Adaptive Interpolation: Methods & Applications

• Adaptive interpolation is a method that adjusts parameters based on local data, error estimates, and problem structure to optimize performance.
• It employs techniques like local adaptivity, mesh refinement, and adaptive basis construction to handle non-uniform sampling and discontinuities effectively.
• Applications span geoscience imputation, image super-resolution, and neural operator learning, showcasing improved accuracy and resource efficiency.

Adaptive interpolation comprises a wide spectrum of strategies that adjust interpolation procedures or parameters based on local data, estimated error, or problem-specific structure to optimize accuracy, stability, or resource usage. Adaptive interpolation is critical in many scientific and engineering applications where global uniformity or equispacing is either infeasible or inefficient, as in non-uniform data sampling, discontinuous phenomena, high-dimensional models, or when computational resources must be concentrated on difficult regions. Adaptive approaches manifest in selection or refinement of grid points, local adjustment of interpolant parameters, meta-learned data-augmentation strategies, adaptive basis construction, and robust rational schemes for discontinuous or complex functions.

1. Local Adaptivity: Data-Dependent Weighting and Parameter Selection

Adaptive interpolation frequently arises in the context of spatial or pointwise local error minimization. For example, in scattered data interpolation or missing-value imputation in geoscience, local adaptivity is essential due to variable point densities and anisotropies. Adaptive Radial Basis Function (RBF) interpolation operates by, for each query location, selecting a local neighborhood of points and adapting the RBF “shape factor” based on estimated local sample density, rather than employing a global, fixed parameter. Specifically, after determining the k-nearest neighborhood, the algorithm computes local-to-global density ratios, maps this to a normalized density indicator, and uses it to select from a discrete set of basis shape parameters. This process yields significantly lower root-mean-squared error compared to global RBFs or non-adaptive Inverse Distance Weighting, albeit at increased computational cost due to per-query system solves (Gao et al., 2019).

A similar principle underlies gradient-based adaptive interpolation for super-resolution image reconstruction, where pixel interpolation coefficients are adaptively weighted not only by Euclidean distance but also by the magnitude of local gradients, penalizing contributions from strong edge locations. This yields sharper restoration and superior robustness to registration errors – a critical advantage for real-time or noise-prone imaging modalities (0903.3995).

2. Adaptive Mesh and Knot Refinement

Adaptivity in mesh or knot selection is central to modern high-accuracy interpolation, sparse surrogates, and uncertainty quantification. The “AutoKnots” algorithm exemplifies this, constructing spline interpolants for expensive functions by automatically selecting both the number and placement of knots so as to respect user-specified pointwise and integral error tolerances. Each step refines intervals where midpoint or low-order quadrature errors exceed a threshold, directly adding new knots only where necessary. This drastically reduces the number of function evaluations compared to uniform meshing, without the need for manual knot configuration. A secondary “refine” loop targets large, suspicious intervals, mitigating undersampling in plateau regions (Vitenti et al., 2024).

In computational chemistry, adaptivity is crucial to interpolating high-dimensional potential energy surfaces. Adaptive partition-of-unity RBF interpolants employ hierarchical local error estimation, introducing new nodes only in regions where the error model (fit to already computed values) predicts potentially unacceptably high uncertainty. Each iteration further refines the local mesh hierarchy, while the partition-of-unity assembly and local patching ensure computational tractability and memory stability compared to monolithic global interpolators (Kowalewski et al., 2016).

3. Adaptive Basis and Expansion Construction

Dimension-adaptive and anisotropy-adaptive algorithms efficiently handle high-dimensional problems by refining the interpolation basis nonuniformly according to error indicators or estimated functional anisotropy.

For polynomial chaos expansions and stochastic collocation, adaptive Leja-based sequences combined with hierarchical Newton bases yield interpolants that naturally expand along the most significant directions, driven by the decay of hierarchical surpluses (coefficients) across a downward-closed multi-index set. This process exploits the natural structure and nesting of Leja sequences, ensuring only “useful” basis functions are added, while mapping basis polynomials uniquely to collocation nodes to maintain interpolatory properties (Loukrezis et al., 2019).

In the field of periodic functions, dimensionally adaptive sparse trigonometric interpolation estimates the anisotropy of the target function directly via regression on current Fourier coefficients, using the results to select the next set of degrees (and thus nodes) in each coordinate direction. The framework adapts to the inherent smoothness of each variable, achieving quasi-optimal convergence matching the best hyperbolic cross for the given function class (Morrow et al., 2019).

