Refined rates of convergence for target-data dependent greedy generalized interpolation with Sobolev kernels

Abstract: Greedy methods have recently been successfully applied to generalized kernel interpolation, or the recovery of a function from data stemming from the evaluation of linear functionals, including the approximation of solutions of linear PDEs by symmetric collocation. When applied to kernels generating Sobolev spaces as their native Hilbert spaces, some of these greedy methods can provide the same error guarantee of generalized interpolation on quasi-uniform points. More importantly, certain target-data-adaptive methods even give a dimension- and smoothness-independent improvement in the speed of convergence over quasi-uniform points, thus offering advantages for high-dimensional problems. These convergence rates however contain a spurious logarithmic term that limits this beneficial effect. The goal of this note is to remove this factor, and this is possible by using estimates on metric entropy numbers.

• This paper refines the convergence rates of PDE-$eta$-greedy algorithms in generalized kernel interpolation, eliminating a logarithmic term that was previously thought to be intrinsic to the method.
• The improved error analysis uses metric entropy numbers instead of covering number arguments, revealing no inherent logarithmic penalty in the interpolation process.
• This advancement applies to any $eta ext{ value } [0,1]$, providing adaptivity gains in approximation rates for PDEs in multiple dimensions.

Overview and contribution

This paper by Haasdonk, Santin, Wenzel, and Winkle sharpens the error analysis of PDE-$\beta$-greedy algorithms for generalized kernel interpolation with Sobolev kernels (2601.20407). Generalized interpolation concerns recovering a function $u \in W_2^\tau(\Omega)$ from data of the form $L_i u(x) = f_i(x)$, where the $L_i$ are linear bounded (differential) operators; symmetric collocation for PDEs is the canonical instance. Prior work [Wenzel2025] established that adaptive greedy selection rules yield a dimension- and smoothness-independent improvement in the convergence rate over interpolation on quasi-uniform points, but the resulting bounds carried a spurious logarithmic factor $$\log(n)^{\frac{2(\tau - \bar m) - \bar d}{2\bar d \max(1,\beta)}}$$. The main result removes this logarithmic term entirely, showing it was an artifact of the proof technique rather than intrinsic to the method.

The improvement is achieved purely through a refined analytical argument based on metric entropy numbers; no algorithmic changes are introduced, and numerical validation is deferred to the experiments already published in [Wenzel2025].

Setting: generalized interpolation with Sobolev kernels

The framework considers a strictly positive definite kernel $k$ on $\Omega \subset \mathbb{R}^d$ whose native space embeds into $W_2^\tau(\Omega)$, with Matérn and Wendland kernels as standard examples. Data are generated by operators $$L_i: W_2^\tau(\Omega) \to W_2^{\tau - m_i}(\Omega_i)$$ satisfying a boundedness (trace-type) inequality, where each $\Omega_i$ is either a subset of $u \in W_2^\tau(\Omega)$0 or a smooth compact manifold of dimension $u \in W_2^\tau(\Omega)$1. The condition $u \in W_2^\tau(\Omega)$2 ensures that $u \in W_2^\tau(\Omega)$3 is continuous on $u \in W_2^\tau(\Omega)$4, so the composed functionals $u \in W_2^\tau(\Omega)$5 are continuous on the native space and possess Riesz representers $u \in W_2^\tau(\Omega)$6.

The generalized interpolant $u \in W_2^\tau(\Omega)$7 is the minimal-norm element of the native space satisfying the constraints at selected functionals, equivalently the orthogonal projection of $u \in W_2^\tau(\Omega)$8 onto the span of the selected representers. Error transfer from residuals to $u \in W_2^\tau(\Omega)$9 relies on a stability assumption of maximum-principle type,

$L_i u(x) = f_i(x)$0

which holds for elliptic problems and more general collocation settings. The framework also accommodates parametric PDEs via slice-wise well-posedness over a parameter domain.

The selection rule analyzed is the PDE-$L_i u(x) = f_i(x)$1-greedy criterion, which maximizes

$L_i u(x) = f_i(x)$2

interpolating between pure $L_i u(x) = f_i(x)$3-greedy ($L_i u(x) = f_i(x)$4, target-independent) and $L_i u(x) = f_i(x)$5-greedy ($L_i u(x) = f_i(x)$6, fully target-adaptive). A useful structural lemma shows that any such greedy iteration selects linearly independent functionals automatically: if $L_i u(x) = f_i(x)$7 then $L_i u(x) = f_i(x)$8, and if all $L_i u(x) = f_i(x)$9 vanish the selected functionals span all of $L_i$0. This guarantees an invertible collocation system without additional point-separation assumptions.

Entropy number estimates

The technical core is a bound on dyadic entropy numbers of the absolute convex hull of the set $L_i$1, where $L_i$2. The argument proceeds in two steps.

