Polynomial Reconstruction & Sampling (PRAS)

• Polynomial Reconstruction and Sampling (PRAS) is a framework comprising deterministic, randomized, and AI-aided techniques for recovering polynomial signals from sampled data with high accuracy.
• It employs structured sampling methods such as poised lattice generation, QR pivoting, and group-theoretic algorithms to ensure unisolvent and well-conditioned interpolation matrices.
• The approach integrates Monte Carlo strategies and error-resilient algorithms, including CRT-based methods, to improve robustness and efficiency in applications across numerical analysis, signal processing, and physics.

Polynomial Reconstruction and Sampling (PRAS) encompasses a spectrum of methodologies and theoretical constructions enabling the stable recovery or approximation of polynomial (or polynomial-parameterized) signals, functions, or distributions from sampled data. The scope includes deterministic constructive schemes, randomized and Monte Carlo strategies, error-resilient algorithms, and AI-driven search techniques, applied across domains such as numerical analysis, signal processing, computational physics, and statistical modeling.

1. Deterministic and Structured Sampling for Polynomial Approximation

Deterministic strategies for polynomial reconstruction leverage the algebraic and geometric structure of the underlying function spaces, enabling stable sampling and reconstruction with minimal redundancy.

For multivariate polynomial interpolation on Cartesian grids, the poised lattice generation (PLG) problem is a central challenge: selecting a subset $\mathcal{T} \subset K \subset \mathbb{Z}^D$ such that interpolation in $\Pi_n$ (space of degree-$n$ polynomials in $D$ variables, with $\dim \Pi_n = \binom{n+D}{D}$) is unisolvent. The AI-aided PLG algorithm (Zhang et al., 7 Aug 2024) solves this by restricting to triangular lattices—index sets defined by $\sum_{i=1}^D k_i \leq n$—and utilizes group-theoretic isomorphisms between triangular lattices and $D$-permutations to efficiently search the feasible space. This deep algebraic structure ensures that the interpolation matrix has a determinantal factorization guaranteeing nonsingularity. The method organizes the AI search space via depth-first backtracking and staged "test sets," achieving high computational efficiency even on large grids.

In polynomial least-squares approximation, deterministic subsampling of tensor grids via QR with column pivoting (effectively subsampled quadratures) achieves near-optimal conditioning (Seshadri et al., 2016). By pruning columns corresponding to high total order basis functions—often the source of numerical instability—one can balance approximation accuracy and stability. For function recovery in reproducing kernel Hilbert spaces (RKHS), two-step subsampling schemes, starting from structured quadrature points (such as rank-1 lattices on the torus) and proceeding with carefully weighted random and deterministic reductions, enable dimension-independent error decay and efficient FFT-based computation (Bartel et al., 2022).

2. Randomized and Monte Carlo Polynomial Reconstruction

Randomized sampling schemes, including Monte Carlo and quasi-Monte Carlo methods, are prominent for capturing distributions or mixing operations that are analytically intractable. In radiative-transfer modeling, the Polynomial Reconstruction and Sampling (PRAS) Monte Carlo method reconstructs the cumulative distribution function (CDF) of opacity by fitting polynomials or B-splines to sorted high-resolution cross-section data, then sampling these functionals to generate opacity mixtures for correlated-k methods (Lee, 9 Aug 2025). The method controls accuracy via the degree of the polynomial fit, the density of spline knots (especially to resolve steep gradients), and the sample count $n_s$; convergence to the exact randomly overlapped distribution is achieved as $n_s \to \infty$ and with perfect fits. Quasi-Monte Carlo improvements, such as Latin Hypercube Sampling, further enhance accuracy per sample.

Notably, for deterministic sampling of sparse trigonometric polynomials, algebraically constructed node sets exploiting Weil's exponential sum bounds enable robust recovery with statistical RIP-like properties, supporting precise reconstruction via greedy algorithms like orthogonal matching pursuit (Xu, 2010).

3. Error-Resilient and Robust Polynomial Reconstruction

Robustness to errors is critical in practical polynomial reconstruction scenarios. The multi-level Chinese Remainder Theorem (CRT) for polynomials allows polynomials to be reconstructed even in presence of residue errors of bounded degree (not just bounded number), trading off allowable polynomial degree (dynamic range) with error tolerance per the decomposition $m_i(x) = m(x)\Gamma_i(x)$ and recursive subresultants $\sigma_i(x)$ (Xiao et al., 2017). The closed-form algorithm ensures that only the least significant coefficients (corresponding to the error degree bound $\tau$) may be erroneous, and is computationally efficient for DSP and coding applications.