4. Adaptive Rational, Nonlinear, and Discontinuity-Handling Schemes

Adaptive interpolation is indispensable for functions with singularities, jumps, or local irregularities. Adaptive rational interpolation methods generalize the WENO (Weighted Essentially Non-Oscillatory) paradigm by constructing explicit, locally optimized weighting schemes that maximally exploit jump-free stencils. At each point, these methods evaluate smoothness indicators (typically large absolute differences to detect jumps), and assign weights so that the interpolant “locks” onto the nearest smooth region, guaranteeing the largest possible local polynomial degree and order. As a result, close to discontinuities, accuracy degrades only to the largest possible order permitted by the smooth subsystem, rather than defaulting to a fixed minimal order—a limitation typical for standard WENO (Arandiga et al., 2020).

Adaptive Thiele rational interpolation similarly avoids ill-conditioning and breakdowns by greedily selecting the node ordering in a continued-fraction interpolation so as to prevent vanishing denominators, using current interpolation error residuals as a guide. This algorithmic adaptivity ensures robust rational approximation—even for problematic node configurations where classical implementations fail entirely (Celis, 2023).

5. Meta-Learning and Data-Driven Adaptive Strategies

In deep learning, adaptive interpolation emerges as a strategy to counteract the deficiencies of stochastic data-mixing methods such as MixUp. Rather than sampling the mixing parameter λ from a fixed distribution, the MetaMixUp approach parameterizes λ using a learnable policy network, with its parameters optimized by meta-gradients computed via a small held-out validation set. This bi-level meta-learning loop ensures that the mixing process is tailored to the data distribution; it steers λ away from underfitting (manifold intrusion) and over-smoothing, hence improving generalization in both supervised and semi-supervised settings. Empirically, the learned λ-distribution exhibits nontrivial class-pair dependence and departs significantly from sampling-based baselines (Mai et al., 2019).

6. Adaptive Interpolation in Neural and Operator Learning Applications

Nested within neural architectures, adaptive interpolation manifests as data-driven or logic-driven parameterization of kernels or receptive fields. Notably, video frame interpolation approaches such as adaptive separable convolution or adaptive Fourier neural operators deploy per-pixel, learnable kernels that adapt spatially to content and motion, balancing the support size for memory constraints. For example, by factorizing 2D convolution kernels into outer products of 1D kernels, adaptive separable convolution methods train deep nets to predict tailored motion-aware resampling operators, with advantageous scaling and capacity for arbitrary motion (Niklaus et al., 2017). Recent operator-based methods further exploit global spectral adaptivity; adaptive Fourier neural interpolation operators combine local convolutional adaptation with learned Fourier-domain token mixing, allowing for scale-invariant, global-context interpolation across arbitrary input resolutions (Viswanath et al., 2022).

7. Theoretical Guarantees, Error Control, and Implementation

Adaptive interpolation strategies are often grounded in error models or a posteriori estimates. The tightness of error control is dictated by the specific algorithmic framework: for local polynomial or Shepard-type interpolation, explicit Lebesgue constants, fill distances, and smoothness indicators govern selection criteria for local degree or radius expansions (Cavoretto et al., 2023). In model reduction, adaptive interpolatory projection frameworks employ efficient a posteriori error estimators—interpolated via RBF surrogates over the parameter domain—to accelerate greedy selection of interpolation points and sharply reduce offline computational requirements (Chellappa et al., 2020).

A summary table illustrating representative adaptive interpolation strategies and their application domains:

Method	Principal Adaptivity Mechanism	Typical Application Domain
Adaptive RBF	Local shape parameter via density ratio	Missing value imputation, geo-informatics
AutoKnots spline	Knot refinement via error threshold	Cosmological functional modeling
Leja / Newton poly PCE	Surplus-driven basis/index refinement	Polynomial chaos, uncertainty quant.
Adaptive rational/WENO	Jump-adapted local smoothness weights	Discontinuous function approximation
MetaMixUp	Meta-learned mixing via validation loss	Deep learning regularization, SSL
Adaptive separable conv	Per-pixel kernel estimation	Video/image interpolation

Adaptive interpolation is a broad, highly active area characterized by tight coupling of interpolation parameters, basis selection, or node placement to data-driven, model-driven, or error-driven heuristics or optimization. It has become indispensable across domains where function smoothness, sampling density, or data complexity is heterogeneous, or where resource efficiency and robustness to irregularities are paramount.