论文标题
使用笛卡尔网格上的小波在一般有限域上的有效函数近似
Efficient function approximation on general bounded domains using wavelets on a cartesian grid
论文作者
论文摘要
傅立叶扩展是一种近似方法，可减轻傅立叶系列的周期性要求，并在近似功能时避免吉布斯现象。我们描述了使用HyperCube上的常规小波底座的类似扩展方法，以近似该立方体的子集上的功能。这些子集可能具有一般形状。这种构建本质上与冗余有关，这导致严重的不良条件，但是最近的理论表明，使用正则化和过度采样可以实现高精度和数值稳定性。正则化最小二乘求解器，例如截断的奇异值分解，这些分解适用于解决所得不良条件和瘦的线性系统通常具有立方计算成本。我们比较了几种对这种复杂性有所改善的算法。这些改进受益于离散小波变换的稀疏性和结构。我们提出了一种方法，该方法需要1-D和$ \ Mathcal O（N^{3（d-1）/d}）$中的$ \ MATHCAL O（n）$ operations in $ d $ -d，$ d，$ d> 1 $。我们通过实验表明，直接稀疏的QR求解器似乎更及时，但产生了更大的扩展系数。
Fourier extension is an approximation method that alleviates the periodicity requirements of Fourier series and avoids the Gibbs phenomenon when approximating functions. We describe a similar extension approach using regular wavelet bases on a hypercube to approximate functions on subsets of that cube. These subsets may have a general shape. This construction is inherently associated with redundancy which leads to severe ill-conditioning, but recent theory shows that nevertheless high accuracy and numerical stability can be achieved using regularization and oversampling. Regularized least squares solvers, such as the truncated singular value decomposition, that are suited to solve the resulting ill-conditioned and skinny linear system generally have cubic computational cost. We compare several algorithms that improve on this complexity. The improvements benefit from the sparsity in and the structure of the discrete wavelet transform. We present a method that requires $\mathcal O(N)$ operations in 1-D and $\mathcal O(N^{3(d-1)/d})$ in $d$-D, $d>1$. We experimentally show that direct sparse QR solvers appear to be more time-efficient, but yield larger expansion coefficients.
