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Generalized MLS and LS-SVR for Allen-Cahn Equations on Spheres

Vahid Mohammadi, Mehdi Dehghan, Stefano De Marchi

arXiv:2610.30714v1math.NAcs.NA

TL;DR

The paper addresses numerical solution of Allen-Cahn and conservative Allen-Cahn equations on spheres using a GMLS-based LS-SVR framework. It connects the dual approximant to smoothing and kernel ridge regression in an RKHS, and uses implicit time splitting for the equations. The framework is shown to yield unique intermediate solutions, subject to the stated full-rank condition for the RKHS construction.

  • Problem

    The paper notes that the LS-SVR interpretation as smoothing or kernel ridge regression in an associated RKHS had not been considered for this sphere-based approach.

    Source: §1 · p. 3

  • Method

    The method combines GMLS-based LS-SVR collocation on spheres with implicit time splitting, solving the linear part numerically and treating the nonlinear part analytically.

    Source: §1 · p. 4

  • Results

    The proposed LS-SVR is equivalent to smoothing or kernel ridge regression in its associated RKHS, and its dual formulation has a unique solution for γ > 0.

    Source: §1 · p. 4

  • Takeaways & Limitations

    The framework is applied to both Allen-Cahn models on spheres, with a correction procedure used to preserve mass for the conservative model.

    Source: §1 · p. 1, §7 · p. 25

    The RKHS construction assumes that the matrix A has full rank for distinct points on the sphere.

    Source: §4 · p. 8

Abstract

from arXiv · show

This work extends least-squares support vector regression (LS-SVR) based on the generalized moving least squares (GMLS) approximation to the numerical solution of partial differential equations on spheres. We show that the proposed approach constructs a kernel-based approximant in its dual form and establishes its connection with the associated reproducing kernel Hilbert space (RKHS). Furthermore, solving the PDE through the proposed LS-SVR framework is shown to be equivalent to solving a smoothing problem or a kernel ridge regression problem in the corresponding RKHS. As applications, the method is employed to solve the Allen-Cahn equation and its conservative counterpart on spheres. An implicit first-order operator-splitting scheme is used to separate each model into linear and nonlinear subproblems. The linear subproblem is discretized by the backward differentiation formula and approximated at each time step using LS-SVR derived from a GMLS-based collocation formulation, while the nonlinear part is treated analytically. The resulting scheme is shown to admit a unique numerical solution at every time step. For the conservative Allen-Cahn model, a correction procedure is incorporated to preserve mass. Numerical experiments demonstrate the accuracy and effectiveness of the proposed framework.

1 Introduction

The paper introduces a GMLS-based LS-SVR method for solving Allen-Cahn and conservative Allen-Cahn equations on spheres, and relates its dual approximant to smoothing and kernel ridge regression in an RKHS. An implicit time-splitting scheme separates linear and nonlinear work, with a correction for mass conservation in the conservative model.

1 Introduction

  • Earlier surface-PDE approaches include narrow-band finite differences and restricted radial-basis-function methods for Allen-Cahn, and finite differences for conservative Allen-Cahn.The paper describes these prior approaches alongside its meshfree formulation.Source: §1 · pp. 2, 4
  • The proposed LS-SVR on spheres is equivalent to a smoothing problem or kernel ridge regression in its associated RKHS.The paper also develops it for the Allen-Cahn and conservative Allen-Cahn equations using GMLS.Source: §1 · pp. 3–4
  • The first GMLS-based LS-SVR approach for these nonlinear phase-field equations on spheres approximates the Laplace-Beltrami operator more easily than mesh-dependent methods.The approximation does not require a triangulation or background cell, and does not require quasi-Neumann boundary conditions on the narrow-band boundary.Source: §1 · p. 4
  • Implicit time splitting yields a fully discrete scheme with linear and nonlinear subproblems, treating the nonlinear part analytically without nonlinear iterative solvers.The linear part is solved using the GMLS collocation formulation at each time step.Source: §1 · p. 4
  • The dual regression problem has a positive definite coefficient matrix, giving a unique approximate intermediate solution at each time step.For the conservative model, a Lagrange multiplier is used to satisfy mass conservation.Source: §1 · pp. 2, 4

2 The GMLS procedure on spheres

GMLS on the sphere uses scattered nodes and local weighted minimization over spherical harmonics to approximate differential operators directly. Its construction is meshfree and provides a sparse differentiation matrix.

2 The GMLS procedure on spheres

  • The method is meshfree: it builds the trial approximation from scattered spherical nodes rather than a triangulation or background cell.The node sets are assumed quasi-uniform, with mesh ratio bounded by a constant independent of the set.Source: §2 · pp. 4–5
  • GMLS approximates the Laplace-Beltrami operator by applying it to spherical harmonics of degree at most m, with coefficients obtained from a weighted minimization.The minimization is over the spherical harmonics space on the local spherical cap.Source: §2 · p. 5
  • Local coefficients use nodes within a spherical cap, with geodesic distance defining the cap radius and weights.The construction assumes unisolvency on each spherical cap.Source: §2 · p. 5
  • Applying GMLS to the differential operator at the node set produces a sparse differentiation matrix; point evaluation instead yields an interpolant with the delta-Kronecker property.The interpolant reproduces the nodal values through coefficients that are one at the matching node and zero at the others.Source: §2 · pp. 5–6

3 The LS-SVR on spheres

The paper extends LS-SVR to collocation problems on spheres by using GMLS to form the differential-operator constraints. Its dual solution is a linear system with a unique solution for positive regularization parameter γ.

3 The LS-SVR on spheres

  • The LS-SVR formulation on spheres treats a linear differential equation as a collocation regression problem with weights on basis functions and errors at the training nodes.The differential operator and forcing term define the constraints at the nodes.Source: §3 · p. 6
  • GMLS supplies the operator matrix A, and the Lagrangian coefficients α are obtained from the dual linear system.The matrix in the system is B = A A^T + (1/γ)I_N.Source: §3 · p. 7
  • For every γ > 0, the LS-SVR minimization problem has a unique solution because its coefficient matrix is invertible.The proof establishes positive definiteness of the matrix.Source: §3 · p. 7
  • The dual approximate solution is kernel-based, with K(x, x_j) defined by an inner product involving the GMLS basis functions and the differential operator.This gives the approximate solution its dual representation.Source: §3 · p. 7

4 An RKHS

The section constructs an RKHS from the GMLS-derived kernel and connects the LS-SVR optimization to smoothing and kernel ridge regression formulations.

4 An RKHS

  • The constructed kernel reproduces functions in the associated RKHS, under the stated full-rank condition on A.The feature-map representation is isometrically isomorphic to the RKHS.Source: §4 · p. 8
  • The kernel-based interpolant matches the given data and has minimum RKHS norm among interpolants.The interpolation result assumes A has no loss of rank.Source: §4 · p. 10

4.1 Smoothing in RKHS

  • The unique solution of the LS-SVR minimization problem is also the solution of the RKHS smoothing problem.The section gives an upper bound for the smoothed approximation at the data points and recommends β ≥ 2 when choosing γ for a proper convergence rate.Source: §4.1 · pp. 11–12

4.2 Connection to the KRR

  • The regularized least-squares approximation and LS-SVR problems have the same solution; as γ tends to infinity, the solution approaches the unregularized one.With L = Id and γ = +∞, the approximate solution interpolates the given data.Source: §4.2 · p. 14

5 Application to two phase field models on S2

The section applies first-order implicit time splitting and GMLS-based LS-SVR collocation to the Allen–Cahn (AC) and conservative Allen–Cahn (CAC) models on S2. The resulting discrete schemes have unique solutions at every time step, and the CAC time discretization preserves total mass.

5 Application to two phase field models on S2

  • Both phase-field equations are discretized using first-order implicit time splitting, followed by LS-SVR spatial discretization of the resulting semi-discrete problems.Source: §5 · p. 15, §5.2 · p. 16

5.1 An implicit time-splitting technique

  • The proposed time discretization for the CAC model preserves total mass in time.Source: §5.1 · p. 15

5.2.1 AC model

  • For AC, GMLS collocation yields sparse differentiation matrices, and the LS-SVR weights are computed using KKT conditions in a dual formulation.Source: §5.2.1 · p. 16

5.2.1 AC model; 5.2.2 CAC model

  • For each n ≥ 0, each of the two discrete schemes has a unique solution Un+1: one is given in (5.10)–(5.13), and the other in (5.16)–(5.20).Source: §5.2.1 · p. 16, §5.2.2 · p. 18

5.2.2 CAC model

  • For CAC, the LS-SVR collocation procedure follows the AC procedure, and its correction coefficient is computed using integration over 16,382 spherical t-design points.Source: §5.2.2 · p. 17

6 Numerical results

The numerical tests assess convergence, accuracy for motion by mean curvature, and the conservative Allen–Cahn model on the sphere. They report the conservative model’s cap evolution and compare time- and space-discretization errors.

