Kansa-like Method for Meshfree Collocation

• Kansa-like Method is a meshfree, kernel-based collocation approach that approximates PDE solutions using radial basis functions and enforces the differential operator at selected collocation points.
• It supports both classical square formulations and enhanced over-tested or least-squares variants, addressing challenges like ill-conditioning and stability through strategic oversampling and basis augmentation.
• The method applies to elliptic, parabolic, fractional, and biological models on complex domains such as spheres and manifolds, offering rigorous error estimates and energy-conserving schemes.

Searching arXiv for the cited Kansa-like method papers to ground the article in current arXiv records. Kansa-like method denotes a family of meshfree, kernel-based strong-form collocation schemes in which an approximate solution is expanded in translates of a kernel or radial basis function and the governing differential operator is enforced at collocation points. In the classical square formulation, the trial and test sets coincide; in later asymmetric, over-tested, or constrained variants, the test set is enlarged, the collocation matrix becomes rectangular, and the discrete problem is solved by least squares or by a constrained optimization procedure. In the cited literature, this framework is developed for second-order strongly elliptic PDEs on spheres and manifolds, parabolic surface diffusion, Hamiltonian wave equations, spatiotemporal fractional diffusion, and systems of ODEs in biology (Hangelbroek et al., 20 Jul 2025, Chen et al., 2021, Li et al., 6 Jul 2025, Sun et al., 2016, Salehi et al., 2017).

1. Classical strong-form collocation

In its basic form, the method seeks an approximation of the form

$u_N(x)=\sum_{j=1}^N \xi_j\,\phi(\|x-x_j\|;c),$

where $\phi(r;c)$ is an RBF depending on a scalar shape parameter $c>0$, $\{x_j\}_{j=1}^N$ are distinct centers, and the coefficients $\{\xi_j\}$ are unknown. For a differential operator problem

$L[u](x)=f(x),\qquad B[u](x)=g(x),$

the strong form is collocated by enforcing

$L[u_N](x_i)=f(x_i),\qquad B[u_N](x_i)=g(x_i),$

at interior and boundary or initial points. This yields the algebraic system

$A(c)\,\Xi = F,$

with entries obtained by applying $L$ or $B$ directly to the basis functions at the collocation points (Salehi et al., 2017).

On compact manifolds, including the sphere, the same idea is expressed with a zonal kernel $\phi(r;c)$0, $\phi(r;c)$1, and a trial space

$\phi(r;c)$2

Writing

$\phi(r;c)$3

classical Kansa collocation imposes

$\phi(r;c)$4

so that the square matrix has entries

$\phi(r;c)$5

with $\phi(r;c)$6 in the classical case. The theoretical treatment on spheres emphasizes that this square Kansa matrix need not be invertible, nor well-conditioned, unless the kernel $\phi(r;c)$7 and operator $\phi(r;c)$8 satisfy special compatibility (Hangelbroek et al., 20 Jul 2025).

A persistent issue in the classical form is the interaction between approximation quality and conditioning. In Gaussian-RBF implementations, small $\phi(r;c)$9 produces very flat, globally supported basis functions and severe ill-conditioning, whereas large $c>0$0 gives strongly peaked bases and poorer global approximation (Salehi et al., 2017). This trade-off is one of the main motivations for later Kansa-like extensions.

2. Asymmetric, over-tested, and least-squares formulations

A central development is the replacement of the square collocation system by an overdetermined one. In the asymmetric formulation for elliptic PDEs on manifolds, the trial set $c>0$1 is retained, but the test set is enlarged to $c>0$2 with $c>0$3, producing

$c>0$4

Here

$c>0$5

and $c>0$6 may be kernel translates, a Lagrange basis, or a local Lagrange basis. The analysis introduces the notion of an $c>0$7-norming set $c>0$8 for a finite-dimensional space $c>0$9, meaning that

$\{x_j\}_{j=1}^N$0

When the trial-related space satisfies a Bernstein inequality, a quasi-uniform minimal $\{x_j\}_{j=1}^N$1-net $\{x_j\}_{j=1}^N$2 of cardinality $\{x_j\}_{j=1}^N$3 can be chosen so that this norming property holds with $\{x_j\}_{j=1}^N$4 (Hangelbroek et al., 20 Jul 2025).

