §4 · pp. 6–7
- Figure 3 plots DSM and VR solutions for n=100 and δrel=0.02.Source: Figure 3 · p. 7
- The DSM approach competes favorably with variational regularization in both computation time and accuracy.Source: §4 · p. 7
- Future work includes a posteriori selection of α, efficient computation of integral (5), and improved Volterra-equation discretization for DSM.Source: §4 · p. 7