Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required to achieve quasi-optimal approximation in a given polynomial subspace scales, up to a logarithmic factor, linearly in the dimension of this space. However, in many applications, the computation of samples includes a numerical discretization error. Thus, obtaining polynomial approximations with a single level method can become prohibitively expensive, as it requires a sufficiently large number of samples, each computed with a sufficiently small discretization error. As a solution to this problem, we propose a multilevel method that utilizes samples computed with different accuracies and is able to match the accuracy of single-level approximations with reduced computational cost. We derive complexity bounds under certain assumptions about polynomial approximability and sample work. Furthermore, we propose an adaptive algorithm for situations where such assumptions cannot be verified a priori. Finally, we provide an efficient algorithm for the sampling from optimal distributions and an analysis of computationally favorable alternative distributions. Numerical experiments underscore the practical applicability of our method.
The export option will allow you to export the current search results of the entered query to a file. Different
formats are available for download. To export the items, click on the button corresponding with the preferred download format.
By default, clicking on the export buttons will result in a download of the allowed maximum amount of items.
For anonymous users the allowed maximum amount is 50 search results.
To select a subset of the search results, click "Selective Export" button and make a selection of the items you want to export.
The amount of items that can be exported at once is similarly restricted as the full export.
After making a selection, click one of the export format buttons. The amount of items that will be exported is indicated in the bubble next to export format.