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A mesh adaptation method is proposed for solving optimal control problems with non-smooth control. The original optimal control problem (OCP) is transcribed into a nonlinear programming (NLP) problem by using the Runge-Kutta discretization method, in which the NLP can be solved by using standard nonlinear programming codes. The method employs collocations from the dyadic background points, which used for the second-generation wavelet (SGW) translation simultaneously. The SGW is used to approximate the control variables and get the wavelet coefficients once they are obtained. In regions contain discontinuities, the magnitude of the relevant wavelet coefficients is large than other regions. The corresponding dyadic background points are reserved as the collocation points. Furthermore, the approximation error of the control and/or state variables can be predicted by a given threshold. Thus, the accuracy and efficiency can be balanced in a simple way. Finally, the method is demonstrated by three numerical examples from the open literature.

We consider the problem of estimating a density of probability from indirect data in the spherical convolution model. We aim at building an estimate of the unknown density as a linear combination of functions of an overcomplete dictionary. The procedure is devised through a well-calibrated @?"1-penalized criterion. The spherical deconvolution setting has been barely studied so far, and the two main approaches to this problem, namely the SVD and the hard thresholding ones considered only one basis at a time. The dictionary approach allows to combine various bases and thus enhances estimates sparsity. We provide an oracle inequality under global coherence assumptions. Moreover, the calibrated procedure that we put forward gives quite satisfying results in the numerical study when compared with other procedures.

In this paper, we propose novel adaptive wavelet filter bank structures based on the lifting scheme. The filter banks are nonseparable, based on quincunx sampling, with their properties being pixel-wise adapted according to the local image features. Despite being adaptive, the filter banks retain a desirable number of primal and dual vanishing moments. The adaptation is introduced in the predict stage of the filter bank with an adaptation region chosen independently for each pixel, based on the intersection of confidence intervals (ICI) rule. The image denoising results are presented for both synthetic and real-world images. It is shown that the obtained wavelet decompositions perform well, especially for synthetic images that contain periodic patterns, for which the proposed method outperforms the state of the art in image denoising.

A dynamic adaptive numerical method for solving partial differential equations on the sphere is developed. The method is based on second generation spherical wavelets on almost uniform nested spherical triangular grids, and is an extension of the adaptive wavelet collocation method to curved manifolds. Wavelet decomposition is used for grid adaption and interpola- tion. An OðN Þ hierarchical finite difference scheme based on the wavelet multilevel decomposition is used to approximate Laplace–Beltrami, Jacobian and flux-divergence operators. The accuracy and efficiency of the method is demonstrated using linear and nonlinear examples relevant to geophysical flows. Although the present paper considers only the sphere, the strength of this new method is that it can be extended easily to other curved manifolds by considering appropriate coarse approximations to the desired manifold (here we used the icosahedral approximation to the sphere at the coarsest level). NSERC

Dynamically adaptive numerical methods have been developed to efficiently solve differential equations whose solutions are intermittent in both space and time. These methods combine an adjustable time step with a spatial grid that adapts to spatial intermittency at a fixed time. The same time step is used for all spatial locations and all scales: this approach clearly does not fully exploit space–time intermittency. We propose an adaptive wavelet collocation method for solving highly intermittent problems (e.g. turbulence) on a simultaneous space–time computational domain which naturally adapts both the space and time resolution to match the solution. Besides generating a near optimal grid for the full space–time solution, this approach also allows the global time integration error to be controlled. The efficiency and accuracy of the method is demonstrated by applying it to several highly intermittent (1D + t)-dimensional and (2D + t)-dimensional test problems. In particular, we found that the space–time method uses roughly 18 times fewer space–time grid points and is roughly 4 times faster than a dynamically adaptive explicit time marching method, while achieving similar global accuracy. JMA and NKRK would like to acknowledge support from NSERC and SHARCNET. Partial support for OVV was provided by the National Science Foundation (NSF) under grants no. EAR-0327269 and ACI- 0242457 and National Aeronautics and Space Administration (NASA) under grant no. NAG-1-02116.