Non-equispaced B-spline wavelets

This paper has three main contributions. The first is the construction of wavelet transforms from B-spline scaling functions defined on a grid of non-equispaced knots. The new construction extends the equispaced, biorthogonal, compactly supported Cohen-Daubechies-Feauveau wavelets. The new construction is based on the factorisation of wavelet transforms into lifting steps. The second and third contributions are new insights on how to use these and other wavelets in statistical applications. The second contribution is related to the bias of a wavelet representation. It is investigated how the fine scaling coefficients should be derived from the observations. In the context of equispaced data, it is common practice to simply take the observations as fine scale coefficients. It is argued in this paper that this is not acceptable for non-interpolating wavelets on non-equidistant data. Finally, the third contribution is the study of the variance in a non-orthogonal wavelet transform in a new framework, replacing the numerical condition as a measure for non-orthogonality. By controlling the variances of the reconstruction from the wavelet coefficients, the new framework allows us to design wavelet transforms on irregular point sets with a focus on their use for smoothing or other applications in statistics.Comment: 42 pages, 2 figure

Fast Isogeometric Boundary Element Method based on Independent Field Approximation

An isogeometric boundary element method for problems in elasticity is presented, which is based on an independent approximation for the geometry, traction and displacement field. This enables a flexible choice of refinement strategies, permits an efficient evaluation of geometry related information, a mixed collocation scheme which deals with discontinuous tractions along non-smooth boundaries and a significant reduction of the right hand side of the system of equations for common boundary conditions. All these benefits are achieved without any loss of accuracy compared to conventional isogeometric formulations. The system matrices are approximated by means of hierarchical matrices to reduce the computational complexity for large scale analysis. For the required geometrical bisection of the domain, a strategy for the evaluation of bounding boxes containing the supports of NURBS basis functions is presented. The versatility and accuracy of the proposed methodology is demonstrated by convergence studies showing optimal rates and real world examples in two and three dimensions.Comment: 32 pages, 27 figure

An exploration of the IGA method for efficient reservoir simulation

Novel numerical methods present exciting opportunities to improve the efficiency of reservoir simulators. Because potentially significant gains to computational speed and accuracy may be obtained, it is worthwhile explore alternative computational algorithms for both general and case-by-case application to the discretization of the equations of porous media flow, fluid-structure interaction, and/or production. In the present work, the fairly new concept of isogeometric analysis (IGA) is evaluated for its suitability to reservoir simulation via direct comparison with the industry standard finite difference (FD) method and 1st order standard finite element method (SFEM). To this end, two main studies are carried out to observe IGA’s performance with regards to geometrical modeling and ability to capture steep saturation fronts. The first study explores IGA’s ability to model complex reservoir geometries, observing L2 error convergence rates under a variety of refinement schemes. The numerical experimental setup includes an 'S' shaped line sink of varying curvature from which water is produced in a 2D homogenous domain. The accompanying study simplifies the domain to 1D, but adds in multiphase physics that traditionally introduce difficulties associated with modeling of a moving saturation front. Results overall demonstrate promise for the IGA method to be a particularly effective tool in handling geometrically difficult features while also managing typically challenging numerical phenomena.Petroleum and Geosystems Engineerin

Un método Wavelet-Galerkin para ecuaciones diferenciales parciales parabólicas

In this paper an Adaptive Wavelet-Galerkin method for the solution ofparabolic partial differential equations modeling physical problems withdifferent spatial and temporal scales is developed. A semi-implicit timedifference scheme is applied andB-spline multiresolution structure on theinterval is used. As in many cases these solutions are known to presentlocalized sharp gradients, local error estimators are designed and an ef-ficient adaptive strategy to choose the appropriate scale for each time isdeveloped. Finally, experiments were performed to illustrate the applica-bility and efficiency of the proposed method.En este trabajo se desarrolla un método Wavelet-Galerkin Adaptativopara la resolución de ecuaciones diferenciales parabólicas que modelanproblemas físicos, con diferentes escalas en el espacio y en el tiempo. Seutiliza un esquema semi-implícito en diferencias temporales y la estructuramultirresolución de las B-splines sobre intervalo.Como es sabido que enmuchos casos las soluciones presentan gradientes localmente altos, se handiseñado estimadores locales de error y una estrategia adaptativa eficientepara elegir la escala apropiada en cada tiempo. Finalmente, se realizaronexperimentos que ilustran la aplicabilidad y la eficiencia del método pro-puestoFil: Vampa, Victoria Cristina. Universidad Nacional de La Plata. Facultad de Ingeniería; ArgentinaFil: Martín, María Teresa. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de La Plata. Facultad de Ingeniería; Argentin