The loss of orthogonality in the Gram-Schmidt orthogonalization process

Giraud, Luc; Langou, Julien; Rozložńık, Miroslav

doi:10.1016/j.camwa.2005.08.009

Cited by 101 publications

(87 citation statements)

References 12 publications

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“…Repeating the procedure twice is generally sufficient to obtain a basis which is orthonormal up to machine precision. This result is in accordance with what reported in [9]. For all T P T h , it can be shown that the set of elementary basis functions tϕ T i u iPD T resulting from the MGS algorithm solves…”

Section: Orthonormal Hierarchical Polynomial Bases In the Physical Framesupporting

confidence: 81%

“…We stress that, unlike the latter approach, our procedure yields a diagonal unit mass matrix on curved elements, independently of the nature of the reference-to-physical frame mapping. The generality of this result is of some importance, as it simplifies both the matrix assembly stage and the computation of the lifting operators defined by (9). For further details on the computation of lifting operators we refer to [1, §3.2].…”

Section: Numerical Examplesmentioning

confidence: 99%

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On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations

Bassi

Botti

Colombo

et al. 2012

Journal of Computational Physics

226

183

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In this work we show that the flexibility of the discontinuous Galerkin (dG) discretization can be fruitfully exploited to implement numerical solution strategies based on the use of elements with very general shapes. Thanks to the freedom in defining the mesh topology, we propose a new h-adaptive technique based on agglomeration coarsening of a fine mesh. The possibility to enhance the error distribution over the computational domain is investigated on a Poisson problem with the goal of obtaining a mesh independent discretization.The main building block of our dG method consists of defining discrete polynomial spaces directly on physical frame elements. For this purpose we orthonormalize with respect to the L 2 -product a set of monomials relocated in a specific element frame and we introduce an easy way to reduce the cost related to numerical integration on agglomerated meshes. To complete the dG formulation for second order problems, two extensions of the BR2 scheme to arbitrary polyhedral grids, including an estimate of the stabilization parameter ensuring the coercivity property, are here proposed.

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Section: Orthonormal Hierarchical Polynomial Bases In the Physical Framesupporting

confidence: 81%

Section: Numerical Examplesmentioning

confidence: 99%

On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations

Bassi

Botti

Colombo

et al. 2012

Journal of Computational Physics

226

183

View full text Add to dashboard Cite

show abstract

“…Recent work [11,16,17,18] demonstrates that the modified Gram-Schmidt algorithm yields a good precision for vectors which are not "too ill-conditioned". In particular, it is demonstrated in [17] that the modified Gram-Schmidt algorithm with reorthogonalization yields errors which are small multiples of the machine round-off error.…”

Section: Reorthonormalization Of the Polynomial Basismentioning

confidence: 99%

“…It is not convenient to illustrate the modified Gram-Schmidt method as a formula in (6.1) due to recursive nature of its computational implementation, and instead we give a step-by-step program of the algorithm as in [18] in a fashion which resembles modern computer languages:…”

Section: Reorthonormalization Of the Polynomial Basismentioning

confidence: 99%

“…As mentioned before, the classical Gram-Schmidt method is numerically unstable, and here we use a suitably modified version of the Gram-Schmidt method with reorthogonalization from [18], tailored for a polynomial basis. It is not convenient to illustrate the modified Gram-Schmidt method as a formula in (6.1) due to recursive nature of its computational implementation, and instead we give a step-by-step program of the algorithm as in [18] in a fashion which resembles modern computer languages:…”

Section: Reorthonormalization Of the Polynomial Basismentioning

confidence: 99%

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The multidimensional maximum entropy moment problem: a review of numerical methods

Abramov¹

2010

Communications in Mathematical Sciences

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Abstract. Recently the author developed a numerical method for the multidimensional momentconstrained maximum entropy problem, which is practically capable of solving maximum entropy problems in the two-dimensional domain with moment constraints of order up to 8, in the threedimensional domain with moment constraints of order up to 6, and in the four-dimensional domain with moment constraints of order up to 4, corresponding to the total number of moment constraints of 44, 83 and 69, respectively. In this work, the author brings together key algorithms and observations from his previous works as well as other literature in an attempt to present a comprehensive exposition of the current methods and results for the multidimensional maximum entropy moment problem.Key words. maximum entropy problem, moment constraints, numerical methods.AMS subject classifications. 49M, 65K, 90C. IntroductionThe moment-constrained maximum entropy problem yields an estimate of a probability density with highest uncertainty among all densities satisfying supplied moment constraints. The moment constrained maximum entropy problem arises in a variety of settings in solid state physics [7,22,23,39], econometrics [34,40], statistical description of gas flows [27,33], geophysical applications such as weather and climate prediction [4,5,21,25,28,29,35,36], and many other areas. The approximation of the probability density itself is obtained by maximizing the Shannon entropy under the constraints established by measured moments (phase space-averaged monomials of problem variables) [32]. A standard formalism [41] transforms the constrained maximum entropy problem into the unconstrained minimization problem of the dual objective function. More details on theoretical aspects of the maximum entropy moment problem can be found in [8,15,14,38].Recently, the author developed new algorithms for the multidimensional momentconstrained maximum entropy problem [1,2,3]. While the method in [1] is somewhat primitive and is only capable of solving two-dimensional maximum entropy problems with moments of order up to 4, the improved algorithm in [2] uses a suitable orthonormal polynomial basis in the space of Lagrange multipliers to improve convergence of its iterative optimization process. It is capable of practically solving two-dimensional problems with moments of order up to 8, three-dimensional problems with moments of order up to 6, and four-dimensional problems of order up to 4, totalling 44, 83 and 69 moment constraints, respectively, not counting the normalization constraint for a probability density. In [3], further improvements were introduced to the algorithm for the multidimensional moment-constrained maximum entropy problem, such as multiple Broyden-Fletcher-Goldfarb-Shanno (BFGS) iterations [9,10,12,19,37] to adaptively progress between points of polynomial reorthonormalization, as opposed to single Newton steps introduced in [2], and suitable constraint rescaling to reduce difference in magnitude between high and low order moment constraints. ...

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References

2012

Optical Communication With Chaotic Lasers

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The loss of orthogonality in the Gram-Schmidt orthogonalization process

Cited by 101 publications

References 12 publications

On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations

On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations

The multidimensional maximum entropy moment problem: a review of numerical methods

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