Francesca Pitolli

A hierarchical Krylov–Bayes iterative inverse solver for MEG with physiological preconditioning

The inverse problem of MEG aims at estimating electromagnetic cerebral activity from measurements of the magnetic fields outside the head. After formulating the problem within the Bayesian framework, a hierarchical conditionally Gaussian prior model is introduced, including a physiologically inspired prior model that takes into account the preferred directions of the source currents. The hyperparameter vector consists of prior variances of the dipole moments, assumed to follow a non-conjugate gamma distribution with variable scaling and shape parameters. A point estimate of both dipole moments and their variances can be computed using an iterative alternating sequential updating algorithm, which is shown to be globally convergent. The numerical solution is based on computing an approximation of the dipole moments using a Krylov subspace iterative linear solver equipped with statistically inspired preconditioning and a suitable termination rule. The shape parameters of the model are shown to control the focality, and furthermore, using an empirical Bayes argument, it is shown that the scaling parameters can be naturally adjusted to provide a statistically well justified depth sensitivity scaling. The validity of this interpretation is verified through computed numerical examples. Also, a computed example showing the applicability of the algorithm to analyze realistic time series data is presented.

Ternary shape-preserving subdivision schemes

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.

A multiscale collocation method for fractional differential problems

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.

An iterative algorithm with joint sparsity constraints for magnetic tomography

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.

Bayes Meets Krylov: Statistically Inspired Preconditioners for CGLS

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.

Brain Activity Mapping from MEG Data via a Hierarchical Bayesian Algorithm with Automatic Depth Weighting

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.

Priorconditioned CGLS-Based Quasi-MAP Estimate, Statistical Stopping Rule, and Ranking of Priors

We consider linear discrete ill-posed problems within the Bayesian framework, assuming a Gaussian additive noise model and a Gaussian prior whose covariance matrices may be known modulo multiplicative scaling factors. In that context, we propose a new pointwise estimator for the posterior density, the priorconditioned CGLS-based quasi-MAP (qMAP) as a computationally attractive approximation of the classical maximum a posteriori (MAP) estimate, in particular when the effective rank of the matrix

Approximation by GP box-splines on a four-direction mesh

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Bayes meets Krylov: preconditioning CGLS for underdetermined systems

is much smaller than the dimension of the unknown. Exploiting the Bayesian paradigm and the connection between standard normal random variables and the

Electric current density imaging via an accelerated iterative algorithm with joint sparsity constraints

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