High Performance Computing for Stability Problems - Applications to Hydrodynamic Stability and Neutron Transport Criticality

In this work we examine two kinds of applications in terms of stability and perform numerical evaluations and benchmarks on parallel platforms. We consider the applicability of pseudospectra in the field of hydrodynamic stability to obtain more information than a traditional linear stability analysis can provide. Furthermore, we treat the neutron transport criticality problem and highlight the Davidson method as an attractive alternative to the so far widely used power method in that context

Structure-Preserving Model Reduction of Physical Network Systems

This paper considers physical network systems where the energy storage is naturally associated to the nodes of the graph, while the edges of the graph correspond to static couplings. The first sections deal with the linear case, covering examples such as mass-damper and hydraulic systems, which have a structure that is similar to symmetric consensus dynamics. The last section is concerned with a specific class of nonlinear physical network systems; namely detailed-balanced chemical reaction networks governed by mass action kinetics. In both cases, linear and nonlinear, the structure of the dynamics is similar, and is based on a weighted Laplacian matrix, together with an energy function capturing the energy storage at the nodes. We discuss two methods for structure-preserving model reduction. The first one is clustering; aggregating the nodes of the underlying graph to obtain a reduced graph. The second approach is based on neglecting the energy storage at some of the nodes, and subsequently eliminating those nodes (called Kron reduction).</p

The Synaptic Weight Matrix: Dynamics, Symmetry Breaking, and Disorder

A key role in simplified models of neural circuitry (Wilson and Cowan, 1972) is played by the matrix of synaptic weights, also called connectivity matrix, whose elements describe the amount of influence the firing of one neuron has on another. Biologically, this matrix evolves over time whether or not sensory inputs are present, and symmetries possessed by the internal dynamics of the network may break up spontaneously, as found in the development of the visual cortex (Hubel and Wiesel, 1977). In this thesis, a full analytical treatment is provided for the simplest case of such a biological phenomenon, a single dendritic arbor driven by correlation-based dynamics (Linsker, 1988). Borrowing methods from the theory of Schrödinger operators, a complete study of the model is performed, analytically describing the break-up of rotational symmetry that leads to the functional specialization of cells. The structure of the eigenfunctions is calculated, lower bounds are derived on the critical value of the correlation length, and explicit expressions are obtained for the long-term profile of receptive fields, i.e. the dependence of neural activity on external inputs. The emergence of a functional architecture of orientation preferences in the cortex is another crucial feature of visual information processing. This is discussed through a model consisting of large neural layers connected by an infinite number of Hebb-evolved arbors. Ohshiro and Weiliky (2006), in their study of developing ferrets, found correlation profiles of neural activity in contradiction with previous theories of the phenomenon (Miller, 1994; Wimbauer, 1998). The theory proposed herein, based upon the type of correlations they measured, leads to the emergence of three different symmetry classes. The contours of a highly structured phase diagram are traced analytically, and observables concerning the various phases are estimated in every phase by means of perturbative, asymptotic and variational methods. The proper modeling of axonal competition proves to be key to reproducing basic phenomenological features. While these models describe the long-term effect of synaptic plasticity, plasticity itself makes the connectivity matrix highly dependent on particular histories, hence its stochasticity cannot be considered perturbatively. The problem is tackled by carrying out a detailed treatment of the spectral properties of synaptic-weight matrices with an arbitrary distribution of disorder. Results include a proof of the asymptotic compactness of random spectra, calculations of the shape of supports and of the density profiles, a fresh analysis of the problem of spectral outliers, a study of the link between eigenvalue density and the pseudospectrum of the mean connectivity, and applications of these general results to a variety of biologically relevant examples. The strong non-normality of synaptic-weight matrices (biologically engineered through Dale’s law) is believed to play important functional roles in cortical operations (Murphy and Miller, 2009; Goldman, 2009). Accordingly, a comprehensive study is dedicated to its effect on the transient dynamics of large disordered networks. This is done by adapting standard field-theoretical methods (such as the summation of ladder diagrams) to the non-Hermitian case. Transient amplification of activity can be measured from the average norm squared; this is calculated explicitly for a number of cases, showing that transients are typically amplified by disorder. Explicit expressions for the power spectrum of response are derived and applied to a number of biologically plausible networks, yielding insights into the interplay between disorder and non-normality. The fluctuations of the covariance of noisy neural activity are also briefly discussed. Recent optogenetic measurements have raised questions on the link between synaptic structure and the response of disordered networks to targeted perturbations. Answering to these developments, formulae are derived that establish the operational regime of networks through their response to specific perturbations, and a minimal threshold is found to exist for counterintuitive responses of an inhibitory-stabilized circuit such as have been reported in Ozeki et al. (2016), Adesnik (2016), Kato et al. (2017). Experimental advances are also bringing to light unsuspected differences between various neuron types, which appear to translate into different roles in network function (Pfeffer et al., 2013; Tremblay et al., 2016). Accordingly, the last part of the thesis focuses on networks with an arbitrary number of neuronal types, and predictions are provided for the response of networks with a multipopulation structure to targeted input perturbations