The same oversampling principle appears in the surface-diffusion setting. With trial centers $\{x_j\}_{j=1}^N$5 and collocation nodes $\{x_j\}_{j=1}^N$6, the semi-discrete method of lines produces

$\{x_j\}_{j=1}^N$7

with $\{x_j\}_{j=1}^N$8. Because the system is overdetermined, it is not solved exactly; instead one minimizes the spatial $\{x_j\}_{j=1}^N$9 residual at fixed $\{\xi_j\}$0, or the space-time functional

$\{\xi_j\}$1

subject to an initial condition obtained by a discrete Tikhonov-regularized fit of $\{\xi_j\}$2 (Chen et al., 2021).

The cited analyses make stability the principal justification for oversampling. In the elliptic manifold theory, if $\{\xi_j\}$3 is an $\{\xi_j\}$4-norming set and the trial basis is a Riesz basis with stability ratio $\{\xi_j\}$5, then

$\{\xi_j\}$6

Hence $\{\xi_j\}$7 is injective and $\{\xi_j\}$8 is invertible on $\{\xi_j\}$9 (Hangelbroek et al., 20 Jul 2025). In the surface-diffusion theory, discrete sampling inequalities and a time-dependent stability estimate control the $L[u](x)=f(x),\qquad B[u](x)=g(x),$0, $L[u](x)=f(x),\qquad B[u](x)=g(x),$1, and $L[u](x)=f(x),\qquad B[u](x)=g(x),$2 components of the error, and the paper explicitly contrasts this with the square Kansa method, which is described as ill-conditioned and subject to solvability and stability difficulties (Chen et al., 2021).

3. Kernel spaces, trial spaces, and geometric operators

Kansa-like methods are defined by a trial space together with a strong-form discretization of the operator. For surface diffusion on a smooth, closed, orientable Riemannian manifold $L[u](x)=f(x),\qquad B[u](x)=g(x),$3, the trial kernel is chosen from a positive-definite ambient-space kernel $L[u](x)=f(x),\qquad B[u](x)=g(x),$4 whose Fourier transform decays like $L[u](x)=f(x),\qquad B[u](x)=g(x),$5. Restricting it to $L[u](x)=f(x),\qquad B[u](x)=g(x),$6 gives

$L[u](x)=f(x),\qquad B[u](x)=g(x),$7

which reproduces $L[u](x)=f(x),\qquad B[u](x)=g(x),$8. The trial space is

$L[u](x)=f(x),\qquad B[u](x)=g(x),$9

and the approximate solution is

$L[u_N](x_i)=f(x_i),\qquad B[u_N](x_i)=g(x_i),$0

The Laplace–Beltrami operator is implemented extrinsically through the closest-point extension and the projection $L[u_N](x_i)=f(x_i),\qquad B[u_N](x_i)=g(x_i),$1: $L[u_N](x_i)=f(x_i),\qquad B[u_N](x_i)=g(x_i),$2 (Chen et al., 2021).

A closely related manifold construction appears in the Trajectory-Based RBF Collocation method for surface advection-diffusion. There the trial centers $L[u_N](x_i)=f(x_i),\qquad B[u_N](x_i)=g(x_i),$3 and collocation points $L[u_N](x_i)=f(x_i),\qquad B[u_N](x_i)=g(x_i),$4 lie on a smooth, compact manifold $L[u_N](x_i)=f(x_i),\qquad B[u_N](x_i)=g(x_i),$5, and the approximation takes the form

$L[u_N](x_i)=f(x_i),\qquad B[u_N](x_i)=g(x_i),$6

The numerical tests use the Whittle–Matérn–Sobolev kernel

$L[u_N](x_i)=f(x_i),\qquad B[u_N](x_i)=g(x_i),$7

and the description explicitly states that no additional shape parameter is introduced beyond the native length-scale $L[u_N](x_i)=f(x_i),\qquad B[u_N](x_i)=g(x_i),$8. The surface Laplacian is again treated extrinsically,

$L[u_N](x_i)=f(x_i),\qquad B[u_N](x_i)=g(x_i),$9

now coupled to a characteristic back-tracking map in time (Li et al., 26 Jan 2026).

On spheres and other closed compact manifolds, the elliptic theory uses either positive-definite or strictly conditionally positive-definite zonal kernels. In the conditionally positive-definite case, the trial space is augmented by spherical harmonics and side conditions of the form $A(c)\,\Xi = F,$0 are imposed for all harmonics of degree at most $A(c)\,\Xi = F,$1 (Hangelbroek et al., 20 Jul 2025). This indicates that the term “Kansa-like method” covers a broader algebraic structure than a single choice of kernel or basis.

