Multiresolution modeling with operator-customized wavelets derived from finite elements

Amaratunga, K. S. P.; Raghunathan, Sudarshan

doi:10.1016/j.cma.2005.05.012

Cited by 27 publications

(25 citation statements)

References 9 publications

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“…It was remarked in Reference [21] that in an adaptive refinement setting where only a small fraction of wavelets are retained at each level, the approximate or partial orthogonalization procedure…”

Section: Construction Of Scale-orthogonal Waveletsmentioning

confidence: 99%

“…As argued in Reference [21], it is possible to construct a valid multiresolution analysis of V provided the interpolation functions { j,k } k∈K(j ) are complete and compatible (see also Reference [20]). We can then simply choose the scaling functions j,k to be the finite element interpolation function j,k .…”

Section: Construction Of Scaling Functionsmentioning

confidence: 99%

“…A desirable property in the design of fast adaptive multiresolution solvers is to have no interaction between basis functions at different levels of resolution [21,22]. To determine the solution at level j + 1, we therefore arrive at a system of equations of the form…”

Section: Interaction Matrices and Scale-orthogonal Waveletsmentioning

confidence: 99%

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A combined approach for goal‐oriented error estimation and adaptivity using operator‐customized finite element wavelets

Raghunathan

Amaratunga

Grätsch

2005

Numerical Meth Engineering

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SUMMARYWe describe how wavelets constructed out of finite element interpolation functions provide a simple and convenient mechanism for both goal-oriented error estimation and adaptivity in finite element analysis. This is done by posing an adaptive refinement problem as one of compactly representing a signal (the solution to the governing partial differential equation) in a multiresolution basis. To compress the solution in an efficient manner, we first approximately compute the details to be added to the solution on a coarse mesh in order to obtain the solution on a finer mesh (the estimation step) and then compute exactly the coefficients corresponding to only those basis functions contributing significantly to a functional of interest (the adaptation step). In this sense, therefore, the proposed approach is unified, since unlike many contemporary error estimation and adaptive refinement methods, the basis functions used for error estimation are the same as those used for adaptive refinement. We illustrate the application of the proposed technique for goal-oriented error estimation and adaptivity for second and fourth-order linear, elliptic PDEs and demonstrate its advantages over existing methods.

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Section: Construction Of Scale-orthogonal Waveletsmentioning

confidence: 99%

Section: Construction Of Scaling Functionsmentioning

confidence: 99%

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A combined approach for goal‐oriented error estimation and adaptivity using operator‐customized finite element wavelets

Raghunathan

Amaratunga

Grätsch

2005

Numerical Meth Engineering

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“…If linear combinations of several primitive wavelets are allowed to construct wavelets in addition to the lifting scheme, more degrees of freedom can be obtained in the wavelet design [8,9,16]. …”

Section: Undo Liftingmentioning

confidence: 99%

“…D'Heedene et al [8] have presented a wavelet-Ritz-Galerkin method for solving PDEs over irregularly spaced meshes on bounded domains based on the lifting scheme, where the relation between operatororthogonality and wavelet vanishing moments has been discussed. Amaratunga and Sudarshan [9] and Sudarshan et al [10] have described a multiresolution modelling with operator-customized wavelets and demonstrated a combined approach for goal-oriented error estimation and adaptivity, where operator-customized wavelets can be constructed from general finite element interpolation functions based on lifting scheme or Gram-Schmidt orthogonalization.…”

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confidence: 99%

Multiresolution analysis for finite element method using interpolating wavelet and lifting scheme

Chen

Xiang

et al. 2007

Commun. Numer. Meth. Engng.

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SUMMARYBased on interpolating wavelet transform and lifting scheme, a multiresolution analysis for finite element method is developed. By designing appropriate finite element interpolation functions using interpolating wavelet and lifted interpolating wavelet on the interval, the finite element equation may be scale decoupled via eliminating all coupling in the stiffness matrix of element across scales, and then resolved in different spaces independently. The coarse solution can be obtained by solving the equation in the coarse approximation space, and refined by adding details, which can be obtained by solving the equations in the corresponding detail spaces, respectively. The method is well suited to the construction of adaptive algorithm and is powerful in analysing the field problems with changes in gradients and singularities. The numerical examples are given to verify the effectiveness of such a method.

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Multiresolution Daubechies finite wavelet domain method for transient dynamic wave analysis in elastic solids

Nastos

Saravanos

2021

Numerical Meth Engineering

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The multiresolution capability provided by the family of Daubechies wavelets is exploited to develop a new computational approach, termed as multiresolution finite wavelet domain method for the fast and hierarchical prediction of transient dynamic problems with focus on wave propagation in one-and two-dimensional structural configurations. In the developed method, both scaling and wavelet functions are employed as basis functions for the approximation of displacements in the governing dynamic equations. Because of the orthogonal and the multiresolution properties of Daubechies wavelets, a two-scale system of mass uncoupled dynamic equations, representing the coarse and the fine spatial wave component solutions is derived. Numerical results concerning wave propagation in one-and two-dimensional elastic structures demonstrate substantial computational gains of the method in comparison with other well-established numerical methods. Moreover, additional benefits of the involved coarse and fine solutions enabling the analysis and characterization of the resultant wave response are revealed and discussed.

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Multiresolution modeling with operator-customized wavelets derived from finite elements

Cited by 27 publications

References 9 publications

A combined approach for goal‐oriented error estimation and adaptivity using operator‐customized finite element wavelets

A combined approach for goal‐oriented error estimation and adaptivity using operator‐customized finite element wavelets

Multiresolution analysis for finite element method using interpolating wavelet and lifting scheme

Multiresolution Daubechies finite wavelet domain method for transient dynamic wave analysis in elastic solids

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