For rational function solutions to polynomial linear systems corrupted by evaluation errors, algorithms inspired by Interleaved Reed-Solomon code decoding improve the error threshold beyond classical bounds, exploiting simultaneous polynomial reconstruction and the error locator polynomial framework (Guerrini et al., 2019). This methodology reduces the number of necessary evaluation points and leverages probabilistic guarantees on recovery, directly impacting symbolic-numeric computation reliability.

In infinite or overcomplete systems, such as frame or redundant sampling schemes, exact reconstruction in the presence of erasures is addressed by bridging (solving for combinations of non-erased coefficients to replace missing data) or explicit inversion of partial reconstruction operators, provided minimal redundancy (Larson et al., 2014). The analysis of nilpotency and redundancy properties further quantifies reconstructibility.

4. AI-Aided and Algebraic Methods for Lattice Generation and High-Order Discretization

Advances in artificial intelligence have been leveraged to navigate exponentially sized solution spaces in polynomial reconstruction. The AI-aided PLG algorithm (Zhang et al., 7 Aug 2024) organizes the search over feasible triangulated stencils via group-theoretic insights, notably establishing an isomorphism between $D$-permutations and triangular lattices. The search tree is dynamically pruned using test sets tied to algebraic structure, ensuring only locally poised candidate stencils are considered at each increment. This yields minimal, unisolvent node sets for interpolation, foundational to constructing high-order finite difference (PLG-FD) discretizations suitable even for irregular geometries.

This methodology has broad applications in PDE discretization for arbitrary domains, solving longstanding limitations of legacy finite difference schemes on non-tensor product regions. The triangulated stencils deliver fourth- and sixth-order accuracy (as shown, e.g., for elliptic or mixed-derivative equations) with computational overhead amortized to negligible levels for large-scale systems.

5. Stability, Conditioning, and Practicality

Stable polynomial reconstruction mandates not only unisolvency of the interpolation matrix but also favorable conditioning. Factorizations of sample matrices leveraging the geometry of node placement (as in triangular lattices), or cross-correlation structures (in shift-invariant generalized sampling (0806.2084), or B-spline RKHS construction (Devaraj et al., 2019)), are central to ensuring both numerical stability and error control.

The deterministic selection of samples (via QR pivots or via algebraic node construction) facilitates reproducibility, avoids the run-to-run variability of random designs, and often matches or surpasses the performance of purely randomized methods. In contrast, randomized approaches excel for mixing distributions, such as the convolution of opacity CDFs, leveraging the central limit properties to converge rapidly as sample count increases.

6. Implications and Applications

PRAS methodologies have found adoption in diverse applications:

• High-order finite difference and reconstruction schemes for PDEs on irregular and complex domains, critical in computational fluid dynamics, porous media, and advection-dominated problems.
• Signal processing tasks, including edge detection, image reconstruction, and FRI (finite rate-of-innovation) modeling, particularly for binary images with algebraic boundaries (Fatemi et al., 2015).
• Radiative-transfer and atmospheric modeling, enabling robust and high-precision retrievals from spectroscopic data (e.g., for JWST exoplanet studies (Lee, 9 Aug 2025)).
• Coding theory, fault-tolerant digital algebra, and symbolic computation, where error-resilient reconstruction is required under hardware and transmission constraints.
• Sparse recovery and compressive sensing in shift-invariant and generalized sampling settings, broadening the class of physical signals recoverable from limited data (Vlašić et al., 2020).

7. Limitations, Perspectives, and Future Directions

While advances in PRAS have addressed many open challenges, certain limitations remain:

• The robustness of error correction often depends on detailed algebraic structure (e.g., moduli relationships in CRT-based methods, or frame redundancy in erasure recovery).
• Randomized and MC/Quasi-MC sampling, while powerful for high-dimensional or distributional convolution, may require careful tuning of sample counts and fitting accuracy for application-critical domains.
• Deterministic approaches based solely on node structure may fail to adapt to local function variations unless hybridized with adaptive or function-aware refinement (Seshadri et al., 2016).
• The polynomial reconstruction problem for spectral invariants (such as the characteristic polynomial of a hypergraph) is intractable from local (deck) data in the general case (Cooper et al., 2023), disproving extensions of reconstructibility conjectures from graphs to hypergraphs.

Ongoing and future developments include:

• Incorporating machine learning (e.g., for optimal knot placement or nonparametric CDF fitting in MC-based PRAS).
• Generalizing algebraic constructions to non-polynomial settings, e.g., within RKHS or in semi-algebraic frameworks.
• Integrating AI-driven search for optimized experimental design in high-dimensional parametric uncertainty quantification.

The breadth and rigor of PRAS research continue to enable stable, efficient, and accurate recovery of information-rich polynomial models from sampled data, spanning domains from numerical analysis to atmospheric sciences and computational algebra.