Biomedical subjects

D Sornette

Publications and source records attributed to D Sornette.

At least 19 recordsLinked to original sources

PubMed2003

Self-similar factor approximants.

The problem of reconstructing functions from their asymptotic expansions in powers of a small variable is addressed by deriving an improved type of approximants. The derivation is based on the self-similar approximation theory, which presents the passage from one approximant to another as the motion realized by a dynamical system with the property of group self-similarity. The derived approximants, because of their form, are called self-similar factor approximants. These complement the obtained earlier self-similar exponential approximants and self-similar root approximants. The specific feature of self-similar factor approximants is that their control functions, providing convergence of the computational algorithm, are completely defined from the accuracy-through-order conditions. These approximants contain the Padé approximants as a particular case, and in some limit they can be reduced to the self-similar exponential approximants previously introduced by two of us. It is proved that the self-similar factor approximants are able to reproduce exactly a wide class of functions, which include a variety of nonalgebraic functions. For other functions, not pertaining to this exactly reproducible class, the factor approximants provide very accurate approximations, whose accuracy surpasses significantly that of the most accurate Padé approximants. This is illustrated by a number of examples showing the generality and accuracy of the factor approximants even when conventional techniques meet serious difficulties.

Journal Article↗

PubMed2002

Diffusion of epicenters of earthquake aftershocks, Omori's law, and generalized continuous-time random walk models.

The epidemic-type aftershock sequence (ETAS) model is a simple stochastic process modeling seismicity, based on the two best-established empirical laws, the Omori law (power-law decay approximately 1/t(1+theta) of seismicity after an earthquake) and Gutenberg-Richter law (power-law distribution of earthquake energies). In order to describe also the space distribution of seismicity, we use in addition a power-law distribution approximately 1/r(1+mu) of distances between triggered and triggering earthquakes. The ETAS model has been studied for the last two decades to model real seismicity catalogs and to obtain short-term probabilistic forecasts. Here, we present a mapping between the ETAS model and a class of CTRW (continuous time random walk) models, based on the identification of their corresponding master equations. This mapping allows us to use the wealth of results previously obtained on anomalous diffusion of CTRW. After translating into the relevant variable for the ETAS model, we provide a classification of the different regimes of diffusion of seismic activity triggered by a mainshock. Specifically, we derive the relation between the average distance between aftershocks and the mainshock as a function of the time from the mainshock and of the joint probability distribution of the times and locations of the aftershocks. The different regimes are fully characterized by the two exponents theta and mu. Our predictions are checked by careful numerical simulations. We stress the distinction between the "bare" Omori law describing the seismic rate activated directly by a mainshock and the "renormalized" Omori law taking into account all possible cascades from mainshocks to aftershocks of aftershock of aftershock, and so on. In particular, we predict that seismic diffusion or subdiffusion occurs and should be observable only when the observed Omori exponent is less than 1, because this signals the operation of the renormalization of the bare Omori law, also at the origin of seismic diffusion in the ETAS model. We present predictions and insights provided by the ETAS to CTRW mapping which suggest different ways for studying seismic catalogs. Finally, we discuss the present evidence for our predicted subdiffusion of seismicity triggered by a main shock, stressing the caveats and limitations of previous empirical works.

Journal Article↗

PubMed2002

Occurrence of finite-time singularities in epidemic models of rupture, earthquakes, and starquakes.

We present a new kind of critical stochastic finite-time singularity, relying on the interplay between long-memory and extreme fluctuations. We illustrate it on the well-established epidemic-type aftershock model for aftershocks, based solely on the most solidly documented stylized facts of seismicity (clustering in space and in time and power law Gutenberg-Richter distribution of earthquake energies). This theory accounts for the main observations (power law acceleration and discrete scale invariant structure) of critical rupture of heterogeneous materials, of the largest sequence of starquakes ever attributed to a neutron star, as well as of earthquake sequences.

Journal Article↗

PubMed2002

Classification of possible finite-time singularities by functional renormalization.

