Fast and direct inversion methods for the multivariate nonequispaced fast Fourier transform

The well-known discrete Fourier transform (DFT) can easily be generalized to arbitrary nodes in the spatial domain. The fast procedure for this generalization is referred to as nonequispaced fast Fourier transform (NFFT). Various applications such as MRI, solution of PDEs, etc., are interested in the inverse problem, i.e., computing Fourier coefficients from given nonequispaced data. In this paper we survey different kinds of approaches to tackle this problem. In contrast to iterative procedures, where multiple iteration steps are needed for computing a solution, we focus especially on so-called direct inversion methods. We review density compensation techniques and introduce a new scheme that leads to an exact reconstruction for trigonometric polynomials. In addition, we consider a matrix optimization approach using Frobenius norm minimization to obtain an inverse NFFT

Optimal density compensation factors for the reconstruction of the Fourier transform of bandlimited functions

An inverse nonequispaced fast Fourier transform (iNFFT) is a fast algorithm to compute the Fourier coefficients of a trigonometric polynomial from nonequispaced sampling data. However, various applications such as magnetic resonance imaging (MRI) are concerned with the analogous problem for bandlimited functions, i.e., the reconstruction of point evaluations of the Fourier transform from given measurements of the bandlimited function. In this paper, we review an approach yielding exact reconstruction for trigonometric polynomials up to a certain degree, and extend this technique to the setting of bandlimited functions. Here we especially focus on methods computing a diagonal matrix of weights needed for sampling density compensation

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, too

Structure Exploiting Parameter Estimation and Optimum Experimental Design Methods and Applications in Microbial Enhanced Oil Recovery

In this thesis, we advance efficient methods to solve parameter estimation problems constrained by partial differential equations (PDEs). If PDE constrained parameter estimation problems are solved by derivative based methods, here, the generalized Gauss--Newton method, and multiple shooting, the numerical effort growths drastically with the number of states. The reduced approach couples the computation of the Jacobians and the subsequent block Gaussian elimination using directional derivatives by exploiting the special structure of the constraints which arises from the shooting formulation. Thus, the computational effort is reduced to the one of single shooting. The advantages of the new method in comparison to the common approach are illustrated by means of two academic examples. Furthermore, we are the first to adapt methods of optimum experimental design for parameter estimation to processes of microbial enhanced oil recovery. We consider a nonlinear coupled PDE model which consists of two parts. The first part, the black oil model, describes two phase flow through porous media and a model of convection--diffusion--reaction type depicts the transport and growth effects of bacteria, nutrients, gas and other metabolites in the two phases. A mixed discontinuous Galerkin finite element discretization is applied in space. The discretized model is solved in time by the extended IMPES method. Under the assumption of rotational symmetry, we examine a one dimensional model formulation for parameter estimation and optimum experimental design. We follow the principles of internal numerical differentiation and algorithmic differentiation to evaluate the required derivatives, i.e., the derivatives of the model functions are computed by software tools and we solve the tangential problems with respect to the model parameters and the control variables. By optimum experimental design, a new experiment is planned to reduce the uncertainties of the estimated parameters. The designed experiment differs substantially from the experiments which are usually realized in practice. The confidence intervals for the estimated parameters are reduced by a factor of one hundred. The developed methods for parameter estimation are implemented in the software package PAREMERA which is embedded in the optimum experimental design software VPLAN. The model equations for microbial enhanced oil recovery are implemented in a simulation tool which computes not only the nominal equation but also evaluates the derivatives with respect to parameters and controls up to second order

Within-Season Homing Movements of Displaced Mature Sunapee Trout (Salvelinus Alpinus) in Floods Pond, Maine

Tagging, displacemenat nd recapture, and ultrasonict racking of displaced mature Sunapee trout (Salvelinusa Ipinus) in Floods Pond, Maine, demonstrated that rapid within-season homing occurs in this relict form of Arctic char. Of the trout displaced about 1.8 km from their spawning ground from 1972 to 1975, 9% to 32% were recaptured one to four times within the same spawning season in trap nets set on the spawning ground. Eight of 14 trout tracked ultrasonically in 1975 homed in 2.5 to 10.0 h. Movements of the homing fish were variable; some trout homed paralleling the shoreline, others homed in open water or used a combination of near-shore and open-water movements. Behavior was similar between the sexes and during day and night, although two fish did begin to move just at sundown. Swimming speeds ranged from 15 to 35 cm s- 1 and averaged about 0 .6 body lengths s -1•. Swimming directions were not influenced by wind and wave direction, nor were swimming speeds within individual tracks influenced by cloud cover, wave height, or water depth. Heavy overcast at night m&y have inhibited movement. Sunapee trout are apparently familiar with the entire lake and travel widely within it. Visual features are postulated as orientational cues, though use of such cues is not clearly demonstrated by our experiments

Comparison of the Calibration Standards of Three Commercially Available Multiplex Kits for Human Cytokine Measurement to WHO Standards Reveals Striking Differences

Serum parameters as indicators for the efficacy of therapeutic drugs are currently in the focus of intensive research. The induction of certain cytokines (or cytokine patterns) is known to be related to the status of the immune response e.g. in regulating the TH1/TH2 balance. Regarding their potential value as surrogate parameters in clinical trials and subsequently for the assignment of treatment efficacy, the accurate and reliable determination of cytokines in patient serum is mandatory. Because serum samples are precious and limited, test methods—like the xMAP multiplex technology—that allow for the simultaneous determination of a variety of cytokines from only a small sample aliquot, can offer great advantages

Fast and direct inversion methods for the multivariate nonequispaced fast Fourier transform

The well-known discrete Fourier transform (DFT) can easily be generalized to arbitrary nodes in the spatial domain. The fast procedure for this generalization is referred to as nonequispaced fast Fourier transform (NFFT). Various applications such as MRI and solution of PDEs are interested in the inverse problem, i.e., computing Fourier coefficients from given nonequispaced data. In this article, we survey different kinds of approaches to tackle this problem. In contrast to iterative procedures, where multiple iteration steps are needed for computing a solution, we focus especially on so-called direct inversion methods. We review density compensation techniques and introduce a new scheme that leads to an exact reconstruction for trigonometric polynomials. In addition, we consider a matrix optimization approach using Frobenius norm minimization to obtain an inverse NFFT