This work proposes and analyzes a compressed sensing approach to polynomial approximation of complex-valued functions in high dimensions. In this context, the target function is often smooth and characterized by a rapidly decaying orthonormal expansion, whose most important terms are captured by a lower (or downward closed) set. Motivated by this fact, we present an innovative weighted -minimization procedure with a precise choice of weights for imposing the downward closed preference. Theoretical results reveal that our computational approaches possess a provably reduced sample complexity compared to existing compressed sensing techniques presented in the literature. In addition, the recovery of the corresponding best approximation using these methods is established through an improved bound for the restricted isometry property. Our analysis represents an extension of the approach for Hadamard matrices by J. Bourgain [An improved estimate in the restricted isometry problem, Lecture Notes in Math., vol. 216, Springer, 2014, pp. 65-70] to the general bounded orthonormal systems, quantifies the dependence of sample complexity on the successful recovery probability, and provides an estimate on the number of measurements with explicit constants. Numerical examples are provided to support the theoretical results and demonstrate the computational efficiency of the novel weighted -minimization strategy.

The most effective simulations of the multiphysics coupling of groundwater to surface water must involve employing the best groundwater codes and the best surface water codes. Partitioned methods, which solve the coupled problem by successively solving the subphysics problems, have recently been studied for the Stokes--Darcy coupling with convergence established over bounded time intervals (with constants growing exponentially in $t$). This report analyzes and tests two such partitioned (noniterative, domain decomposition) methods for the fully evolutionary Stokes--Darcy problem. Under a modest time step restriction of the form $\Delta t\leq C$, where $C=C$ (physical parameters), we prove unconditional, long time (over $0\leq t<\infty $) stability of both partitioned methods. From this we derive an optimal error estimate that is uniform in time over $0\leq t<\infty$.