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1

Intrinsic dimensionality detection criterion based on Locally Linear Embedding

Meng L., Breitkopf P.

Computer Science

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2018

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Vol. 19 (3)

345--356

EN

In this work, we revisit the Locally Linear Embedding (LLE) algorithm that is widely employed in dimensionality reduction. With a particular interest to the correspondences of the nearest neighbors in the original and embedded spaces, we observe that, when prescribing low-dimensional embedding spaces, LLE remains merely a weight-preserving rather than a neighborhood-preserving algorithm. Thus, we propose a \neighborhood-preserving ratio" criterion to estimate the minimal intrinsic dimensionality required for neighborhood preservation. We validate its efficiency on sets of synthetic data, including S-curve, Swiss roll, and a dataset of grayscale images.

2

A parallel algorithm of icsym forcomplexsymmetric linear systems in quantum chemistry

Zhang Y., Lv Q., Xiao M., Xie G., Breitkopf P.

Computer Science

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2018

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Vol. 19 (4)

385--401

EN

Computational effort is a common issue for solving large-scale complex symmetric linear systems, particularly in quantum chemistry applications. In order to alleviate this problem, we propose a parallel algorithm of improved conjugate gradient-type iterative (ICSYM). Using three-term recurrence relation and or- thogonal properties of residual vectors to replace the tridiagonalization process of classical CSYM, which allows to decrease the degree of the reduce-operator from two to one communication at each iteration and to reduce the amount of vector updates and vector multiplications. Several numerical examples are implemented to show that high performance of proposed improved version is obtained both in convergent rate and in parallel efficiency.

3

A POD/PGD reduction approach for an efficient parameterization of data - driven material microstructure models

Xia L., Raghavan B., Breitkopf P., Zhang W.

Computer Methods in Materials Science

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2013

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Vol. 13, No. 2

219--225

EN

The general idea here is to produce a high quality representation of the indicator function of different phases of the material while adequately scaling with the storage requirements for high resolution Digital Material Representation (DMR). To this end, we propose a three-stage reduction algorithm combining Proper Orthogonal Decomposition (POD) and Proper Generalized Decomposition (PGD)- first, each snapshot pixel/voxel matrix is decomposed into a linear combination of tensor products of 1D basis vectors. Next a common basis is determined for the entire set of microstructure snapshots. Finally, the analysis of the dimensionality of the resulting nonlinear space yields the minimal set of parameters needed in order to represent the microstructure with sufficient precision. We showcase this approach by constructing a low-dimensional model of a two-phase composite microstructure.

PL

Ogólna idea pracy polega na automatycznej generacji precyzyjnej funkcji reprezentującej topologię i geometrię poszczególnych faz materiału celem uzyskania modelu obliczeniowego o minimalnej liczbie parametrów. W tym celu proponujemy trzystopniowy algorytm redukcji obrazu, łączący cechy dekompozycji POD i PGD. W pierwszym etapie, macierz obrazu reprezentatywnego elementu objętościowego jest rozłożona na liniową kombinację tensorowych produktów jednowymiarowych wektorów bazowych. Następnie budujemy wspólną bazę dla całego zbioru obrazów mikrostruktury. W trzecim etapie, analiza wymiaru otrzymanej rozmaitości topologicznej daje minimalny zestaw parametrów potrzebnych do reprezentowania mikrostruktury z odpowiednią dokładnością. Jako przykład podajemy budowę niskowymiarowego modelu dwufazowej mikrostruktury kompozytu.

4

Consistency approach and diffuse derivation in element free methods based on moving least squares approximation

Breitkopf P., Touzot G., Villon P.

Computer Assisted Mechanics and Engineering Sciences

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1998

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Vol. 5, nr 4

479-501

EN

The paper concerns shape functions formulations in the scope of the recent methods generalizing finite elements and whose common feature is the absence of a mesh. These methods may also be interpreted as a genarelization of the finite difference approach for irregular grids. The shape functions obtained by the Moving Least Squares an by the GFDM (Generalized Finite Difference Method) approach exhibit a number of interesting properties, the most interesting being a local character of the approximation, high degree of continuity and the satisfaction of consistencu contraints neccesary for exact reproduction of polynomials. In the present work we formulate the shape functions directly as solution of the minimization of a weighted quadratic form subjected to the consistency contraints explicity introduced by Lagrange multipliers. This approach gives similar results as the standard moving least squares algorithm applied to the Taylor series expansion where the consistency is automatically satisfied but is more general in the sense, that an explicit specification of wished properties permits an introduction of additional arbitrary conraints other than consistency. It also leads to faster and more robust algorithms by avoiding matrix inversion. On the other hand, the consistency based formulations naturally lead to diffuse (or incomplete) derivatives of the shape functions. They are obtained at a significantly lower cost than full derivatives and their convergence to extact derivatives is demonstrated.