4. Time discretization and structure-preserving variants

For parabolic surface problems, the Kansa-like method is naturally cast as a method of lines. Substituting the trial expansion into

$A(c)\,\Xi = F,$2

and collocating in space gives the overdetermined ODE system

$A(c)\,\Xi = F,$3

where $A(c)\,\Xi = F,$4 and $A(c)\,\Xi = F,$5. Backward Euler yields

$A(c)\,\Xi = F,$6

or equivalently

$A(c)\,\Xi = F,$7

which is solved in the least-squares sense through the normal equations

$A(c)\,\Xi = F,$8

The paper also notes that higher-order backward-difference formulas and quadrature in time lead to decoupled least-squares solves at each time slice, and that an equivalent ODE form is

$A(c)\,\Xi = F,$9

(Chen et al., 2021).

In Hamiltonian wave equations, the Kansa framework is modified so that the discrete evolution inherits energy conservation. After time discretization of

$L$0

the step at time level $L$1 is written as a nonlinear least-squares problem

$L$2

subject to the energy constraint

$L$3

or, in quadratic-plus-nonlinear form,

$L$4

The constrained problem is handled through a Lagrangian and a fast iterative solver based on a generalized SVD of the pair $L$5 together with a one-dimensional Newton iteration for the Lagrange multiplier $L$6. The cited complexity estimates are one GSVD at cost $L$7 and $L$8 per Newton iteration, with a typical requirement of $L$9–$B$0 iterations per step. The discrete energy is maintained to within the solver tolerance, reported as $B$1–$B$2 in experiments, and comparisons with a secant-based Lagrange-multiplier solver show a $B$3–$B$4 speed-up (Li et al., 6 Jul 2025).

The characteristic-splitting surface advection-diffusion method provides a different time-dependent variant. After back-tracking the advection term along characteristics, the diffusion stage is advanced by Crank–Nicolson or BDF. For Crank–Nicolson the coefficients satisfy

$B$5

where the matrix $B$6 evaluates the kernel and its surface Laplacian at the departure points $B$7. The theoretical contribution is an exact equivalence between the operator-split characteristic system and the original surface PDE, so that no operator-splitting error arises; the resulting time discretization is described as unconditionally stable under Crank–Nicolson (Li et al., 26 Jan 2026).

5. Fractional, biological, and surface applications

The method has been adapted to spatiotemporal fractional diffusion by combining Kansa collocation in space with an analytical treatment in time. For the two-dimensional FADE

$B$8

the approximation uses Hardy’s multiquadric kernel

$B$9

and a global expansion

$\phi(r;c)$00

Collocating the spatial operators gives

$\phi(r;c)$01

When $\phi(r;c)$02, setting $\phi(r;c)$03 yields the analytical solution

$\phi(r;c)$04

The implementation therefore avoids time-stepping; the main cost is formation and inversion of $\phi(r;c)$05 and the eigendecomposition of $\phi(r;c)$06, each independent of the final time horizon $\phi(r;c)$07. Reported one-dimensional maximum absolute errors at $\phi(r;c)$08, $\phi(r;c)$09, $\phi(r;c)$10 are $\phi(r;c)$11, $\phi(r;c)$12, $\phi(r;c)$13, and $\phi(r;c)$14 for $\phi(r;c)$15, respectively. In the two-dimensional rectangular test, the MAEs at $\phi(r;c)$16 are $\phi(r;c)$17, $\phi(r;c)$18, $\phi(r;c)$19, and $\phi(r;c)$20 for $\phi(r;c)$21, and random versus perturbed grids are reported to give comparable MAEs within a factor of two (Sun et al., 2016).

In biological ODE models, Kansa collocation has been combined with a genetic strategy for selecting the Gaussian shape parameter

$\phi(r;c)$22

For the HIV CD4$\phi(r;c)$23 T-cell model, the three ODEs are collocated at $\phi(r;c)$24 time points in $\phi(r;c)$25, the three initial conditions are imposed at $\phi(r;c)$26, and the resulting $\phi(r;c)$27 nonlinear equations are solved by Newton–Raphson. The GA searches $\phi(r;c)$28 and, for $\phi(r;c)$29, converges to $\phi(r;c)$30. The reported Gaussian-RBF solution matches an eighth-order Runge–Kutta solution to $\phi(r;c)$31 or better for $\phi(r;c)$32, $\phi(r;c)$33, and $\phi(r;c)$34. For the Influenza SIRC model, four ODEs plus four initial conditions produce $\phi(r;c)$35 equations, the GA searches $\phi(r;c)$36, and for $\phi(r;c)$37 the selected values are $\phi(r;c)$38, $\phi(r;c)$39, and $\phi(r;c)$40, respectively. Relative errors of order $\phi(r;c)$41 are reported even at $\phi(r;c)$42, improving as $\phi(r;c)$43 increases (Salehi et al., 2017).

