Stochastic Map Research Articles

Macroscale and mesoscale analysis of concrete as a multiphase material for biological shields against nuclear radiation

SUMMARYThe overall thermo‐hygro‐mechanical behavior of concrete is to be investigated, because its bearing capacity is required together with its shielding properties, specifically when concrete structures are exposed to high‐energy neutron fluxes, which represent the next generation facilities designed for the production of high energy radioactive ion beams in physics research. Irradiation in the form of either fast and thermal neutrons, primary gamma rays or gamma rays produced as a result of neutron capture, are learnt to affect concrete as well as neutron fluences of the order of 1019 n/cm2 and gamma radiation doses of 1010 rad seem to become critical for concrete strength. The collection of data on concrete samples, variously exposed to neutron radiation, has allowed for defining a law for radiation damage within the FEM research code NEWCON3D, assessing the 3D coupled thermo‐hygro‐mechanical behavior of concrete, modeled as a multiphase porous medium, both at the macroscale and the mesoscale level. The required damage law is thought to be a function of the neutron flux impinging the concrete shielding wall, and a good estimate of this quantity has been provided by means of a Monte Carlo code developed by CERN and the National Institute of Nuclear Physics of Milan, Italy; this code handles radiation transport calculations and represents at this day one of the most reliable procedures for dealing with the interaction of radiation and matter. The suggested procedure for the radiation damage evaluation has allowed for discussing on differences between mesolevel and macrolevel approaches. Stochastic contour maps of the expected radiation field, properly interfaced with the numerical FE code, have allowed for obtaining a more precise evaluation of the radiation damage front as well as its evolution in time. Copyright © 2013 John Wiley & Sons, Ltd.

Distributed Localization and Submapping for Robot Formations using a prior map

In this paper, we present a distributed algorithm which allows a robot formation to jointly improve a prior stochastic map of the environment where it has to accomplish a commanded task. To reduce the computational cost of the global map updates we exploit the fact that a robot formation work in the same map area allowing us to use conditional independence properties over the state distribution. Each robot maintains its own local and global maps which can be improved with the information received when communications among robots take place. Besides, the robots also exchange their positions in order to maintain the structure of the formation. Simulation experiments were conducted showing that, after the synchronization steps, each robot will have exactly the same information about the map and about the location of the robots at its disposal. Our results also demonstrate the achieved precision and efficiency of the proposed distributed algorithm.

Stochastic Cellular Automata: Correlations, Decidability and Simulations

This paper introduces a simple formalism for dealing with deterministic, non- deterministic and stochastic cellular automata in an unified and composable manner. This formalism allows for local probabilistic correlations, a feature which is not present in usual definitions. We show that this feature allows for strictly more behaviors (for instance, number conserving stochastic cellular automata require these local probabilistic correlations). We also show that several problems which are deceptively simple in the usual definitions, become undecidable when we allow for local probabilistic correlations, even in dimension one. Armed with this formalism, we extend the notion of intrinsic simulation between deterministic cellular automata, to the non-deterministic and stochas- tic settings. Although the intrinsic simulation relation is shown to become undecidable in dimension two and higher, we provide explicit tools to prove or disprove the existence of such a simulation between any two given stochastic cellular automata. Those tools rely upon a characterization of equality of stochastic global maps, shown to be equivalent to the existence of a stochastic coupling between the random sources. We apply them to prove that there is no universal stochastic cellular automaton. Yet we provide stochastic cellular automata achieving optimal partial universality, as well as a universal non-deterministic cellular automaton.

Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

Strongly nonlinear stochastic processes can be found in many applications in physics and the life sciences. In particular, in physics, strongly nonlinear stochastic processes play an important role in understanding nonlinear Markov diffusion processes and have frequently been used to describe order-disorder phase transitions of equilibrium and nonequilibrium systems. However, diffusion processes represent only one class of strongly nonlinear stochastic processes out of four fundamental classes of time-discrete and time-continuous processes evolving on discrete and continuous state spaces. Moreover, strongly nonlinear stochastic processes appear both as Markov and non-Markovian processes. In this paper the full spectrum of strongly nonlinear stochastic processes is presented. Not only are processes presented that are defined by nonlinear diffusion and nonlinear Fokker-Planck equations but also processes are discussed that are defined by nonlinear Markov chains, nonlinear master equations, and strongly nonlinear stochastic iterative maps. Markovian as well as non-Markovian processes are considered. Applications range from classical fields of physics such as astrophysics, accelerator physics, order-disorder phase transitions of liquids, material physics of porous media, quantum mechanical descriptions, and synchronization phenomena in equilibrium and nonequilibrium systems to problems in mathematics, engineering sciences, biology, psychology, social sciences, finance, and economics.

