A Fourier Analysis of the Finite Element Variational Method

Strang, Gilbert; Fix, George J.

doi:10.1007/978-3-642-10984-3_7

Cited by 187 publications

(168 citation statements)

References 9 publications

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“…If the poles of the constructed interpolators are of the form α = π x M , x ∈ C, and if M is related to the number of control points, then the definition of the interpolator (9) implies that, as M → ∞, ϕ converges to piecewise-polynomial interpolators that have an (n 0 )th order of approximation (by the Strang-Fix equivalence [25,26]). Such interpolators are of special interest for the construction of particular shapes (see Section 4.3).…”

Section: Order Of Approximationmentioning

confidence: 99%

A family of smooth and interpolatory basis functions for parametric curve and surface representation

Schmitter

Delgado-Gonzalo

Unser

2016

Applied Mathematics and Computation

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a b s t r a c tInterpolatory basis functions are helpful to specify parametric curves or surfaces that can be modified by simple user-interaction. Their main advantage is a characterization of the object by a set of control points that lie on the shape itself (i.e., curve or surface). In this paper, we characterize a new family of compactly supported piecewise-exponential basis functions that are smooth and satisfy the interpolation property. They can be seen as a generalization and extension of the Keys interpolation kernel using cardinal exponential B-splines. The proposed interpolators can be designed to reproduce trigonometric, hyperbolic, and polynomial functions or combinations of them. We illustrate the construction and give concrete examples on how to use such functions to construct parametric curves and surfaces.

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Section: Order Of Approximationmentioning

confidence: 99%

A family of smooth and interpolatory basis functions for parametric curve and surface representation

Schmitter

Delgado-Gonzalo

Unser

2016

Applied Mathematics and Computation

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“…Initially, various sampling kernels have been proposed for exact sampling and reconstruction of FRI signals such as the infinite-support sinc and Gaussian functions [5], [6] and compact support polynomial and exponential reproducing kernels [9]. In order to guarantee exact reconstruction, the sampling kernels are required to satisfy the so-called generalized Strang-Fix conditions [31]. However, these limitations on the choice of the sampling kernels have been successfully removed by relaxing the exact exponential reproduction requirement [8], [16].…”

Section: A Acquisition Of Fri Signalsmentioning

confidence: 99%

“…We first note that this is an appropriate condition since the coefficients of the filter in (13), i.e., , can always be scaled by a constant such as without altering the roots of filter. With this condition, we want to minimize (31) where the term also depends on . Hence, the problem cannot be minimized directly and we propose an iterative scheme, also known as iterative quadratic maximum likelihood (IQML) algorithm [34]- [37], assuming the term remains constant at each iteration of the solution and is obtained using the previous estimate of the solution .…”

Section: ) Quadratic Norm Constraint (Mf-1)mentioning

confidence: 99%

Reconstruction of Finite Rate of Innovation Signals with Model-Fitting Approach

Dogan

Gilliam

Blu

et al. 2015

IEEE Trans. Signal Process.

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Abstract-Finite rate of innovation (FRI) is a recent framework for sampling and reconstruction of a large class of parametric signals that are characterized by finite number of innovations (parameters) per unit interval. In the absence of noise, exact recovery of FRI signals has been demonstrated. In the noisy scenario, there exist techniques to deal with non-ideal measurements. Yet, the accuracy and resiliency to noise and model mismatch are still challenging problems for real-world applications. We address the reconstruction of FRI signals, specifically a stream of Diracs, from few signal samples degraded by noise and we propose a new FRI reconstruction method that is based on a model-fitting approach related to the structured-TLS problem. The model-fitting method is based on minimizing the training error, that is, the error between the computed and the recovered moments (i.e., the FRI-samples of the signal), subject to an annihilation system. We present our framework for three different constraints of the annihilation system. Moreover, we propose a model order selection framework to determine the innovation rate of the signal; i.e., the number of Diracs by estimating the noise level through the training error curve. We compare the performance of the model-fitting approach with known FRI reconstruction algorithms and Cramér-Rao's lower bound (CRLB) to validate these contributions.Index Terms-Annihilating Filter, Cadzow, Cramér-Rao's lower bound (CRLB), finite-rate-of-innovation, iterative quadratic maximum likelihood (IQML), Kumaresan-Tufts, matrix pencil, model fitting, noise, reconstruction, sampling, structured total least squares (STLS), total least squares (TLS).

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“…An alternative to the approximation generated by (4.55) is a local averaging approximation. Since the piecewise constant gradient ∇Φ is most accurate at the centroids of the elements [131], a local element-weighted average can be used to advantage, giving the approximation 60) where |Ω k | is the measure of element k of the patch of elements surrounding node i. Boundary conditions on V can be applied as required.…”

Section: Solving For the Velocitymentioning

confidence: 99%

Velocity-Based Moving Mesh Methods for Nonlinear Partial Differential Equations

Baines

Hubbard

Jimack

2011

Commun. comput. phys.

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Abstract. This article describes a number of velocity-based moving mesh numerical methods for multidimensional nonlinear time-dependent partial differential equations (PDEs). It consists of a short historical review followed by a detailed description of a recently developed multidimensional moving mesh finite element method based on conservation. Finite element algorithms are derived for both mass-conserving and non mass-conserving problems, and results shown for a number of multidimensional nonlinear test problems, including the second order porous medium equation and the fourth order thin film equation as well as a two-phase problem. Further applications and extensions are referenced.

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A Fourier Analysis of the Finite Element Variational Method

Cited by 187 publications

References 9 publications

A family of smooth and interpolatory basis functions for parametric curve and surface representation

A family of smooth and interpolatory basis functions for parametric curve and surface representation

Reconstruction of Finite Rate of Innovation Signals with Model-Fitting Approach

Velocity-Based Moving Mesh Methods for Nonlinear Partial Differential Equations

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