## Simulation and differential equations

*Applied Acoustics*, vol. 97, pp. 140-151, Oct. 2015.

Describes an indoor acoustics simulator that can deal with high bandwidth ultrasonic signals and the multiple reflections due to the room walls and the objects inside it. It also accounts fo the transducer's beam patterns and the ultrasonic wave propagation losses. The simulator can also be used to simulate the Doppler effect for any type of movement of the receiver. The simulator was implemented using Matlab and is freely available. Field tests have confirmed its accuracy, turning it into a valuable tool to study and compare different ultrasonic pulses or ultrasonic location systems under controlled conditions.

*Journal of Computational and Applied Mathematics*, vol. 234, no. 8, pp. 2377-2389, Aug. 2010.

The idea behind the Sparse Point Representation (SPR) method is to retain the function data indicated by significant interpolatory wavelet coefficients, which are defined as interpolation errors in reference to an interpolating subdivision scheme. Typically, a SPR grid is coarse in smooth regions, and refined close to irregularities. Furthermore, the computation of partial derivatives of a function from the information of its SPR content is performed in two steps. The first one is a refinement procedure to extend the SPR by the inclusion of new interpolated point values in a security zone. Then, for points in the refined grid, such derivatives are approximated by uniform finite differences, using a step size proportional to each point local scale. If required neighboring stencils are not present in the grid, the corresponding missing point values are approximated from coarser scales using the interpolating subdivision scheme. Using the cubic interpolation subdivision scheme, we demonstrate that such adaptive finite differences can be formulated in terms of a collocation scheme based on the wavelet expansion associated to the SPR. For this purpose, we prove some results concerning the local behavior of such wavelet reconstruction operators, which stand for SPR grids having appropriate structures. This statement implies that the adaptive finite difference scheme and the one using the step size of the finest level produce the same result at SPR grid points. Consequently, in addition to the refinement strategy, our analysis indicates that some care must be taken concerning the grid structure, in order to keep the truncation error under a certain accuracy limit. Numerical examples based on 2D Maxwell equations are given.

*IEEE Transactions on Magnetics*, vol. 43, no. 3, pp. 1013-1022, Mar. 2007.

This paper discusses the use of sparse point representations (SPR) in computational eletromagnetics. The idea is to represent the solution using only those samples that correspond to significant wavelet coefficients. The paper studies staggered and non-staggered grids for the discretization of the magnetic and electrical fields. Both lead to sparse grids that adapt in space to the local smoothness of the fields, and, at the same time, track the evolution of the fields over time. The conclusion is that schemes based on staggered grids lead to better numerical dispersion, specially for low order schemes and coarse grids; however, for a given accuracy, the adaptive, non-staggered grid scheme requires less computational effort, and its dispersion characteristics can be controlled by varying the order and the grid density. The SPR method combined with non-staggered grids therefore seems to have a good potential in computational electromagnetics.