Starting from a representation of the early time evolution of a dynamical system in terms of the polynomial expression of some observable phi(t) as a function of the time variable in some interval 0 < or = t < or = T, we investigate how to extrapolate/forecast in some optimal stability sense the future evolution of phi(t) for time t>T. Using the functional renormalization of Yukalov and Gluzman, we offer a general classification of the possible regimes that can be defined based on the sole knowledge of the coefficients of a second-order polynomial representation of the dynamics. In particular, we investigate the conditions for the occurrence of finite-time singularities from the structure of the time series, and quantify the critical time and the functional nature of the singularity when present. We also describe the regimes when a smooth extremum replaces the singularity and determine its position and amplitude. This extends previous works by (1) quantifying the stability of the functional renormalization method more accurately, (2) introducing more global constraints in terms of moments, and (3) going beyond the "mean-field" approximation.

Journal Article↗

PubMed2002

Log-periodic route to fractal functions.

Log-periodic oscillations have been found to decorate the usual power-law behavior found to describe the approach to a critical point, when the continuous scale-invariance symmetry is partially broken into a discrete-scale invariance symmetry. For Ising or Potts spins with ferromagnetic interactions on hierarchical systems, the relative magnitude of the log-periodic corrections are usually very small, of order 10(-5). In growth processes [diffusion limited aggregation (DLA)], rupture, earthquake, and financial crashes, log-periodic oscillations with amplitudes of the order of 10% have been reported. We suggest a "technical" explanation for this 4 order-of-magnitude difference based on the property of the "regular function" g(x) embodying the effect of the microscopic degrees of freedom summed over in a renormalization group (RG) approach F(x)=g(x)+mu(-1)F(gamma x) of an observable F as a function of a control parameter x. For systems for which the RG equation has not been derived, the previous equation can be understood as a Jackson q integral, which is the natural tool for describing discrete-scale invariance. We classify the "Weierstrass-type" solutions of the RG into two classes characterized by the amplitudes A(n) of the power-law series expansion. These two classes are separated by a novel "critical" point. Growth processes (DLA), rupture, earthquake, and financial crashes thus seem to be characterized by oscillatory or bounded regular microscopic functions that lead to a slow power-law decay of A(n), giving strong log-periodic amplitudes. If in addition, the phases of A(n) are ergodic and mixing, the observable presents self-affine nondifferentiable properties. In contrast, the regular function of statistical physics models with "ferromagnetic"-type interactions at equilibrium involves unbound logarithms of polynomials of the control variable that lead to a fast exponential decay of A(n) giving weak log-periodic amplitudes and smoothed observables.

Journal Article↗

PubMed2002

Positive feedback, memory, and the predictability of earthquakes.

We review the "critical point" concept for large earthquakes and enlarge it in the framework of so-called "finite-time singularities." The singular behavior associated with accelerated seismic release is shown to result from a positive feedback of the seismic activity on its release rate. The most important mechanisms for such positive feedback are presented. We solve analytically a simple model of geometrical positive feedback in which the stress shadow cast by the last large earthquake is progressively fragmented by the increasing tectonic stress.

Disasters↗

PubMed2001

Self-consistent theory of rupture by progressive diffuse damage.

We analyze a self-consistent theory of crack growth controlled by a cumulative damage variable d(t) dependent on stress history, in the quasistatic regime where the sound wave velocity is taken as infinite. Depending upon the damage exponent m, which controls the rate of damage dd/dt~sigma(m) as a function of local stress sigma, we find two regimes. For 0 /=2, the rupture dynamics is not defined without the introduction of a regularizing scheme. We investigate three regularization schemes involving, respectively, a saturation of damage, a minimum distance of approach to the crack tip, and a fixed stress maximum. In the first and third schemes, the finite-time singularity is replaced by a crack dynamics defined for all times but which is controlled by either the existence of a microscopic scale at which the stress is regularized or by the maximum sustainable stress. In the second scheme, a finite-time singularity is again found. In the first two schemes within this regime m>/=2, the theory has no continuous limit.

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PubMed1996