Showing papers by "Daniel Potts published in 2020"

Transformed rank-1 lattices for high-dimensional approximation

Abstract: This paper describes an extension of Fourier approximation methods for multivariate functions defined on the torus $\mathbb{T}^d$ to functions in a weighted Hilbert space $L_{2}(\mathbb{R}^d, \omega)$ via a multivariate change of variables $\psi:\left(-\frac{1}{2},\frac{1}{2}\right)^d\to\mathbb{R}^d$. We establish sufficient conditions on $\psi$ and $\omega$ such that the composition of a function in such a weighted Hilbert space with $\psi$ yields a function in the Sobolev space $H_{\mathrm{mix}}^{m}(\mathbb{T}^d)$ of functions on the torus with mixed smoothness of natural order $m \in \mathbb{N}_{0}$. In this approach we adapt algorithms for the evaluation and reconstruction of multivariate trigonometric polynomials on the torus $\mathbb{T}^d$ based on single and multiple reconstructing rank-$1$ lattices. Since in applications it may be difficult to choose a related function space, we make use of dimension incremental construction methods for sparse frequency sets. Various numerical tests confirm obtained theoretical results for the transformed methods.

Topography influences species-specific patterns of seasonal primary productivity in a semiarid montane forest.

Abstract: Semiarid forests in the southwestern USA are generally restricted to mountain regions where complex terrain adds to the challenge of characterizing stand productivity. Among the heterogeneous features of these ecosystems, topography represents an important control on system-level processes including snow accumulation and melt. This basic relationship between geology and hydrology affects radiation and water balances within the forests, with implications for canopy structure and function across a range of spatial scales. In this study, we quantify the effect of topographic aspect on primary productivity by observing the response of two codominant native tree species to seasonal changes in the timing and magnitude of energy and water inputs throughout a montane headwater catchment in Arizona, USA. On average, soil moisture on north-facing aspects remained higher during the spring and early summer compared with south-facing aspects. Repeated measurements of net carbon assimilation (Anet) showed that Pinus ponderosa C. Lawson was sensitive to this difference, while Pseudotsuga menziesii (Mirb.) Franco was not. Irrespective of aspect, we observed seasonally divergent patterns at the species level where P. ponderosa maintained significantly greater Anet into the fall despite more efficient water use by P. menziesii individuals during that time. As a result, this study at the southern extent of the geographical P. menziesii distribution suggests that this species could increase water-use efficiency as a response to future warming and/or drying, but at lower rates of production relative to the more drought-adapted P. ponderosa. At the sub-landscape scale, opposing aspects served as a mesocosm of current versus anticipated climate conditions. In this way, these results also constrain the potential for changing carbon sequestration patterns from Pinus-dominated landscapes due to forecasted changes in seasonal moisture availability.

Continuous window functions for NFFT

Abstract: In this paper, we study the error behavior of the nonequispaced fast Fourier transform (NFFT). This approximate algorithm is mainly based on the convenient choice of a compactly supported window function. Here we consider the continuous Kaiser--Bessel, continuous $\exp$-type, $\sinh$-type, and continuous $\cosh$-type window functions with the same support and same shape parameter. We present novel explicit error estimates for NFFT with such a window function and derive rules for the optimal choice of the parameters involved in NFFT. The error constant of a window function depends mainly on the oversampling factor and the truncation parameter. For the considered continuous window functions, the error constants have an exponential decay with respect to the truncation parameter.

Grouped Transformations in High-Dimensional Explainable ANOVA Approximation.

Abstract: Many applications are based on the use of efficient Fourier algorithms such as the fast Fourier transform and the nonequispaced fast Fourier transform. In a high-dimensional setting it is also already possible to deal with special sampling sets such as sparse grids or rank-1 lattices. In this paper we propose fast algorithms for high-dimensional scattered data points with corresponding frequency sets that consist of groups along the dimensions in the frequency domain. From there we propose a fast matrix-vector multiplication, the grouped Fourier transform, that finds theoretical foundation in the context of the analysis of variance (ANOVA) decomposition where there is a one-to-one correspondence from the ANOVA terms to our proposed groups. An application can be found in function approximation for high-dimensional functions where the number of the variable interactions is limited. We tested two different approximation approaches: Classical Tikhonov-regularization, namely, regularized least squares, and the technique of group lasso, which promotes sparsity in the groups. As for the latter, there are no explicit solution formulas which is why we applied the fast iterative shrinking-thresholding algorithm to obtain the minimizer. Numerical experiments in under-, overdetermined, and noisy settings indicate the applicability of our algorithms.

A sparse FFT approach for ODE with random coefficients

Abstract: The paper presents a general strategy to solve ordinary differential equations (ODE), where some coefficient depend on the spatial variable and on additional random variables. The approach is based on the application of a recently developed dimension-incremental sparse fast Fourier transform. Since such algorithms require periodic signals, we discuss periodization strategies and associated necessary deperiodization modifications within the occurring solution steps. The computed approximate solutions of the ODE depend on the spatial variable and on the random variables as well. Certainly, one of the crucial challenges of the high-dimensional approximation process is to rate the influence of each variable on the solution as well as the determination of the relations and couplings within the set of variables. The suggested approach meets these challenges in a full automatic manner with reasonable computational costs, i.e., in contrast to already existing approaches, one does not need to seriously restrict the used set of ansatz functions in advance.