Maximum Likelihood Fusion of Stochastic Maps

The fusion of independently obtained stochastic maps by collaborating mobile agents is considered. The proposed approach includes two parts: generalized likehood ratio matching and maximum likelihood alignment. In particular, an affine invariant hypergraph model is constructed for each stochastic map and a bipartite matching via a linear program is used to establish landmark correspondence between stochastic maps. A maximum likelihood alignment procedure is proposed to estimate rotation, translation and scale parameters in order to construct a global map of the environment. A main feature of the proposed approach is its scalability with respect to the number of landmarks: the matching step has polynomial complexity and the maximum likelihood alignment solution is obtained in closed form. Experimental validation of the proposed fusion approach is performed using the Victoria Park experimental benchmark.

Stochastic Regular Grazing Bifurcations

A grazing bifurcation corresponds to the collision of a periodic orbit with a switching manifold in a piecewise-smooth ODE system and often generates complicated dynamics. The lowest order terms of the induced Poincare map expanded about a regular grazing bifurcation constitute a Nordmark map. In this paper we study a normal form of the Nordmark map in two dimensions with additive Gaussian noise of amplitude $\varepsilon$. We show that this particular noise formulation arises in a general setting and consider a harmonically forced linear oscillator subject to compliant impacts to illustrate the accuracy of the map. Numerically computed invariant densities of the stochastic Nordmark map can take highly irregular forms or, if there exists an attracting period-$n$ solution when $\varepsilon=0$, be well approximated by the sum of $n$ Gaussian densities centered about each point of the deterministic solution, and scaled by $\frac{1}{n}$, for sufficiently small $\varepsilon>0$. We explain the irregular forms an...

Stochastic Cellular Automata: Correlations, Decidability and Simulations

This paper introduces a simple formalism for dealing with deterministic, non-deterministic and stochastic cellular automata in an unified and composable manner. This formalism allows for local probabilistic correlations, a feature which is not present in usual definitions. We show that this feature allows for strictly more behaviors for instance, number conserving stochastic cellular automata require these local probabilistic correlations. We also show that several problems which are deceptively simple in the usual definitions, become undecidable when we allow for local probabilistic correlations, even in dimension one. Armed with this formalism, we extend the notion of intrinsic simulation between deterministic cellular automata, to the non-deterministic and stochastic settings. Although the intrinsic simulation relation is shown to become undecidable in dimension two and higher, we provide explicit tools to prove or disprove the existence of such a simulation between any two given stochastic cellular automata. Those tools rely upon a characterization of equality of stochastic global maps, shown to be equivalent to the existence of a stochastic coupling between the random sources. We apply them to prove that there is no universal stochastic cellular automaton. Yet we provide stochastic cellular automata achieving optimal partial universality, as well as a universal non-deterministic cellular automaton.

Self-dual maps and symmetric bistochastic matrices

The H-unistochastic matrices are a special class of symmetric bistochastic matrices obtained by taking the square of the absolute value of each entry of a Hermitian unitary matrix. We examine the geometric relationship of the convex hull of the n by n H-unistochastic matrices relative to the larger convex set of n by n symmetric bistochastic matrices. We show that any line segment in the convex set of the n by n symmetric bistochastic matrices which passes through the centroid of this convex set must spend at least two-thirds of its length in the convex hull of the n by n H-unistochastic matrices when n is either three or four and we prove a partial result for higher n. A class of completely positive linear maps, called the self-dual doubly stochastic maps, is useful for studying this problem. Some results on self-dual doubly stochastic maps are given including a self-dual version of the Laudau–Streater theorem.

