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Wavelet expansions are a powerful tool for constructing adaptive approximations. For this reason, they find applications in a variety of fields, from signal processing to approximation theory. Wavelets are usually derived from refinable functions, which are the solution of a recursive functional equation called the refinement equation. The analytical expression of refinable functions is known in only a few cases, so if we need to evaluate refinable functions we can make use only of the refinement equation. This is also true for the evaluation of their derivatives and integrals. In this paper, we detail a procedure for computing integrals of wavelet products exactly, up to machine precision. The efficient and accurate evaluation of these integrals is particularly required for the computation of the connection coefficients in the wavelet Galerkin method. We show the effectiveness of the procedure by evaluating the integrals of pseudo-splines.

We present a standalone Matlab software platform complete with visualization for the reconstruction of the neural activity in the brain from MEG or EEG data. The underlying inversion combines hierarchical Bayesian models and Krylov subspace iterative least squares solvers. The Bayesian framework of the underlying inversion algorithm allows to account for anatomical information and possible a priori belief about the focality of the reconstruction. The computational efficiency makes the software suitable for the reconstruction of lengthy time series on standard computing equipment. The algorithm requires minimal user provided input parameters, although the user can express the desired focality and accuracy of the solution. The code has been designed so as to favor the parallelization performed automatically by Matlab, according to the resources of the host computer. We demonstrate the flexibility of the platform by reconstructing activity patterns with supports of different sizes from MEG and EEG data. Moreover, we show that the software reconstructs well activity patches located either in the subcortical brain structures or on the cortex. The inverse solver and visualization modules can be used either individually or in combination. We also provide a version of the inverse solver that can be used within Brainstorm toolbox. All the software is available online by Github, including the Brainstorm plugin, with accompanying documentation and test data.

The MagnetoEncephaloGraphy (MEG) has gained great interest in neurorehabilitation training due to its high temporal resolution. The challenge is to localize the active regions of the brain in a fast and accurate way. In this paper we use an inversion method based on random spatial sampling to solve the real-time MEG inverse problem. Several numerical tests on synthetic but realistic data show that the method takes just a few hundredths of a second on a laptop to produce an accurate map of the electric activity inside the brain. Moreover, it requires very little memory storage. For this reasons the random sampling method is particularly attractive in real-time MEG applications., 15 pages, 15 figures

Meditation practices have been claimed to have a positive effect on the regulation of mood and emotion for quite some time by practitioners, and in recent times there has been a sustained effort to provide a more precise description of the changes induced by meditation on human brain. Longitudinal studies have reported morphological changes in cortical thickness and volume in selected brain regions due to meditation practice, which is interpreted as evidence for effectiveness of it beyond the subjective self reporting. Evidence based on real time monitoring of meditating brain by functional imaging modalities such as MEG or EEG remains a challenge. In this article we consider MEG data collected during meditation sessions of experienced Buddhist monks practicing focused attention (Samatha) and open monitoring (Vipassana) meditation, contrasted by resting state with eyes closed. The MEG data is first mapped to time series of brain activity averaged over brain regions corresponding to a standard Destrieux brain atlas, and further by bootstrapping and spectral analysis to data matrices representing a random sample of power spectral densities over bandwidths corresponding to $\alpha$, $\beta$, $\gamma$, and $\theta$ bands in the spectral range. We demonstrate using linear discriminant analysis (LDA) that the samples corresponding to different meditative or resting states contain enough fingerprints of the brain state to allow a separation between different states, and we identify the brain regions that appear to contribute to the separation. Our findings suggest that cingulate cortex, insular cortex and some of the internal structures, most notably accumbens, caudate and putamen nuclei, thalamus and amygdalae stand out as separating regions, which seems to correlate well with earlier findings based on longitudinal studies., Comment: 34 pages, 15 figures

