Sparse polynomial approximation has become indispensable for approximating smooth, high- or infinite-dimensional functions from limited samples. This is a key task in computational science and engineering, e.g., surrogate modelling in uncertainty quantification where the function is the solution map of a parametric or stochastic differential equation (DE). Yet, sparse polynomial approximation lacks a complete theory. On the one hand, there is a well-developed theory of best $s$-term polynomial approximation, which asserts exponential or algebraic rates of convergence for holomorphic functions. On the other, there are increasingly mature methods such as (weighted) $\ell^1$-minimization for computing such approximations. While the sample complexity of these methods has been analyzed with compressed sensing, whether they achieve best $s$-term approximation rates is not fully understood. Furthermore, these methods are not algorithms per se, as they involve exact minimizers of nonlinear optimization problems. This paper closes these gaps. Specifically, we consider the following question: are there robust, efficient algorithms for computing approximations to finite- or infinite-dimensional, holomorphic and Hilbert-valued functions from limited samples that achieve best $s$-term rates? We answer this affirmatively by introducing algorithms and theoretical guarantees that assert exponential or algebraic rates of convergence, along with robustness to sampling, algorithmic, and physical discretization errors. We tackle both scalar- and Hilbert-valued functions, this being key to parametric or stochastic DEs. Our results involve significant developments of existing techniques, including a novel restarted primal-dual iteration for solving weighted $\ell^1$-minimization problems in Hilbert spaces. Our theory is supplemented by numerical experiments demonstrating the efficacy of these algorithms.

翻译：稀疏多项式逼近已成为从有限样本逼近光滑、高维或无限维函数不可或缺的工具。这是计算科学与工程中的关键任务，例如不确定性量化中基于参数或随机微分方程解映射的代理模型。然而，稀疏多项式逼近仍缺乏完整的理论体系。一方面，最佳s项多项式逼近理论已发展成熟，断言对全纯函数具有指数或代数收敛速率。另一方面，（加权）ℓ¹极小化等日趋成熟的方法被用于计算此类逼近。尽管这些方法的样本复杂度已通过压缩感知理论得到分析，但其能否实现最佳s项逼近速率尚未完全明晰。此外，这些方法本身并非严格意义上的算法，因为它们涉及非线性优化问题精确极小值的求解。本文填补了这些空白。具体而言，我们考虑以下问题：是否存在鲁棒且高效的计算算法，能够从有限样本逼近有限维或无限维、全纯且Hilbert值函数，并达到最佳s项逼近速率？我们通过引入算法与理论保证给出肯定答案，断言指数或代数收敛速率，同时对采样、算法及物理离散误差具有鲁棒性。我们同时处理标量值与Hilbert值函数——这对参数或随机微分方程至关重要。我们的成果包含对现有技术的重大发展，包括一种新颖的用于求解Hilbert空间中加权ℓ¹极小化问题的重启原始-对偶迭代方法。数值实验验证了这些算法的有效性，为理论提供了补充。