Surface applications highlight the geometric flexibility of over-tested Kansa-like methods. On the sphere, the surface-diffusion paper studies

$\phi(r;c)$44

with manufactured solution

$\phi(r;c)$45

Using Wendland-type or Matérn-type $\phi(r;c)$46 with $\phi(r;c)$47, the method shows second-order temporal accuracy in $\phi(r;c)$48 and spectral-like spatial decay in $\phi(r;c)$49 until stagnation. A tensor-diffusion example with

$\phi(r;c)$50

demonstrates anisotropic diffusion, and an Allen–Cahn computation compares the least-squares RBF method with a meshless Galerkin solver of Kunemund–Narcowich–Ward–Wendland, reporting comparable accuracy and superior robustness for long-time integration. The same work emphasizes that surface geometries beyond spheres, including the torus, require no parameterization, only closest-point extension and normal-vector projection (Chen et al., 2021).

6. Error estimates, conditioning, and methodological scope

The modern theory of Kansa-like methods is largely a theory of stability and error control under oversampling. For second-order elliptic PDEs on spheres, the discrete least-squares solution based on a thin-plate spline spherical basis satisfies

$\phi(r;c)$51

provided $\phi(r;c)$52 and $\phi(r;c)$53 are sufficiently smooth and $\phi(r;c)$54 is a suitable $\phi(r;c)$55-norming set with $\phi(r;c)$56. The RRQR-based thinning procedure, which reduces an oversampled system to a square one, leads to

$\phi(r;c)$57

The same framework shows that the reduced matrix $\phi(r;c)$58 can be chosen so that

$\phi(r;c)$59

with $\phi(r;c)$60 (Hangelbroek et al., 20 Jul 2025).

For parabolic surface diffusion, the analysis is organized into regularity of the exact solution, regularization of the discrete initial condition, stability of the discrete least-squares formulation, and consistency with a kernel interpolant. With $\phi(r;c)$61, the regularized initial state satisfies

$\phi(r;c)$62

and the final bound is

$\phi(r;c)$63

where

$\phi(r;c)$64

The stated theorem assumes $\phi(r;c)$65, quasi-uniform $\phi(r;c)$66 and $\phi(r;c)$67, and $\phi(r;c)$68 (Chen et al., 2021).

For manifold advection-diffusion with Whittle–Matérn–Sobolev kernels, the cited estimate is the standard Sobolev-space bound

$\phi(r;c)$69

while the numerical section reports superconvergent rates in $\phi(r;c)$70 (Li et al., 26 Jan 2026). For Hamiltonian wave equations, the oversampled Kansa discretization is stated to recover nearly $\phi(r;c)$71-order in $\phi(r;c)$72 when the oversampling rate $\phi(r;c)$73 and to preserve discrete energy by construction, up to solver tolerance (Li et al., 6 Jul 2025).

Several recurrent limitations are explicit in the literature. Global collocation matrices are dense, so direct dense solution is typical and can incur $\phi(r;c)$74 cost per solve or per precomputation stage (Sun et al., 2016, Li et al., 26 Jan 2026). Conditioning remains sensitive to the kernel choice, the smoothness order $\phi(r;c)$75, the node density $\phi(r;c)$76, and, when present, the shape parameter $\phi(r;c)$77 (Salehi et al., 2017, Li et al., 26 Jan 2026). At the same time, the cited papers consistently present the meshfree character, absence of triangulation, elimination of surface integrals in strong-form discretizations, and direct handling of complex geometries as the main practical scope of Kansa-like methods (Chen et al., 2021, Li et al., 6 Jul 2025).

A plausible implication is that “Kansa-like” is best understood not as a single algorithm, but as a family of asymmetric or strong-form kernel collocation strategies whose defining questions are the same across applications: choice of trial space, selection of test points, stabilization by oversampling or constraints, and the management of dense linear algebra.