Flavio Sartoretto

Automatic Detection of Epileptiform Activity by Single-Level Wavelet Analysis

We describe a new strategy to automatically identify epileptiform activity in EEG. Our scheme is based upon detecting epileptic spikes, via multiresolution analysis, a relatively new tool in signal processing, which allows for dramatic improvements in the efficiency of basic wavelet analysis. We perform a single-level analysis, which is fast and delivers satisfactory results, provided a wise strategy is adopted. Key points are: the identification of suitable wavelets, in order to gain high computational efficiency; the recognition of a proper resolution level; the computation of an appropriate dynamic threshold, in order to pick out the pathological events. Using a suitable wavelet as the model of a threshold-event proved to be a good choice for devising an algorithm which efficiently performs automatic analysis at high-sensitivity levels. The proposed algorithm was implemented into a C++ multiplatform code having an user-friendly interface, which runs on general-purpose PCs. Results obtained on a set of test tracings, show that the sensitivity of the automatic analysis can be as high as 96%, while less than 5% of the overall recording time is marked. The computational complexity of our algorithm is O (N). Its highly efficient implementation allows for the analysis of up to 310 s of 8 channel EEG, by spending one mere CPU second on a standard PC.

An orthogonal accelerated deflation technique for large symmetric eigenproblems

Abstract An improvement in accelerated conjugate gradient iterations is presented for the evaluation of several of the leftmost eigenpairs of large sparse symmetric positive definite matrices. The approach relies on an orthogonal deflation procedure and is based on the subsequent preconditioned conjugate gradient optimization of Rayleigh quotients over the restricted space orthogonal to the set of eigenvectors previously computed. Comparison with the accelerated simultaneous iterations performed over large finite element problems (with size up to 4500) shows that storage requirement is significantly less and CPU times may be reduced by a factor of two or more.

Accelerated simultaneous iterations for large finite element eigenproblems

Abstract An accelerated simultaneous iteration method is presented for the solution of the generalized eigenproblem Ax = λ Bx , where A and B are real sparse symmetric positive definite matrices. The approach is well suited for the determination of the leftmost eigenpairs of problems with large size N . The procedure relies on the optimization of the Rayleigh quotient over a subspace of orthogonal vectors by a conjugate gradient technique effectively preconditioned with the pointwise incomplete Cholesky factorization. The method is applied to the evaluation of the smallest 15 eigenpairs of finite element models with size ranging between 150 and 2300. The numerical experiments show that, while the simultaneous conjugate gradient scheme fails to converge, the accelerated iterations yield accurate results in a number of steps which is much smaller than N . The new approach does not require the a priori estimate of any empirical parameter and appears to be a robust, reliable, and efficient tool for the partial eigensolution of large finite element problems.

Approximate Inverse Preconditioning in the Parallel Solution of Sparse Eigenproblems

A preconditioned scheme for solving sparse symmetric eigenproblems is proposed. The solution strategy relies upon the DACG algorithm, which is a Preconditioned Conjugate Gradient algorithm for minimizing the Rayleigh Quotient. A comparison with the well established ARPACK code, shows that when a small number of the leftmost eigenpairs is to be computed, DACG is more efficient than ARPACK. Effective convergence acceleration of DACG is shown to be performed by a suitable approximate inverse preconditioner (AINV). The performance of such a preconditioner is shown to be safe, i.e. not highly dependent on a drop tolerance parameter. On sequential machines, AINV preconditioning proves a practicable alternative to the effective incomplete Cholesky factorization, and is more efficient tha n Block Jacobi. Due to its parallelizability, the AINV preconditioner is exploited for a parall el implementation of the DACG algorithm. Numerical tests account for the high degree of parallelization attainable on a Cray T3E machine and confirm the satisfactory scalability properties of t he algorithm. A final comparison with PARPACK shows the (relative) higher efficiency of AI NV-DACG.

An improved iterative optimization technique for the leftmost eigenpairs of large symmetric matrices

Abstract An accelerated optimization technique combined with a stepwise deflation procedure is presented for the efficient evaluation of the p ( p ≤ 20) leftmost eigenvalues and eigenvectors of finite element symmetric positive definite (p.d.) matrices of very large size. The optimization is performed on the Rayleigh quotient of the deflated matrices by the aid of a conjugate gradient (CG) scheme effectively preconditioned with the incomplete Cholesky factorization. No “a priori” estimate of acceleration parameters is required. Numerical experiments on large arbitrarily sparse problems taken from the engineering finite elements (f.e.) practice show a very fast convergence rate for any value of p within the explored interval and particularly so for the minimal eigenpair. In this case the number of iterations needed to achieve an accurate solution may be as much as 2 orders of magnitude smaller than the problem size. Several results concerning the spectral behavior of the CG preconditioning matrices are also given and discussed.