Group Representations for Decision Making under Risk and Uncertainty

This paper extends decision making under risk and uncertainty to group theory via representations of invariant behavioural space for prospect theory. First, we predict that canonical specifications for value functions, probability weighting functions, and stochastic choice maps are homomorphic. Second, we derive a continuous singular matrix operator T for affine transformation of a vector space of skewed S-shape value functions V isomorphic to a vector spaceW of inverted S-shaped probability weighting functions. To characterize the transformation, we decompose the operator into shear, scale and translation components. In that milieu, Moore-Penrose psuedoinverse transformation recovers value functions from probability weighting functionals. Removal of 0 from the point spectrum induces nonsingular operators that support group representation of stochastic choice maps in an invariant subspace of the general linear group GL(V). Third, we demonstrate how group theoretic operations on a gamble provide mathematical foundations of probability weighting functions that subsume the Prelec class. Fourth, we predict that a gamble is isomorphic to an invariant cyclic sub-group in weighted probablity space. This result implies that probability weighting functions [and value functions] fluctuate near their extremes, and explain violation of transitivity axioms in decision theory. Moreover, representations include the special unitary group SU(2) and orthogonal group Θ*3. The former includes Pauli’s spin matrices and accounts for skewness. It also provides microfoundations for construction of (1) behavioural stochastic processes from group character in the frequency domain; and (2) Schrodinger-Pauli Hamiltonian to compute, inter alia, time dependent probabilities in decision field theory.

A Comment on the Use of Stochastic Character Maps to Estimate Evolutionary Rate Variation in a Continuously Valued Trait

Phylogenetic comparative biology has progressed considerably in recent years (e.g., Butler and King 2004; Rabosky 2006; Bokma 2008; Alfaro et al. 2009; Stadler 2011; Slater et al. 2012). One of the most important developments has been the application of likelihood-based methods to fit alternative models for trait evolution in a phylogenetic treewith branch lengths proportional to time (e.g., Butler andKing2004;O’Meara et al. 2006; Thomas et al. 2006; Revell and Collar 2009; Beaulieu et al. 2012). An important example of this type of method is O’Meara et al. (2006) “noncensored” test for variation in the evolutionary rate for a continuously valued character trait through time or across the branches of a phylogenetic tree (also see Thomas et al. 2006 for a closely related approach). According to this method, we first hypothesize evolutionary rate regimes on the tree (called “painting” in Butler and King 2004); and then we fit an evolutionary model, specifically the popular Brownian model (Cavalli-Sforza and Edwards 1967; Felsenstein 1973, 1985), in which the instantaneous variance of the Brownian random diffusion process has different values in different parts of the phylogeny (O’Meara et al. 2006). In their original article, O’Meara et al. (2006) did not focus specifically on how to hypothesize evolutionary regimes on the tree. The authors did, however, suggest that to test the hypothesis that a discrete character state had influenced the rate of a continuous character, one could use the approach of (Nielsen 2002; Huelsenbeck et al. 2003; Bollback 2006) to first stochastically map the discretely valued trait, and then “test to see whether the portions of the tree with one state for the discrete character have a different rate of evolution for the continuous character than portions of the tree to which the other discrete state has beenmapped” (O’Meara et al. 2006, p. 931). Indeed, this has become common practice for this and other closely related methods (e.g., Collar et al. 2009, 2010; Revell and Collar 2009; Martin and Wainwright 2011; Price et al. 2011). Normally, then, the set of evolutionary rate estimates from the stochastically mapped trees are averaged across all the trees in the sample. This is presented as a method of obtaining rate estimates that “integrate over uncertainty in...ancestral states” (Collar et al. 2010, p. 1035). Figure 1 illustrates this analysis pipeline. a)

Stochastic mapping of first order reaction networks: A systematic comparison of the stochastic and deterministic kinetic approaches

Stochastic maps are developed and used for first order reaction networks to decide whether the deterministic kinetic approach is appropriate for a certain evaluation problem or the use of the computationally more demanding stochastic approach is inevitable. On these maps, the decision between the two approaches is based on the standard deviation of the expectation of detected variables: when the relative standard deviation is larger than 1%, the use of the stochastic method is necessary. Four different systems are considered as examples: the irreversible first order reaction, the reversible first order reaction, two consecutive irreversible first order reactions, and the unidirectional triangle reaction. Experimental examples are used to illustrate the practical use of the theoretical results. It is shown that the maps do not only depend on particle numbers, but the influence of parameters such as time, rate constants, and the identity of the detected target variable is also an important factor.

