Fast asymptotic algorithm for real-time analysis of multivariate systems and signals by directed transfer function and partial directed coherence measures of connectivity

Connectivity Granger-causality measures in the frequency domain, such as the Directed Transfer Function (DTF) and Partial Directed Coherence (PDC) and their variants, constitute a family of measures that stem from the modeling of multidimensional time series by multivariate autoregressive (MVAR) models. measures have become popular for evaluation of causal interactions in neuronal networks. Surrogate and asymptotic statistical analysis are the two most frequently used methods to quantify the statistical significance of the derived interactions, a critical step for validation of the results. Each method has its own pros and cons, with the recently published asymptotic methodology being faster. The state-of-the-art asymptotic methods, introduced by Baccala et al., run fairly fast on low-dimensional datasets but become impractical for high-dimensional datasets due to the involved computational time and memory demand; the amount of calculations increases exponentially with the number of time series to be analyzed. This is a huge limitation in the application of measures to fields that deal with a large number of concurrently acquired time series from probing of complex systems such as the human brain. In this study, we optimized the original algorithms for fast asymptotic analysis of measures and achieved a reduction of their computation speed by at least three orders of magnitude, thus allowing computation of connectivity measures and their significance in real-time from a plurality of concurrently recorded biological signals. The optimizations were accomplished by a decrease of the dimension of the involved matrices, reduction of the calculation time of complex functions (e.g. eigenvalue estimation and Cholesky factorization), and variable separation. The superior performance of the proposed optimized algorithms in the estimation of the statistical significance and confidence interval of measures of causal interactions is shown with simulation examples

A fast Monte-Carlo method with a Reduced Basis of Control Variates applied to Uncertainty Propagation and Bayesian Estimation

The Reduced-Basis Control-Variate Monte-Carlo method was introduced recently in [S. Boyaval and T. Leli\`evre, CMS, 8 2010] as an improved Monte-Carlo method, for the fast estimation of many parametrized expected values at many parameter values. We provide here a more complete analysis of the method including precise error estimates and convergence results. We also numerically demonstrate that it can be useful to some parametrized frameworks in Uncertainty Quantification, in particular (i) the case where the parametrized expectation is a scalar output of the solution to a Partial Differential Equation (PDE) with stochastic coefficients (an Uncertainty Propagation problem), and (ii) the case where the parametrized expectation is the Bayesian estimator of a scalar output in a similar PDE context. Moreover, in each case, a PDE has to be solved many times for many values of its coefficients. This is costly and we also use a reduced basis of PDE solutions like in [S. Boyaval, C. Le Bris, Nguyen C., Y. Maday and T. Patera, CMAME, 198 2009]. This is the first combination of various Reduced-Basis ideas to our knowledge, here with a view to reducing as much as possible the computational cost of a simple approach to Uncertainty Quantification

A matrix-free ILU realization based on surrogates

Matrix-free techniques play an increasingly important role in large-scale simulations. Schur complement techniques and massively parallel multigrid solvers for second-order elliptic partial differential equations can significantly benefit from reduced memory traffic and consumption. The matrix-free approach often restricts solver components to purely local operations, for instance, the Jacobi- or Gauss--Seidel-Smoothers in multigrid methods. An incomplete LU (ILU) decomposition cannot be calculated from local information and is therefore not amenable to an on-the-fly computation which is typically needed for matrix-free calculations. It generally requires the storage and factorization of a sparse matrix which contradicts the low memory requirements in large scale scenarios. In this work, we propose a matrix-free ILU realization. More precisely, we introduce a memory-efficient, matrix-free ILU(0)-Smoother component for low-order conforming finite elements on tetrahedral hybrid grids. Hybrid grids consist of an unstructured macro-mesh which is subdivided into a structured micro-mesh. The ILU(0) is used for degrees-of-freedom assigned to the interior of macro-tetrahedra. This ILU(0)-Smoother can be used for the efficient matrix-free evaluation of the Steklov-Poincare operator from domain-decomposition methods. After introducing and formally defining our smoother, we investigate its performance on refined macro-tetrahedra. Secondly, the ILU(0)-Smoother on the macro-tetrahedrons is implemented via surrogate matrix polynomials in conjunction with a fast on-the-fly evaluation scheme resulting in an efficient matrix-free algorithm. The polynomial coefficients are obtained by solving a least-squares problem on a small part of the factorized ILU(0) matrices to stay memory efficient. The convergence rates of this smoother with respect to the polynomial order are thoroughly studied