Efficient Solution of Two-Dimensional Wave Propagation Problems by CQ-Wavelet BEM: Algorithm and Applications

In this paper we consider wave propagation problems in two-dimensional unbounded domains, including dissipative effects, reformulated in terms of space-time boundary integral equations. For their solution, we employ a convolution quadrature (CQ) for the temporal and a Galerkin boundary element method (BEM) for the spatial discretization. It is known that one of the main advantages of the CQ-BEMs is the use of the FFT algorithm to retrieve the discrete time integraloperators with an optimal linear complexity in time, up to a logarithmic term. It is also known that a key ingredient for the success of such methods is the efficient and accurate evaluation of all the integrals that define the matrix entries associated to the full space-time discretization. This topic has been successfully addressed when standard Lagrangian basis functions are considered for the space discretization. However, it results that, for such a choice of the basis, the BEM matrices are in general fully populated, a drawback that prevents the application of CQ-BEMs to large-scale problems. In this paper, as a possible remedy to reduce the global complexity of the method, we consider approximant functions of wavelet type. In particular, we propose a numerical procedure that, by taking advantage of the fast wavelet transform, allows us on the one hand to compute the matrix entries associated to the choice of wavelet basis functions by maintaining the accuracy of those associated to the Lagrangian basis ones and, on the other hand, to generate sparse matrices without the need of storing a priori the fully populated ones. Such an approach allows in principle the use of wavelet basis of any type and order, combined with CQ based on any stable ordinary differential equations solver. Several numerical results, showing the accuracy of the solution and the gain in terms of computer memory saving, are presented and discusse

The Effects of Parental Dialogic Reading and Elaborative Reminiscing on Children’s Narratives and Vocabulary

Seven mothers were taught the same language support strategies, in one of two conditions: dialogic reading or elaborative reminiscing with their child. They were also supported in using these strategies over a 6-week period. The frequency, distribution, and diversity of mothers' strategies were examined before and after the intervention, in two tasks: reading a storybook and reminiscing about a past event. Mothers were also interviewed about their experience and perceptions of dialogic reading or elaborative reminiscing. Additionally, children’s narrative and vocabulary skills were examined. Mothers in the dialogic reading condition used strategies more frequently than those in the elaborative reminiscing condition on the storybook and reminiscing tasks, as revealed by descriptive statistics. Surprisingly, mothers in the reminiscing condition used fewer strategies in the reminiscing task at posttest than at pretest. Descriptively, mothers also showed changes at posttest in their distribution of strategy types, notably greater use of WH questions during both the storybook and reminiscing tasks. All mothers increased their diversity of strategy types on the reminiscing task. Following the intervention, the children showed improvements in fictional and personal storytelling and receptive vocabulary. Expressive vocabulary and story comprehension did not show an effect. The findings suggest that dialogic reading or elaborative reminiscing, implemented by mothers, has positive effects on certain aspects of children's narrative and vocabulary skills. Nuances in mothers' strategy use across conditions and their impact on children’s language skills merit further investigation