Nowadays, fractional differential equations are a well established tool to model phenomena from the real world. Since the analytical solution is rarely available, there is a great effort in constructing efficient numerical methods for their solution. In this paper we are interested in solving boundary value problems having space derivative of fractional order. To this end, we present a collocation method in which the solution of the fractional problem is approximated by a spline quasi-interpolant operator. This allows us to construct the numerical solution in an easy way. We show through some numerical tests that the proposed method is efficient and accurate., Proceedings of Approximation Theory XVI, Nashville TN, 2019

In this work we explore the construction and the applications of a special family of level-dependent biorthogonal filters, i.e. filters whose taps depend on the scaling level. Such a family is generated from a class of functions all related through level-dependent (or nonstationary) refinement equations, which contains cardinal polynomial B-splines as a particular case. The greater flexibility offered by the nonstationarity of these filters allows to achieve better results in some image processing problems, such as image compression, when compared to classical biorthogonal B-spline filters.

The aim of this paper is to numerically solve a diffusion differential problem having time derivative of fractional order. To this end we propose a collocation-Galerkin method that uses the fractional splines as approximating functions. The main advantage is in that the derivatives of integer and fractional order of the fractional splines can be expressed in a closed form that involves just the generalized finite difference operator. This allows us to construct an accurate and efficient numerical method. Several numerical tests showing the effectiveness of the proposed method are presented., 15 pages

The solution of linear inverse problems when the unknown parameters outnumber data requires addressing the problem of a nontrivial null space. After restating the problem within the Bayesian framework, a priori information about the unknown can be utilized for determining the null space contribution to the solution. More specifically, if the solution of the associated linear system is computed by the conjugate gradient for least squares (CGLS) method, the additional information can be encoded in the form of a right preconditioner. In this paper we study how the right preconditioner changes the Krylov subspaces where the CGLS iterates live, and we draw a tighter connection between Bayesian inference and Krylov subspace methods. The advantages of a Bayes-meets-Krylov approach to the solution of underdetermined linear inverse problems is illustrated with computed examples.

We construct a new class of approximating functions that are M-refinable and provide shape preserving approximations. The refinable functions in the class are smooth, compactly supported, centrally symmetric and totally positive. Moreover, their refinable masks are associated with convergent subdivision schemes. The presence of one or more shape parameters gives a great flexibility in the applications. Some examples for dilation M = 4 and M = 5 are also given.

The IMACS Series in Computational and Applied Mathematics collects refereed papers on research results presented in scientific events held under the auspices of the International Association for Mathematics and Computers in Simulation (IMACS). The MASCOT2018 Proceedings Book refers to the 15th IMACS/ISGG International Workshop of the MASCOT series of meetings which is yearly organised since 2001. The latest MASCOT 2018, 15th Meeting on Applied Scientific Computing and Tools, Grid Generation, Approximation and Visualization, dealt with mathematical modelling, methodologies and advanced applications, which are the themes of the contributions here collected.

We used a collocation method in refinable spline space to solve a linear dynamical system having fractional derivative in time. The method takes advantage of an explicit derivation rule for the B-spline basis that allows us to efficiently evaluate the collocation matrices appearing in the method. We proof the convergence of the method. Some numerical results are shown., Comment: 19 pages, 9 figures

Magnetoencephalography (MEG) aims at reconstructing the unknown neuroelectric activity in the brain from non-invasive measurements of the magnetic field induced by neural sources. The solution of this ill-posed, ill-conditioned inverse problem is usually dealt with using regularization techniques that are often time-consuming, and computationally and memory storage demanding. In this paper we analyze how a slimmer procedure, random sampling, affects the estimation of the brain activity generated by both synthetic and real sources.