First, a general lemma combines entropy estimates across the $L_i$3 operator families: if $L_i$4 for each $L_i$5, then additivity and monotonicity of entropy numbers give

$L_i$6

with the budget $L_i$7 allocated to each summand. The authors note this allocation is optimized only asymptotically; uneven distributions could improve the constant, and different regimes may hold for moderate $L_i$8 depending on the $L_i$9 and $$\log(n)^{\frac{2(\tau - \bar m) - \bar d}{2\bar d \max(1,\beta)}}$$0.

Second, each individual $$\log(n)^{\frac{2(\tau - \bar m) - \bar d}{2\bar d \max(1,\beta)}}$$1 is handled via the Siegel–van Kempen-type theorem on smoothly parametrized sets: since the map $$\log(n)^{\frac{2(\tau - \bar m) - \bar d}{2\bar d \max(1,\beta)}}$$2 is Lipschitz of smoothness $$\log(n)^{\frac{2(\tau - \bar m) - \bar d}{2\bar d \max(1,\beta)}}$$3 as a $$\log(n)^{\frac{2(\tau - \bar m) - \bar d}{2\bar d \max(1,\beta)}}$$4-valued function — established through the Hölder–Sobolev embedding on manifolds and the operator boundedness — one obtains

$$\log(n)^{\frac{2(\tau - \bar m) - \bar d}{2\bar d \max(1,\beta)}}$$5

Combining both steps yields $$\log(n)^{\frac{2(\tau - \bar m) - \bar d}{2\bar d \max(1,\beta)}}$$6, where $$\log(n)^{\frac{2(\tau - \bar m) - \bar d}{2\bar d \max(1,\beta)}}$$7 attains the minimum exponent among the operator families. Generic bounded domains are covered by embedding them into a ball and extending the kernel, which is possible for commonly used kernels.

Main convergence result

The bridge from entropy numbers to greedy rates follows the Li–Santin–Wenzel simplex-volume technique: the geometric mean of incremental power function values satisfies

$$\log(n)^{\frac{2(\tau - \bar m) - \bar d}{2\bar d \max(1,\beta)}}$$8

Substituting the entropy bound and following the structure of Theorem 5.1 in [Wenzel2025] yields the main theorem: for any $$\log(n)^{\frac{2(\tau - \bar m) - \bar d}{2\bar d \max(1,\beta)}}$$9,

$k$0

with $k$1. This coincides with the earlier rate up to constants but without the logarithmic factor, confirming that the log term in prior analyses was suboptimal.

For $k$2 the rate simplifies to $k$3: at $k$4 ($k$5-greedy) this matches the quasi-uniform-point rate, while every increase in adaptivity contributes a dimension- and smoothness-independent gain of $k$6, maximal at $k$7 ($k$8-greedy). This is precisely the feature that makes these schemes attractive for high-dimensional PDE approximation, and removing the logarithmic penalty strengthens that advantage. The result also extends to piecewise-smooth manifolds $k$9 by splitting operators per piece — though this inflates $\Omega \subset \mathbb{R}^d$0 through a larger effective $\Omega \subset \mathbb{R}^d$1, a cost not present in the earlier estimates, which is a mild trade-off worth noting.

Limitations and open questions

The analysis inherits several assumptions. The stability estimate must hold uniformly for the problem class; it is guaranteed by maximum principles for elliptic problems but must be verified separately otherwise. The entropy bound requires smooth parametrization of the representer sets, hence smooth manifolds or extendable kernels, and the constant $\Omega \subset \mathbb{R}^d$2 grows with $\Omega \subset \mathbb{R}^d$3, degrading practical relevance when many operator pieces are needed. The result is stated for $\Omega \subset \mathbb{R}^d$4 only, since the $\Omega \subset \mathbb{R}^d$5-greedy limit case was already exhaustively treated in [Wenzel2025]. Finally, the paper is purely theoretical: no new experiments are provided, and whether the refined constant-level behavior manifests in practice remains to be confirmed numerically. An open question is whether finer allocations of the entropy budget across the $\Omega \subset \mathbb{R}^d$6 families, or non-asymptotic regime analysis, can further tighten the constants.

Conclusion

The paper delivers a clean refinement of the convergence theory for PDE-$\Omega \subset \mathbb{R}^d$7-greedy generalized interpolation with Sobolev kernels: by replacing covering-number arguments with metric entropy estimates of smoothly parametrized convex hulls, the previously observed logarithmic deterioration is shown to be removable. The resulting rates preserve the optimal quasi-uniform baseline at $\Omega \subset \mathbb{R}^d$8 while retaining the full dimension- and smoothness-independent adaptive gain for $\Omega \subset \mathbb{R}^d$9, now without penalty. Given the generality of the continuity and stability assumptions used, the same argument should apply beyond PDE collocation to broader classes of generalized interpolation problems.