Toni Volkmer

Approximation of multivariate periodic functions by trigonometric polynomials based on rank-1 lattice sampling

In this paper, we present algorithms for the approximation of multivariate periodic functions by trigonometric polynomials. The approximation is based on sampling of multivariate functions on rank-1 lattices. To this end, we study the approximation of periodic functions of a certain smoothness. Our considerations include functions from periodic Sobolev spaces of generalized mixed smoothness. Recently an algorithm for the trigonometric interpolation on generalized sparse grids for this class of functions was investigated by Griebel and Hamaekers (2014). The main advantage of our method is that the algorithm is based mainly on a single one-dimensional fast Fourier transform, and that the arithmetic complexity of the algorithm depends only on the cardinality of the support of the trigonometric polynomial in the frequency domain. Therefore, we investigate trigonometric polynomials with frequencies supported on hyperbolic crosses and energy norm based hyperbolic crosses in more detail. Furthermore, we present an algorithm for sampling multivariate functions on perturbed rank-1 lattices and show the numerical stability of the suggested method. Numerical results are presented up to dimension d = 10 , which confirm the theoretical findings.

Approximation of multivariate periodic functions by trigonometric polynomials based on sampling along rank-1 lattice with generating vector of Korobov form

In this paper, we present error estimates for the approximation of multivariate periodic functions in periodic Hilbert spaces of isotropic and dominating mixed smoothness by trigonometric polynomials. The approximation is based on sampling of the multivariate functions on rank-1 lattices. We use reconstructing rank-1 lattices with generating vectors of Korobov form for the sampling and generalize the technique from Temlyakov (1986), in order to show that the aliasing error of that approximation is of the same order as the error of the approximation using the partial sum of the Fourier series. The main advantage of our method is that the computation of the Fourier coefficients of such a trigonometric polynomial, which we use as approximant, is based mainly on a one-dimensional fast Fourier transform, cf. Kammerer et?al. (2013), Kammerer (2014). This means that the arithmetic complexity of the computation depends only on the cardinality of the support of the trigonometric polynomial in the frequency domain. Numerical results are presented up to dimension d = 10 .

Efficient Spectral Estimation by MUSIC and ESPRIT with Application to Sparse FFT

In spectral estimation, one has to determine all parameters of an exponential sum for finitely many (noisy) sampled data of this exponential sum. Frequently used methods for spectral estimation are MUSIC (MUltiple SIgnal Classification) and ESPRIT (Estimation of Signal Parameters via Rotational Invariance Technique). For a trigonometric polynomial of large sparsity, we present a new sparse fast Fourier transform by shifted sampling and using MUSIC resp. ESPRIT, where the ESPRIT based method has lower computational cost. Later this technique is extended to a new reconstruction of a multivariate trigonometric polynomial of large sparsity for given (noisy) values sampled on a reconstructing rank-1 lattice. Numerical experiments illustrate the high performance of these procedures.

Impact of run-time reconfiguration on design and speed - A case study based on a grid of run-time reconfigurable modules inside a FPGA

This paper examines the feasibility of utilizing a grid of run-time reconfigurable (RTR) modules on a dynamically and partially reconfigurable (DPR) FPGA. The aim is to create a homogeneous array of RTR regions on a FPGA, which can be reconfigured on demand during run-time. We study its setup, implementation and performance in comparison with its static counterpart. Such a grid of partially reconfigurable regions (PRR) on a FPGA could be used as an accelerator for computers to offload compute kernels or as an enhancement of functionality in the embedded market which uses FPGAs. An in-depth look at the methodology of creating run-time reconfigurable modules and its tools is shown. Due to the lack of the tools in handling hundreds of dynamically reconfigurable regions a framework is presented which supports the user in the creation process of the design. A case study which uses state of the art Xilinx Virtex-5 FPGAs compares the run-time reconfigurable implementation and achievable clock speeds of a grid with up to 47 reconfigurable module regions with its static counterpart. For this examination a high performance module is used, which finds patterns in a bit stream (pattern matcher). This module is replicated for each partially reconfigurable region. Particularly, design considerations for the controller, which manages the modules, are introduced. Beyond this, the paper also addresses further challenges of the implementation of such a RTR grid and limitations of the reconfigurability of Xilinx FPGAs.

