arXiv subjects

Daniel Potts

Publications and source records attributed to Daniel Potts.

At least 37 records · Page 2Linked to original sources

arXiv2022

The uniform sparse FFT with application to PDEs with random coefficients

We develop the uniform sparse Fast Fourier Transform (usFFT), an efficient, non-intrusive, adaptive algorithm for the solution of elliptic partial differential equations with random coefficients. The algorithm is an adaption of the sparse Fast Fourier Transform (sFFT), a dimension-incremental algorithm, which tries to detect the most important frequencies in a given search domain and therefore adaptively generates a suitable Fourier basis corresponding to the approximately largest Fourier coefficients of the function. The usFFT does this w.r.t. the stochastic domain of the PDE simultaneously for multiple fixed spatial nodes, e.g., nodes of a finite element mesh. The key idea of joining the detected frequency sets in each dimension increment results in a Fourier approximation space, which fits uniformly for all these spatial nodes. This strategy allows for a faster and more efficient computation due to a significantly smaller amount of samples needed, than just using other algorithms, e.g., the sFFT for each spatial node separately. We test the usFFT for different examples using periodic, affine and lognormal random coefficients in the PDE problems.

math.NA↗

arXiv2022

Nonuniform fast Fourier transforms with nonequispaced spatial and frequency data and fast sinc transforms

In this paper we study the nonuniform fast Fourier transform with nonequispaced spatial and frequency data (NNFFT) and the fast sinc transform as its application. The computation of NNFFT is mainly based on the nonuniform fast Fourier transform with nonequispaced spatial nodes and equispaced frequencies (NFFT). The NNFFT employs two compactly supported, continuous window functions. For fixed nonharmonic bandwidth, it is shown that the error of the NNFFT with two sinh-type window functions has an exponential decay with respect to the truncation parameters of the used window functions. As an important application of the NNFFT, we present the fast sinc transform. The error of the fast sinc transform is estimated as well.

math.NA↗

arXiv2021

Grouped Transformations and Regularization in High-Dimensional Explainable ANOVA Approximation

In this paper we propose a tool for high-dimensional approximation based on trigonometric polynomials where we allow only low-dimensional interactions of variables. In a general high-dimensional setting, it is already possible to deal with special sampling sets such as sparse grids or rank-1 lattices. This requires black-box access to the function, i.e., the ability to evaluate it at any point. Here, we focus on scattered data points and grouped frequency index sets along the dimensions. From there we propose a fast matrix-vector multiplication, the grouped Fourier transform, for high-dimensional grouped index sets. Those transformations can be used in the application of the previously introduced method of approximating functions with low superposition dimension based on the analysis of variance (ANOVA) decomposition where there is a one-to-one correspondence from the ANOVA terms to our proposed groups. The method is able to dynamically detected important sets of ANOVA terms in the approximation. In this paper, we consider the involved least-squares problem and add different forms of regularization: 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. Moreover, we discuss the possibility of incorporating smoothness information into the least-squares problem. Numerical experiments in under-, overdetermined, and noisy settings indicate the applicability of our algorithms. While we consider periodic functions, the idea can be directly generalized to non-periodic functions as well.

math.NA↗

arXiv2021

Interpretable transformed ANOVA approximation on the example of the prevention of forest fires

The distribution of data points is a key component in machine learning. In most cases, one uses min-max normalization to obtain nodes in $[0,1]$ or Z-score normalization for standard normal distributed data. In this paper, we apply transformation ideas in order to design a complete orthonormal system in the $\mathrm{L}_2$ space of functions with the standard normal distribution as integration weight. Subsequently, we are able to apply the explainable ANOVA approximation for this basis and use Z-score transformed data in the method. We demonstrate the applicability of this procedure on the well-known forest fires data set from the UCI machine learning repository. The attribute ranking obtained from the ANOVA approximation provides us with crucial information about which variables in the data set are the most important for the detection of fires.

stat.ML↗

arXiv2021

Fast Hyperbolic Wavelet Regression meets ANOVA

We use hyperbolic wavelet regression for the fast reconstruction of high-dimensional functions having only low dimensional variable interactions. Compactly supported periodic Chui-Wang wavelets are used for the tensorized hyperbolic wavelet basis. In a first step we give a self-contained characterization of tensor product Sobolev-Besov spaces on the $d$-torus with arbitrary smoothness in terms of the decay of such wavelet coefficients. In the second part we perform and analyze scattered-data approximation using a hyperbolic cross type truncation of the basis expansion for the associated least squares method. The corresponding system matrix is sparse due to the compact support of the wavelets, which leads to a significant acceleration of the matrix vector multiplication. In case of i.i.d. samples we can even bound the approximation error with high probability by loosing only $\log$-terms that do not depend on $d$ compared to the best approximation. In addition, if the function has low effective dimension (i.e. only interactions of few variables), we qualitatively determine the variable interactions and omit ANOVA terms with low variance in a second step in order to increase the accuracy. This allows us to suggest an adapted model for the approximation. Numerical results show the efficiency of the proposed method.

math.NA↗

arXiv2021

A note on transformed Fourier systems for the approximation of non-periodic signals

A variety of techniques have been developed for the approximation of non-periodic functions. In particular, there are approximation techniques based on rank-$1$ lattices and transformed rank-$1$ lattices, including methods that use sampling sets consisting of Chebyshev- and tent-transformed nodes. We compare these methods with a parameterized transformed Fourier system that yields similar $\ell_2$-approximation errors.

math.NA↗

arXiv2021

Uniform error estimates for nonequispaced fast Fourier transforms

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. So far, various window functions have been used and new window functions have recently been proposed. We present novel error estimates for NFFT with compactly supported, continuous window functions and derive rules for convenient choice from the parameters involved in NFFT. The error constant of a window function depends mainly on the oversampling factor and the truncation parameter.

math.NA↗

arXiv2021

Learning multivariate functions with low-dimensional structures using polynomial bases

In this paper we propose a method for the approximation of high-dimensional functions over finite intervals with respect to complete orthonormal systems of polynomials. An important tool for this is the multivariate classical analysis of variance (ANOVA) decomposition. For functions with a low-dimensional structure, i.e., a low superposition dimension, we are able to achieve a reconstruction from scattered data and simultaneously understand relationships between different variables.

math.NA↗

arXiv2021

Interpretable Approximation of High-Dimensional Data

In this paper we apply the previously introduced approximation method based on the ANOVA (analysis of variance) decomposition and Grouped Transformations to synthetic and real data. The advantage of this method is the interpretability of the approximation, i.e., the ability to rank the importance of the attribute interactions or the variable couplings. Moreover, we are able to generate an attribute ranking to identify unimportant variables and reduce the dimensionality of the problem. We compare the method to other approaches on publicly available benchmark datasets.

stat.ML↗

arXiv2021

Approximation of high-dimensional periodic functions with Fourier-based methods

In this paper we propose an approximation method for high-dimensional $1$-periodic functions based on the multivariate ANOVA decomposition. We provide an analysis on the classical ANOVA decomposition on the torus and prove some important properties such as the inheritance of smoothness for Sobolev type spaces and the weighted Wiener algebra. We exploit special kinds of sparsity in the ANOVA decomposition with the aim to approximate a function in a scattered data or black-box approximation scenario. This method allows us to simultaneously achieve an importance ranking on dimensions and dimension interactions which is referred to as attribute ranking in some applications. In scattered data approximation we rely on a special algorithm based on the non-equispaced fast Fourier transform (or NFFT) for fast multiplication with arising Fourier matrices. For black-box approximation we choose the well-known rank-1 lattices as sampling schemes and show properties of the appearing special lattices.

math.NA↗

arXiv2021

Continuous window functions for NFFT

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.

math.NA↗

arXiv2021

Efficient multivariate approximation on the cube

We combine a periodization strategy for weighted $L_{2}$-integrands with efficient approximation methods in order to approximate multivariate non-periodic functions on the high-dimensional cube $\left[-\frac{1}{2},\frac{1}{2}\right]^{d}$. Our concept allows to determine conditions on the $d$-variate torus-to-cube transformations ${ψ:\left[-\frac{1}{2},\frac{1}{2}\right]^{d}\to\left[-\frac{1}{2},\frac{1}{2}\right]^{d}}$ such that a non-periodic function is transformed into a smooth function in the Sobolev space $\mathcal H^{m}(\mathbb{T}^{d})$ when applying $ψ$. We adapt some $L_{\infty}(\mathbb{T}^{d})$- and $L_{2}(\mathbb{T}^{d})$-approximation error estimates for single rank-$1$ lattice approximation methods and adjust algorithms for the fast evaluation and fast reconstruction of multivariate trigonometric polynomials on the torus in order to apply these methods to the non-periodic setting. We illustrate the theoretical findings by means of numerical tests in up to $d=5$ dimensions.

math.NA↗

arXiv2020

Fast Cross-validation in Harmonic Approximation

Finding a good regularization parameter for Tikhonov regularization problems is a though yet often asked question. One approach is to use leave-one-out cross-validation scores to indicate the goodness of fit. This utilizes only the noisy function values but, on the downside, comes with a high computational cost. In this paper we present a general approach to shift the main computations from the function in question to the node distribution and, making use of FFT and FFT-like algorithms, even reduce this cost tremendously to the cost of the Tikhonov regularization problem itself. We apply this technique in different settings on the torus, the unit interval, and the two-dimensional sphere. Given that the sampling points satisfy a quadrature rule our algorithm computes the cross-validations scores in floating-point precision. In the cases of arbitrarily scattered nodes we propose an approximating algorithm with the same complexity. Numerical experiments indicate the applicability of our algorithms.

math.NA↗

arXiv2019

Transformed rank-1 lattices for high-dimensional approximation

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 $L_{2}(\mathbb{R}^d, ω)$ via a multivariate change of variables $ψ:\left(-\frac{1}{2},\frac{1}{2}\right)^d\to\mathbb{R}^d$. We establish sufficient conditions on $ψ$ and $ω$ such that the composition of a function in such a weighted Hilbert space with $ψ$ 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.

math.NA↗

arXiv2019

A sparse FFT approach for ODE with random coefficients

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 occuring 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.

math.NA↗

arXiv2019

Direct inversion of the nonequispaced fast Fourier transform

Various applications such as MRI, solution of PDEs, etc. need to perform an inverse nonequispaced fast Fourier transform (NFFT), i. e., compute $M$ Fourier coefficients from given $N$ nonequispaced data. In the present paper we consider direct methods for the inversion of the NFFT. We introduce algorithms for the setting $M=N$ as well as for the underdetermined and overdetermined cases. For the setting $M=N$ a direct method of complexity $\mathcal O(N\log N)$ is presented which utilizes Lagrange interpolation and the fast summation. For the remaining cases, we use the matrix representation of the NFFT to deduce our algorithms. Thereby, we are able to compute an inverse NFFT up to a certain accuracy by dint of a modified adjoint NFFT in $\mathcal O(M\log M+N)$ arithmetic operations. Finally, we show that these approaches can also be explained by means of frame approximation.

math.NA↗

arXiv2018

NFFT meets Krylov methods: Fast matrix-vector products for the graph Laplacian of fully connected networks

The graph Laplacian is a standard tool in data science, machine learning, and image processing. The corresponding matrix inherits the complex structure of the underlying network and is in certain applications densely populated. This makes computations, in particular matrix-vector products, with the graph Laplacian a hard task. A typical application is the computation of a number of its eigenvalues and eigenvectors. Standard methods become infeasible as the number of nodes in the graph is too large. We propose the use of the fast summation based on the nonequispaced fast Fourier transform (NFFT) to perform the dense matrix-vector product with the graph Laplacian fast without ever forming the whole matrix. The enormous flexibility of the NFFT algorithm allows us to embed the accelerated multiplication into Lanczos-based eigenvalues routines or iterative linear system solvers and even consider other than the standard Gaussian kernels. We illustrate the feasibility of our approach on a number of test problems from image segmentation to semi-supervised learning based on graph-based PDEs. In particular, we compare our approach with the Nyström method. Moreover, we present and test an enhanced, hybrid version of the Nyström method, which internally uses the NFFT.

cs.LG↗

arXiv2018

Accelerating the calculation of dipolar interactions in particle based simulations with open boundary conditions by means of the P2NFFT method

Magnetic gels are soft elastic materials consisting of magnetic particles embedded in a polymer network. Their shape and elasticity can be controlled by an external magnetic field, which gives rise to both, engineering and biomedical applications. Computer simulations are a commonly used tool to study these materials. A well-known bottleneck of these simulations is the demanding calculation of dipolar interactions. Under periodic boundary conditions established algorithms are available for doing this, however, at the expense of restricting the way in which the gels can deform in an external magnetic field. Moreover, the magnetic properties depend on the sample shape, ruling out periodic boundary conditions entirely for some research questions. In this article we will employ the recently developed dipolar variant of the P$^2$NFFT method that is able to calculate dipolar interactions under open boundary conditions with an $N \log N$ scaling in the number of particles, rather than the expensive $N^2$ scaling of a direct summation of pair forces. The dipolar P$^2$NFFT method has been implemented within the ScaFaCoS library. The molecular dynamics software ESPResSo has been extended to make use of the library. After a short summary of the method, we will discuss its value for studying magnetic soft matter systems. A particular focus is put on developing a tuning strategy to reach the best performance of the method at a predefined accuracy, and lastly applying the method to a magnetic gel model. Here, adapting to the gel's change in shape during the course of a simulation is of particular interest.

cond-mat.soft↗