On transverse stability of random dynamical system

In this paper, we study the transverse stability of random dynamical systems (RDS). Suppose a RDS on a Riemann manifold possesses a non-random invariant submanifold, what conditions can guarantee that a random attractor of the RDS restrained on the invariant submanifold is a random attractor with respect to the whole manifold? By the linearization technique, we prove that if all the normal Lyapunov exponents with respect to the tangent space of the submanifold are negative, then the attractor on the submanifold is also a random attractor of the whole manifold. This result extends the idea of the transverse stability analysis of deterministic dynamical systems in [1,3]. As an explicit example, we discuss the complete synchronization in network of coupled maps with both stochastic topologies and maps, which extends the well-known master stability function (MSF) approach for deterministic cases to stochastic cases.

Intrinsic Simulations between Stochastic Cellular Automata

The paper proposes a simple formalism for dealing with deterministic, non-deterministic and stochastic cellular automata in a unifying and composable manner. Armed with this formalism, we extend the notion of intrinsic simulation between deterministic cellular automata, to the non-deterministic and stochastic settings. We then provide explicit tools to prove or disprove the existence of such a simulation between two stochastic cellular automata, even though the intrinsic simulation relation is shown to be undecidable in dimension two and higher. The key result behind this is the caracterization of equality of stochastic global maps by the existence of a coupling between the random sources. We then prove that there is a universal non-deterministic cellular automaton, but no universal stochastic cellular automaton. Yet we provide stochastic cellular automata achieving optimal partial universality.

Phase-locking, quasiperiodicity and chaos in periodically driven noisy neuronal models: a spectral approach

In auditory experiments neurons are often driven by periodic or periodically modulated inputs. The response of the neurons is often periodic and phase-locked to the stimulus. However, this is not always the case. It is well-known that even simple deterministic models can exhibit quasiperiodic behavior, with no clear relationship between the phases of the stimulus and response. Moreover, the biological situation is further complicated by presence of noise in the periodic inputs and intrinsic cellular properties, e.g. firing rate adaptation. We develop analytical tools for detection of parameter regimes corresponding to different locking behaviors in more detailed stochastic biophysical models, so that these regimes can be avoided or chosen as necessary. In an effort to develop a mathematical theory applicable to the above biologically-motivated project on phase-locking and related phenomena, we have developed a spectral approach to stochastic circle maps. A stochastic circle map is defined as a Markov chain on the circle. This abstract class of objects includes a wide range of models for firing times of periodically forced noisy neuronal models. We define and analyze some dynamic properties of the system such as stochastic phase locking and stochastic bifurcation cascades. The main tool is analysis of the spectrum and eigenspaces of the transition operator of the Markov chain. We relate the dynamics of the stochastic system to those of a few identifiable finite state deterministic dynamical systems, the dynamics of which can in turn be deduced from the spectral properties of the corresponding matrices. We also show that the firing regime of the stochastic system can be determined by the shape of its eigenvalue “cloud.”

Experimental investigation of noise induced triggering in thermoacoustic systems

We experimentally investigate the effect of noise on the stability of a thermoacoustic system operating in a bistable region. The thermoacoustic system chosen for investigation is a ducted non-premixed flame. The system is excited by random fuel flow rate fluctuations which affects the system dynamics as parametric noise. Under the influence of noise, the system undergoes transition from stable to oscillatory state. In particular, transition is observed even when the amplitude of the noise is significantly less than the triggering amplitude of the corresponding deterministic system. While triggering from noise of low amplitudes, phase portraits reveal that the system evolves transiently towards an unstable periodic orbit, before eventually growing to a stable periodic orbit. A stochastic stability map is constructed to represent the probability of the system to undergo transition. It is also observed that the amplitude of the oscillatory state is affected by the noise level.