Computing Spectral Measures and Spectral Types

Spectral measures arise in numerous applications such as quantum mechanics, signal processing, resonances, and fluid stability. Similarly, spectral decompositions (pure point, absolutely continuous and singular continuous) often characterise relevant physical properties such as long-time dynamics of quantum systems. Despite new results on computing spectra, there remains no general method able to compute spectral measures or spectral decompositions of infinite-dimensional normal operators. Previous efforts focus on specific examples where analytical formulae are available (or perturbations thereof) or on classes of operators with a lot of structure. Hence the general computational problem is predominantly open. We solve this problem by providing the first set of general algorithms that compute spectral measures and decompositions of a wide class of operators. Given a matrix representation of a self-adjoint or unitary operator, such that each column decays at infinity at a known asymptotic rate, we show how to compute spectral measures and decompositions. We discuss how these methods allow the computation of objects such as the functional calculus, and how they generalise to a large class of partial differential operators, allowing, for example, solutions to evolution PDEs such as Schr\"odinger equations on $L^2(\mathbb{R}^d)$. Computational spectral problems in infinite dimensions have led to the SCI hierarchy, which classifies the difficulty of computational problems. We classify computation of measures, measure decompositions, types of spectra, functional calculus, and Radon--Nikodym derivatives in the SCI hierarchy. The new algorithms are demonstrated to be efficient on examples taken from OPs on the real line and the unit circle (e.g. giving computational realisations of Favard's theorem and Verblunsky's theorem), and are applied to evolution equations on a 2D quasicrystal

Reports of planetary geology and geophysics program, 1987

This is a compilation of abstracts of reports from Principal Investigators of NASA's PLanetary Geology and Geophysics program, Office of Space Science and Applications. The purpose is to document in summary form research work conducted in this program during 1987. Each report reflects significant accomplishments in the area of the author's funded grant or contract

IST Austria Thesis

The eigenvalue density of many large random matrices is well approximated by a deterministic measure, the self-consistent density of states. In the present work, we show this behaviour for several classes of random matrices. In fact, we establish that, in each of these classes, the self-consistent density of states approximates the eigenvalue density of the random matrix on all scales slightly above the typical eigenvalue spacing. For large classes of random matrices, the self-consistent density of states exhibits several universal features. We prove that, under suitable assumptions, random Gram matrices and Hermitian random matrices with decaying correlations have a 1/3-Hölder continuous self-consistent density of states ρ on R, which is analytic, where it is positive, and has either a square root edge or a cubic root cusp, where it vanishes. We, thus, extend the validity of the corresponding result for Wigner-type matrices from [4, 5, 7]. We show that ρ is determined as the inverse Stieltjes transform of the normalized trace of the unique solution m(z) to the Dyson equation −m(z) −1 = z − a + S[m(z)] on C N×N with the constraint Im m(z) ≥ 0. Here, z lies in the complex upper half-plane, a is a self-adjoint element of C N×N and S is a positivity-preserving operator on C N×N encoding the first two moments of the random matrix. In order to analyze a possible limit of ρ for N → ∞ and address some applications in free probability theory, we also consider the Dyson equation on infinite dimensional von Neumann algebras. We present two applications to random matrices. We first establish that, under certain assumptions, large random matrices with independent entries have a rotationally symmetric self-consistent density of states which is supported on a centered disk in C. Moreover, it is infinitely often differentiable apart from a jump on the boundary of this disk. Second, we show edge universality at all regular (not necessarily extreme) spectral edges for Hermitian random matrices with decaying correlations

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal