Greedy Trial Subspace Selection in Meshfree Time-Stepping Scheme with Applications in Coupled Bulk-Surface Pattern Formations

Abstract

Combining kernel-based collocation methods with time-stepping methods to solve parabolic partial differential equations can potentially introduce challenges in balancing temporal and spatial discretization errors. Typically, using kernels with high orders of smoothness on some sufficiently dense set of trial centers provides high spatial approximation accuracy that can exceed the accuracy of finite difference methods in time. The paper proposes a greedy approach for selecting trial subspaces in the kernel-based collocation method applied to time-stepping to balance errors in both well-conditioned and ill-conditioned scenarios. The approach involves selecting trial centers using a fast block-greedy algorithm with new stopping criteria that aim to balance temporal and spatial errors. Numerical simulations of coupled bulk-surface pattern formations, a system involving two functions in the domain and two on the boundary, illustrate the effectiveness of the proposed method in reducing trial space dimensions while maintaining accuracy.

Figures (14)

• Figure 1: For Example \ref{['eg: 2D RD PDE']}, the relative root mean error profiles of solving a heat equation by meshfree time-stepping method with various $\triangle t$ (\ref{['fig:Unstable2Dsqr']}) without and with the greedy algorithm in default settings; (\ref{['fig:2DComGdyTau']}) with greedy algorithm using different tolerance in stopping criteria. Colored numbers in (\ref{['fig:2DComGdyTau']}) are the number of selected columns in ${{A}}$ by the greedy algorithm.
• Figure 2: Greedy algorithm stops by large condition number, 128 columns are selected in the column space -- A schematic demonstration of residual value and condition numbers of submatrices of ${{A}}$ with expending columns ${\mathbcal{n}}\to{\mathbcal{n}}'$ and (a) selected rows ${\mathbcal{m}}$ and (b) all rows $\mathbb{N}_m$ in the column space.
• Figure 3: Greedy algorithm stops by small residual, 128 columns are selected in the column space --A schematic demonstration of residual value and condition numbers of submatrices of ${{A}}$ with expending columns ${\mathbcal{n}}\to{\mathbcal{n}}'$ and (a) selected rows ${\mathbcal{m}}$, and (b) all rows $\mathbb{N}_m$ in the column space.
• Figure 4: Example \ref{['eg: 2D RD PDE new']}, error profiles for solving a 2D heat equation using a meshfree time-stepping method with greedy algorithm and the MS kernel with $\mu=6$. Different values of $\varepsilon$ and $\triangle t$ were used, and the shaded areas in each plot show the error range for $n\in[500,1000]$ data points, with the median as a dashed line. The stopping criteria that terminate the greedy algorithm are shown as black circles for all columns being selected, red circles for SC-$1'$, and blue circles for SC-$2'$.
• Figure 5: Example \ref{['eg: 2D RD PDE new']}, (a) error profiles for the solution of a 3D heat equation using a meshfree time-stepping method with the new greedy stopping criteria for different numbers of data points $n$. (b) error function for $n = 7000$, $\triangle t=0.01$.