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A meshless based method for solution of integral equations: Improving the error analysis
Davoud Mirzaei
TL;DR
The paper addresses error analysis for an MLS-based collocation method for integral equations. It develops revised approximation and operator-analysis results, including convergence and error bounds. The final estimation reflects both quadrature and MLS approximation errors, while the analysis assumes quadrature convergence and requires uniform control of the interpolation projections.
Problem
The paper studies how to establish error bounds for a discrete MLS collocation method for Fredholm integral equations, requiring solvability, operator properties, and MLS approximation estimates.
Method
The method combines MLS approximation with quadrature, interpolation projections, and collectively compact operator analysis to derive the final error bound.
Results
The analysis shows that both quadrature and MLS approximation errors contribute to the final estimate; with high-order quadrature, MLS approximation error dominates for sufficiently smooth kernels.
Takeaways & Limitations
Uniformly bounded projections together with MLS convergence yield uniformly convergent interpolants, supporting the collocation error analysis.
Takeaways & Limitations
The analysis assumes quadrature convergence for every continuous function and leaves the uniform O(1) bound for the projection operators to be proved theoretically.
Abstract
from arXiv · showhide
This draft concerns the error analysis of a collocation method based on the moving least squares (MLS) approximation for integral equations, which improves the results of [2] in the analysis part. This is mainly a translation from Persian of some parts of Chapter 2 of the author's PhD thesis in 2011.
1 Introduction
The paper revises the MLS collocation error analysis for second-kind Fredholm integral equations, improving the presentation of the approximation and its error estimation relative to.
- The study revises an MLS-based meshless method previously applied to second-kind integral equations.The earlier work presented error analysis for Fredholm integral equations; this paper develops a revised analysis.
2 MLS approximation
This section defines the moving least squares approximation from locally weighted data sites and establishes its stability, regularity, convergence, and error properties under geometric and smoothness assumptions. It also describes shifted and scaled polynomial bases for more stable numerical implementation.
- Construction: The MLS approximation represents u using values at centers xj, with locally supported weights determined by the evaluation point and a threshold δ.The active centers are those within δ of the evaluation point, and the weights are generated from a nonnegative kernel supported in the unit ball.
- Construction: P_m-unisolvency makes the matrix A(x) symmetric positive definite, ensuring unique local coefficient determination under the stated assumptions.The kernel is assumed continuous, nonnegative, and positive on B(0, 1/2).
- Regularity: If the kernel K belongs to C^k(R^d), the MLS shape functions and approximation inherit C^n regularity with n = min{k, m}.The radial-kernel formulation uses a profile positive on [0, 1/2], supported in [0, 1], with a continuous nonnegative even extension.
- Approximation properties: For quasi-uniform data sites in compact domains satisfying an interior cone condition, MLS shape functions provide stable local polynomial reproduction of degree m.The associated stability constants are independent of the data-site set under the stated conditions.
- Approximation properties: The MLS approximation converges uniformly for every continuous function on a compact domain as the fill distance h_X,Ω tends to zero for quasi-uniform sets.The proof uses stable local polynomial reproduction, compactness, and the modulus of continuity of u.
- Error analysis: For u ∈ W^{m+1}_∞(Ω) on a compact domain with Lipschitz boundary, the section states an MLS error estimate for sufficiently fine quasi-uniform data sites.The estimate applies when h_X,Ω ≤ h0 and the quasi-uniformity constant remains fixed.
- Implementation: Shifted and scaled polynomial bases are recomputed at each evaluation point, producing a more stable algorithm and overcoming reported instability for quadratic bases.The text specifically links this basis change to instability in Tables 1, 4, and 6 of [2, Section 6].
3 The MLS collocation method
The method discretizes a second-kind Fredholm integral equation using MLS trial functions, numerical quadrature, and collocation at test points. Solving the resulting linear system produces discrete and iterated collocation approximations whose error bounds are analyzed later.
- The target equation has an unknown function u, parameter λ, continuous right-hand side f, and continuous kernel κ on a compact domain Ω.
- The MLS approximation is constructed over quasi-uniform trial points X and belongs to the finite-dimensional space V_N.
- A projection operator P_N maps continuous functions into V_N by interpolation at test points Y.
- Collocation imposes vanishing residuals at the test points, yielding an abstract discrete collocation scheme.
- Numerical quadrature replaces the integral using integration points and weights, and the resulting equations are solved for the approximation values at the trial points.
- The usual choice M = N produces a square linear system, while oversampling with M > N may help when solvability is problematic.
4 Error Analysis
The error analysis establishes convergence and solvability of the MLS collocation scheme by combining MLS approximation bounds with collectively compact operator theory. The final estimate depends on both quadrature and MLS errors, with high-order quadrature leaving MLS error dominant for sufficiently smooth kernels.
- The analysis requires solvability of the integral equation, properties of integration operators and projections, and an MLS approximation error bound.These ingredients are needed to obtain the final error bound for the discrete collocation method.
- A uniform bound on the projection operators, together with pointwise convergence of the integration operators, yields a collectively compact, pointwise-convergent family {F_NP_N}.The argument uses boundedness of {P_N} and collective compactness of {F_N}.
- Under quasi-uniform sampling and the stated stability condition, P_Nu converges uniformly to u as the fill distance tends to zero.The projection bound follows when ||Φ_N^-1||_∞ is O(1) independently of N or h_X,Ω.
- The analysis still requires theoretical proof that ||Φ_N^-1||_∞ remains O(1) independently of the fill distance, even when M=N.This assertion is identified as remaining to be proved.
- For sufficiently large N, the discrete operator inverse exists and is uniformly bounded when the continuous equation is uniquely solvable and the operator-family assumptions hold.The theorem also provides an error estimate for the iterative and discrete collocation solutions.
- Both quadrature and MLS approximation error bounds contribute to the final estimate; high-order quadrature makes MLS error dominant for sufficiently smooth kernels.The paper points to for numerical results.