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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.

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

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

We introduce a multiscale collocation method to numerically solve differential problems involving both ordinary and fractional derivatives of high order. The proposed method uses multiresolution analyses (MRA) as approximating spaces and takes advantage of a finite difference formula that allows us to express both ordinary and fractional derivatives of the approximating function in a closed form. Thus, the method is easy to implement, accurate and efficient. The convergence and the stability of the multiscale collocation method are proved and some numerical results are shown.

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.

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.

We study the approximation properties of the class of nonstationary refinable ripplets introduced in \cite{GP08}. These functions are solution of an infinite set of nonstationary refinable equations and are defined through sequences of scaling masks that have an explicit expression. Moreover, they are variation-diminishing and highly localized in the scale-time plane, properties that make them particularly attractive in applications. Here, we prove that they enjoy Strang-Fix conditions and convolution and differentiation rules and that they are bell-shaped. Then, we construct the corresponding minimally supported nonstationary prewavelets and give an iterative algorithm to evaluate the prewavelet masks. Finally, we give a procedure to construct the associated nonstationary biorthogonal bases and filters to be used in efficient decomposition and reconstruction algorithms. As an example, we calculate the prewavelet masks and the nonstationary biorthogonal filter pairs corresponding to the $C^2$ nonstationary scaling functions in the class and construct the corresponding prewavelets and biorthogonal bases. A simple test showing their good performances in the analysis of a spike-like signal is also presented. Keywords: total positivity, variation-dimishing, refinable ripplet, bell-shaped function, nonstationary prewavelet, nonstationary biorthogonal basis, Comment: 30 pages, 10 figures

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.

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.

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.

The magnetoencephalography (MEG) aims at reconstructing the unknown neuroelectric activity in the brain from the measurements of the neuromagnetic field in the outer space. The localization of neuroelectric sources from MEG data results in an ill-posed and ill-conditioned inverse problem that requires regularization techniques to be solved. In this paper we propose a new inversion method based on random spatial sampling that is suitable to localize focal neuroelectric sources. The method is fast, efficient and requires little memory storage. Moreover, the numerical tests show that the random sampling method has a high spatial resolution even in the case of deep source localization from noisy magnetic data.

In this paper a new class of nonstationary subdivision schemes is proposed to construct functions having all the main properties of B-splines, namely compact support, central symmetry and total positivity. We show that the constructed nonstationary subdivision schemes are asympotically equivalent to the stationary subdivision scheme associated with a B-spline of suitable degree, but the resulting limit function has smaller support than the B-spline although keeping its regularity.

The aim of this paper is to study the shape preserving properties of some positive operators constructed by means of B-bases on a finite interval. These bases are obtained by the integer translates of totally positive compactly supported refinable functions. We shall prove that the constructed B-bases generate, on the interval, multiresolution analyses which reproduce polynomials up to a certain degree.

Magnetoencephalopgraphy (MEG) is a non-invasive functional imaging modality for mapping cerebral electromagnetic activity from measurements of the weak magnetic field that it generates. It is well known that the MEG inverse problem, i.e. the problem of identifying electric currents from the induced magnetic fields, is a severely underdetermined problem and, without complementary prior information, no unique solution can be found. In the literature, many regularization techniques were proposed. In particular, optimization-based methods usually explain the data by superficial sources even when the activity is deep in the brain. A way to make easier the identification of deep focal sources is the use of depth weighting. We revisit the MEG inverse problem, regularization and depth weighting from a Bayesian point of view by hierarchical models: The primary unknown is the discretized current density inside the head, and we postulate a conditionally Gaussian anatomical prior model. In this model, each current element, or dipole, has a preferred, albeit not fixed, direction that is extracted from the anatomical data of the subject. The variance of each dipole is not fixed a priori, but modeled itself as a random variable described by its hyperprior density. The hypermodel is then used to build a fast iterative algorithm with the novel feature that their parameters are determined using an empirical Bayes approach. The hypermodel provides a very natural Bayesian interpretation for sensitivity weighting, and the parameters in the hyperprior provide a tool for controlling the focality of the solution, thus leading to a flexible algorithm that can handle both sparse and distributed sources. To demonstrate the effects of different parameter selections under optimal conditions, we test the algorithm on synthetic but realistic data. The tests show that the hierarchical Bayesian models combined with linear algebraic methods provide a versatile framework to develop robust and flexible numerical methods, and are able to overcome some of the limitations of standard regularization techniques, for instance deep source localization. The proposed algorithm is computationally efficient, gives a direct control of how well the computed estimates satisfy the data and is designed to easily accommodate different types of prior information.