Fast and exact reconstruction of arbitrary multivariate algebraic polynomials in Chebyshev form

We describe a fast method for the evaluation of an arbitrary high-dimensional multivariate algebraic polynomial in Chebyshev form at the nodes of an arbitrary rank-1 Chebyshev lattice. Our main focus is on conditions on rank-1 Chebyshev lattices allowing for the exact reconstruction of such polynomials from samples along such lattices. We present an algorithm for constructing suitable rank-1 Chebyshev lattices based on a component-by-component approach. Moreover, we give a method for the fast and exact reconstruction.

Computational Methods for the Fourier Analysis of Sparse High-Dimensional Functions

A straightforward discretisation of high-dimensional problems often leads to a curse of dimensions and thus the use of sparsity has become a popular tool. Efficient algorithms like the fast Fourier transform (FFT) have to be customised to these thinner discretisations and we focus on two major topics regarding the Fourier analysis of high-dimensional functions: We present stable and effective algorithms for the fast evaluation and reconstruction of multivariate trigonometric polynomials with frequencies supported on an index set \(\mathcal{I}\subset \mathbb{Z}^{d}\).

An on Chip Network inside a FPGA for Run-Time Reconfigurable Low Latency Grid Communication

In this paper a low latency, on chip communication network (NoC) for a run-time reconfigurable (RTR) grid inside dynamically and partially reconfigurable (DPR) FPGAs is proposed, which supports the arbitrary placement of run-time reconfigurable modules (RTRM) inside the grid. The dedicated, fully meshed, silicon network should support the arrangement of communication channels between the RTRMs within the different partially reconfigurable regions (PRRs) on the FPGA. The design of the network guarantees a low latency communication of RTRMs without mutual interference of each other. In comparison with an implementation using FPGA resources the dedicated silicon network could save an huge amount of resources in terms of transistors. The new degree of parallel communication provided for a RTR grid with arbitrarily placeable RTRMs offers new application fields for DPR capable FPGAs. Multiple user applications with inter-communicating offload compute kernels can be loaded on a host coupled FPGA accelerator, a real-time (RT) system with concurrent communication tasks are possible and enhancing the functionality on demand for embedded systems is conceivable. A case study was conducted for proof of concept and for the verification of the run-time environment system, which manages the configurable network.

Multivariate sparse FFT based on rank-1 Chebyshev lattice sampling

We present a method for the fast reconstruction of high-dimensional sparse algebraic polynomials in Chebyshev form and for the fast approximation of multivariate non-periodic functions from samples, when the frequency locations belonging to the non-zero or largest Chebyshev coefficients are unknown. We only assume that we have given a generally very large index set of possible frequencies, e.g. a d-dimensional full grid. We determine the frequency locations in a dimension-incremental way from samples along reconstructing rank-1 Chebyshev lattices. We demonstrate the high performance of the proposed method in numerical examples in up to 15 dimensions.

Design and Performance of a Grid of Asynchronously Clocked Run-Time Reconfigurable Modules on a FPGA

This paper examines the feasibility of utilizing a grid of asynchronously clocked run-time reconfigurable modules (RTRMs) on a dynamically and partially reconfigurable (DPR) FPGA. In contrast to a synchronously clocked grid studied in research, the design, the implementation, the performance and the resource utilization of an asynchronously clocked grid is shown. Such a run-time reconfigurable (RTR) grid on a FPGA can be utilized to dynamically offload compute functions on a host coupled system, providing multi-user and multi-context execution on behalf of user demands. For embedded systems it can be utilized as a highly dynamical platform by providing functional enhancement by module replacement during run-time. The presented platform leverages synthesis and development constraints and is able to increase the overall throughput by allowing multiple clock domains within the grid. The performance and the additional resource utilization of handling multiple clock domains is compared to synchronously clocked grids. As proof of concept a case study with a grid of 47 RTRMs is conducted on state of the art Virtex-5 FPGAs.