Computational experience with sequential and parallel, preconditioned Jacobi--Davidson for large, sparse symmetric matrices

The Jacobi-Davidson (JD) algorithm was recently proposed for evaluating a number of the eigenvalues of a matrix. JD goes beyond pure Krylov-space techniques; it cleverly expands its search space, by solving the so-called correction equation, thus in principle providing a more powerful method. Preconditioning the Jacobi-Davidson correction equation is mandatory when large, sparse matrices are analyzed. We considered several preconditioners: Classical block-Jacobi, and IC(0), together with approximate inverse (AINV or FSAI) preconditioners. The rationale for using approximate inverse preconditioners is their high parallelization potential, combined with their efficiency in accelerating the iterative solution of the correction equation. Analysis was carried on the sequential performance of preconditioned JD for the spectral decomposition of large, sparse matrices, which originate in the numerical integration of partial differential equations arising in physical and engineering problems. It was found that JD is highly sensitive to preconditioning, and it can display an irregular convergence behavior. We parallelized JD by data-splitting techniques, combining them with techniques to reduce the amount of communication data. Our own parallel, preconditioned code was executed on a dedicated parallel machine, and we present the results of our experiments. Our JD code provides an appreciable parallel degree of computation. Its performance was also compared with those of PARPACK and parallel DACG.

Parallel preconditioning of a sparse eigensolver

We exploit an optimization method, called deflation-accelerated conjugate gradient (DACG), which sequentially computes the smallest eigenpairs of a symmetric, positive definite, generalized eigenproblem, by conjugate gradient (CG) minimizations of the Rayleigh quotient over deflated subspaces. We analyze the effectiveness of the AINV and FSAI approximate inverse preconditioners, to accelerate DACG for the solution of finite element and finite difference eigenproblems. Deflation is accomplished via CGS and MGS orthogonalization strategies whose accuracy and efficiency are tested. Numerical tests on a Cray T3E Supercomputer were performed, showing the high degree of parallelism attainable by the code. We found that for our DACG algorithm, AINV and FSAI are both effective preconditioners. They are more efficient than Block–Jacobi.

FSAIPACK: A Software Package for High-Performance Factored Sparse Approximate Inverse Preconditioning

The Factorized Sparse Approximate Inverse (FSAI) is an efficient technique for preconditioning parallel solvers of symmetric positive definite sparse linear systems. The key factor controlling FSAI efficiency is the identification of an appropriate nonzero pattern. Currently, several strategies have been proposed for building such a nonzero pattern, using both static and dynamic techniques. This article describes a fresh software package, called FSAIPACK, which we developed for shared memory parallel machines. It collects all available algorithms for computing FSAI preconditioners. FSAIPACK allows for combining different techniques according to any specified strategy, hence enabling the user to thoroughly exploit the potential of each preconditioner, in solving any peculiar problem. FSAIPACK is freely available as a compiled library at http://www.dmsa.unipd.it/~janna/software.html, together with an open-source command language interpreter. By writing a command ASCII file, one can easily perform and test any given strategy for building an FSAI preconditioner. Numerical experiments are discussed in order to highlight the FSAIPACK features and evaluate its computational performance.

The Spatial Representation of Angles

We investigated whether angle magnitude, similarly to numerical quantities (i.e., the spatial-numerical association of response codes effect), is associated to the side of response execution. In addition, we investigated whether this association has the properties of a spatially oriented mental line, since angles are taught in a right-to-left progression. We tested two groups of participants: civil engineering students (high familiarity with angles) and psychology students (low familiarity with angles). In Experiment 1, participants were asked to judge the continuity of the angles’ arms (continuous vs. dashed). Magnitude of the angles was task-irrelevant. In Experiment 2, they were asked to judge whether the presented angles were smaller or larger than a right angle (90°). Therefore, the angle magnitude was relevant for performing the task. Overall, engineering students responded faster with their left hand to large angles and with their right hand to small angles. Conversely, psychology students did not show any reliable differences between left- and right-hand responses. In the case of engineering students, the spatial association has a right-to-left (counter clockwise) direction, suggesting the influence of education and practice on the mental representation of angle magnitude.

Solution to Large Symmetric Eigenproblems by an Accelerated Conjugate Gradient Technique

Abstract The computation of the smallest eigenvalues and eigenvectors of large numerical problems is a very important task in a number of engineering applications. The eigensolution to finite element or finite difference linear models provides the shape of the normal modes of vibration and the corresponding natural frequencies of mechanical, structural and hydrodynamical systems. In the present paper the leftmost eigenpairs of large sparse symmetric positive definite matrices are assessed by an efficient numerical technique which combines a deflation procedure together with an optimization approach wherein the Rayleigh quotient is minimized by an accelerated conjugate gradient scheme. The acceleration is achieved by the aid of a preconditioning matrix given by the incomplete Cholesky factorization of the discretized model. The results from finite element matrices show that the p (with p equal to 10÷15) smallest eigenvalues and eigenvectors are evaluated by the iterative deflating method after a number of iterations which turns out to be some orders of magnitude smaller than the problem size N. Several numerical experiments emphasize the promising features of the proposed approach.