A two-parameter class of refinable functions is considered and Gaussian quadrature rules having these functions as weight functions. A discretization method is described for generating the recursion coefficients of the required orthogonal polynomials. Numerical results are also presented.

We analyze the shape-preserving properties of ternary subdivision schemes generated by bell-shaped masks. We prove that any bell-shaped mask, satisfying the basic sum rules, gives rise to a convergent monotonicity preserving subdivision scheme, but convexity preservation is not guaranteed. We show that to reach convexity preservation the first order divided difference scheme needs to be bell-shaped, too. Finally, we show that ternary subdivision schemes associated with certain refinable functions with dilation 3 have shape-preserving properties of higher order.

We analyze the properties of a class of shape-preserving refinable functions with dilation M=3. We give an algorithm to construct totally positive bases with optimal shape-preserving properties on a finite interval. Bernstein-type bases on [0,1] are also treated. Moreover, semiorthogonal wavelets associated with these refinable functions are constructed. Finally, a detailed example is described.

EEG/MEG devices record external signals which are generated by the neuronal electric activity of the brain. The localization of the neuronal sources requires the solution of the neuroelectromagnetic inverse problem which is highly ill-posed and ill-conditioned. We provide an iterative thresholding algorithm for recovering neuroeletric current densities within the brain through combined EEG/MEG data. We use a joint sparsity constraint to promote solutions localized in small brain area, assuming that the vector components of the current densities possess the same sparse spatial pattern. At each iteration step, the EEG/MEG forward problem is numerically solved by a Galerkin boundary element method. Some numerical experiments on the localization of current dipole sources are also given. The numerical results show that joint sparsity constraints outperform classical regularization methods based on quadratic constraints.

Magnetic tomography is an ill-posed and ill-conditioned inverse problem since, in general, the solution is non-unique and the measured magnetic field is affected by high noise. We use a joint sparsity constraint to regularize the magnetic inverse problem. This leads to a minimization problem whose solution can be approximated by an iterative thresholded Landweber algorithm. The algorithm is proved to be convergent and an error estimate is also given. Numerical tests on a bidimensional problem show that our algorithm outperforms Tikhonov regularization when the measurements are distorted by high noise.

We analyze the properties of a new class of totally positive refinable functions obtained from nonstationary subdivision schemes. We show that the corresponding system of the integer translates is linearly independent, satisfies a Whitney-Schoenberg condition, reproduces polynomials up to a certain degree and generates a multiresolution analysis. Finally, pre-wavelets and bases on the interval are constructed.

International audience; On a four-directional uniform mesh of the plane, we define and study C0 Gori–Pitolli (abbreviated as GP) quadratics and C1 GP quartics generated by integer translates of GP box-splines generalizing respectively the classical C1 quadratic box-spline and one of the C2 quartic box-splines. More specifically, we study the approximation of a bivariate function f by some quasi-interpolants and interpolants with values in these generalized spline spaces

The WW and ZZ mechanisms for production of heavy Higgs bosons are further studied in detail. The exact analytic distribution Ekdσd3k with k being the Higgs momentum is derived. In particular, the rapidity and transverse momentum distributions of the Higgs are obtained, which are important for the experimental reconstruction of the Higgs signal through its decay products. The relation between the exact results and some approximations for the total cross section and the kT distribution are discussed. Finally extensive numerical applications to e+e−, e−p and pp reactions in the TeV or multi-TeV energy domain are presented.

International audience; Fractures in the Earth's subsurface have a strong impact in many physical and chemical phenomena, as their properties are very different from those of the surrounding rocks. They are generally organized as multi-scale structures, which can be modeled by Discrete Fracture Networks (DFNs) that may contain hundreds of thousands of ellipses in the tridi-mensional space. This paper presents our approach to generate meshes of such large DFNs and to simulate single-phase flow problems using these meshes.