6 Numerical results

  • The experiments use quasi-uniform minimum-energy and phyllotaxis-spiral nodes, with fill distance O(N^-1/2), and local spherical caps for trial points.The reported implementation also uses QR factorization for local normal systems and RCM-reordered PCG with ILU(0) preconditioning.Source: §6 · pp. 18–19

6.1 Convergence test

  • For LS-SVR based on the GMLS approximation, discrete maximum errors show observed convergence approximately twice as accurate as the theoretical rate; γ=h^-3, h^-4, and h^-5 yield orders 3, 4, and 5, respectively.Source: §6.1 · p. 19

6.2 Test 1 (Accuracy test): Motion by mean curvature on the unit sphere

  • At T = 0.2, the mean-curve motion test reports average convergence rates of about 1.35 in time and 2.93 in space.The time-discretization errors use 10,201 PTS points; spatial errors use Δt = 10^-4.Source: §6.2 · p. 21

6.3 Test 2

  • The larger spherical cap grows while the smaller shrinks until disappearing; the same results were obtained in using 2,803,221 points and a closest-point approach.For N = 25,600, approximate mass and radii R1 and R2 are reported over time.Source: §6.3 · pp. 22–23

7 Concluding remarks

The concluding remarks connect the GMLS-based LS-SVR approximation on spheres to an RKHS and to smoothing or kernel ridge regression. The approach is applied to Allen–Cahn models, with numerical tests supporting its stated accuracy and suitability.

7 Concluding remarks

  • The LS-SVR dual approximation defines a kernel and an associated RKHS, where its minimization problem is connected to smoothing and kernel ridge regression.The connection is established for the GMLS-based approximation problem on spheres.Source: §7 · p. 25
  • An implicit first-order time splitting separates each phase-field equation into linear and nonlinear problems, with LS-SVR/GMLS applied to the linear part at each step.The intermediate solution is obtained from a sparse symmetric positive-definite linear system; the nonlinear stage is handled by substitution into the semi-discrete scheme.Source: §7 · p. 25
  • The authors report that their simulations illustrate the error bounds and support stable, accurate numerical solutions for the studied phase-field models on spheres.Source: §7 · p. 25

Appendix A

Appendix A states mass-conservation properties for the continuous conservative Allen–Cahn model and its time discretization.

Appendix A

  • Lemma 7.1 states that solutions of the conservative Allen–Cahn equation satisfy a mass-conservation property.Source: Appendix A · p. 29
  • The proof tests the equation with the constant function 1 and invokes the Green–Beltrami identity on the sphere.Source: Appendix A · p. 29
  • Lemma 7.2 states that the time discretization defined by relations (5.3)–(5.5) is mass conservative.The proof cites relation (5.6).Source: Appendix A · p. 30

Appendix B

Appendix B presents the numerical schemes and computational procedures for the Allen–Cahn and conservative Allen–Cahn equations on the sphere.

Appendix B

  • The appendix provides algorithms for GMLS approximation and for the two phase-field numerical schemes.It identifies Algorithm 1 as the GMLS procedure and Algorithms 2 and 3 as the scheme procedures.Source: Appendix B · pp. 30–32
  • The GMLS procedure takes trial and test points on the sphere and a cap radius, then constructs sparse approximation matrices.Its steps form local matrices and compute approximations using the weight function and QR factorization.Source: Appendix B · p. 31
  • The scheme algorithms take point sets, cap radius, model parameters, final time, and time step, and output an approximate solution at final time.For the CAC scheme, the listed steps also compute the correction coefficient by numerical integration.Source: Appendix B · pp. 31–32
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