A recently proposed iterated alternating sequential (IAS) MEG inverse solver algorithm, based on the coupling of a hierarchical Bayesian model with computationally efficient Krylov subspace linear solver, has been shown to perform well for both superficial and deep brain sources. However, a systematic study of its ability to correctly identify active brain regions is still missing. We propose novel statistical protocols to quantify the performance of MEG inverse solvers, focusing in particular on how their accuracy and precision at identifying active brain regions. We use these protocols for a systematic study of the performance of the IAS MEG inverse solver, comparing it with three standard inversion methods, wMNE, dSPM, and sLORETA. To avoid the bias of anecdotal tests towards a particular algorithm, the proposed protocols are Monte Carlo sampling based, generating an ensemble of activity patches in each brain region identified in a given atlas. The performance in correctly identifying the active areas is measured by how much, on average, the reconstructed activity is concentrated in the brain region of the simulated active patch. The analysis is based on Bayes factors, interpreting the estimated current activity as data for testing the hypothesis that the active brain region is correctly identified, versus the hypothesis of any erroneous attribution. The methodology allows the presence of a single or several simultaneous activity regions, without assuming that the number of active regions is known. The testing protocols suggest that the IAS solver performs well with both with cortical and subcortical activity estimation.

We present a collocation method based on fractional B-splines for the solution of fractional differential problems. The key-idea is to use the space generated by the fractional B-splines, i.e., piecewise polynomials of noninteger degree, as approximating space. Then, in the collocation step the fractional derivative of the approximating function is approximated accurately and efficiently by an exact differentiation rule that involves the generalized finite difference operator. To show the effectiveness of the method for the solution of nonlinear dynamical systems of fractional order, we solved the fractional Lotka-Volterra model and a fractional predator-pray model with variable coefficients. The numerical tests show that the method we proposed is accurate while keeping a low computational cost.

MASCOT2018 is the 15th edition of the IMACS/ISGG workshop focused on methods and tools for the complete cycle of numerical modelling and simulation, from the development of theoretically well based models and methods to the deep investigation of crucial applications, from geometry and model construction to its implementation, from data analysis and powerful design of computational tools, with advanced capabilities. Aim of the workshop is the presentation and discussion of innovative technologies dealing with all aspects of computational processes. We recall, but without limiting to them, methodologies for effective grid generation, advanced approximation methods, identification and application of efficient PDE numerical solution algorithms, the development of powerful tools for image processing and scientific visualization. The workshop would bring together developers and users of computational methods and software tools to go deeper into advantages of new methodologies, discuss applications and results, illustrate educational approaches, identify new needs and innovative development directions in a growing variety of problems in applied mathematics

We construct a collocation method based on the fractional B-splines to solve a nonlinear differential problem that involves fractional derivatives, i.e., the fractional-order logistic equation. The use of the fractional B-splines allows us to express the fractional derivatives of the approximating function in an analytical form. Thus, the fractional collocation method is easy to implement, accurate, and efficient. Several numerical tests illustrate the efficiency of the proposed collocation method.

Integrals involving refinable functions are of interest in several applications ranging from discretization of PDEs to wavelet analysis. We present a procedure to construct quadrature rules with assigned nodes for these integrals. The process requires in input the refinement mask coefficients and the sequence of nodes only. The corresponding weights are computed by an iterative procedure that does not involve the solution of linear systems. The proposed approach is deeply based on the strong connection between balanced measures and integrals of refinable functions.

We provide fast and accurate adaptive algorithms for the spatial resolution of current densities in MEG. We assume that vector components of the current densities possess a sparse expansion with respect to preassigned wavelets. Additionally, different components may also exhibit common sparsity patterns. We model MEG as an inverse problem with joint sparsity constraints, promoting the coupling of non-vanishing components. We show how to compute solutions of the MEG linear inverse problem by iterative thresholded Landweber schemes. The resulting adaptive scheme is fast, robust, and significantly outperforms the classical Tikhonov regularization in resolving sparse current densities. Numerical examples are included.

In this paper, we present a new class of operators, which are refinable, quasi-interpolatory and satisfy some interpolation conditions. The refinability is achieved by using as functional bases the B-bases corresponding to totally positive refinable functions. We analyze the main properties of the constructed refinable operators and give some convergence results. Some examples are also displayed.