Modelação baseada em dados para exploração de água subterrânea em terrenos cristalinos fraturados, nordeste do Brasil

It is not possible, using numerical methods, to model groundwater flow and transport in the fractured crystalline rock of northeastern Brazil. As an alternative, the usefulness of self-organizing map (SOM), k-means clustering, and Davies-Bouldin techniques to conceptualize the hydrogeology was evaluated. Also estimated was the well yield and groundwater quality across the Jua region. This process relies on relations in the underlying multivariate density function associated with a sparse local set of hydrogeologic (electrical conductivity, geology, temperature, and well yield) and a complete regional set of airborne geophysical (electromagnetic, magnetic, and radiometric) and satellite spectrometric measurements. Resampling of the regional well yield and electrical conductivity estimates provides sufficient resolution to construct variograms for stochastic modeling of the hydrogeologic variables. The combination of these stochastic maps provides a way to identify potential drilling targets for future groundwater development. The data-driven estimation approach, when applied to available airborne electromagnetic and water-well hydrogeologic measurements, provides a low-cost alternative to numerical groundwater flow modeling. In addition to fractured rock environments, the alternative modeling framework can provide spatial parameter estimates and associated variograms for constraints to improve the traditional calibration of equivalent groundwater-porous-media models.

Spectral and correlation properties of rings of delay-coupled elements: Comparing linear and nonlinear systems

The dynamical properties of delay-coupled systems are currently of great interest. So far the analysis has concentrated primarily on identical synchronization properties. Here we study the dynamics of rings of delay-coupled nodes, a topology that cannot show identical synchronization, and compare its properties to those of linear stochastic maps. We find that, in the long delay limit, the correlation functions and spectra of delay-coupled rings of nonlinear systems obey the same scaling laws as linear systems, indicating that important properties of the emerging solution result from network topology.

Inversions of Infinitely Divisible Distributions and Conjugates of Stochastic Integral Mappings

The dual of an infinitely divisible distribution on ℝd without Gaussian part defined in Sato (ALEA Lat. Am. J. Probab. Math. Statist. 3:67–110, 2007) is renamed to the inversion. Properties and characterization of the inversion are given. A stochastic integral mapping is a mapping μ=Φ f ρ of ρ to μ in the class of infinitely divisible distributions on ℝd, where μ is the distribution of an improper stochastic integral of a nonrandom function f with respect to a Lévy process on ℝd with distribution ρ at time 1. The concept of the conjugate is introduced for a class of stochastic integral mappings and its close connection with the inversion is shown. The domains and ranges of the conjugates of three two-parameter families of stochastic integral mappings are described. Applications to the study of the limits of the ranges of iterations of stochastic integral mappings are made.

SPECO: Stochastic Perturbation based Clock tree Optimization considering temperature uncertainty

Modern computing system applications or workloads can bring significant non-uniform temperature gradient on-chip, and hence can cause significant temperature uncertainty during clock-tree synthesis. Existing designs of clock-trees have to assume a given time-invariant worst-case temperature map but cannot deal with a set of temperature maps under a set of workloads. For robust clock-tree synthesis considering temperature uncertainty, this paper presents a new problem formulation: Stochastic PErturbation based Clock Optimization (SPECO). In SPECO algorithm, one nominal clock-tree is pre-synthesized with determined merging points. The impact from the stochastic temperature variation is modeled by perturbation (or small physical displacement) of merging points to offset the induced skews. Because the implementation cost is reduced but the design complexity is increased, the determination of optimal positions of perturbed merging points requires a computationally efficient algorithm.In this paper, one Non-Monte-Carlo (NMC) method is deployed to generate skew and skew variance by one-time analysis when a set of stochastic temperature maps is already provided. Moreover, one principal temperature–map analysis is developed to reduce the design complexity by clustering correlated merging points based on the subspace of the correlation matrix. As a result, the new merging points can be efficiently determined level by level with both skew and its variance reduced. The experimental results show that our SPECO algorithm can effectively reduce the clock-skew and its variance under a number of workloads with minimized wire-length overhead and computational cost.

Weak Drifts of Infinitely Divisible Distributions and Their Applications

Weak drift of an infinitely divisible distribution μ on ℝ d is defined by analogy with weak mean; properties and applications of weak drift are given. When μ has no Gaussian part, the weak drift of μ equals the minus of the weak mean of the inversion μ′ of μ. Applying the concepts of having weak drift 0 and of having weak drift 0 absolutely, the ranges, the absolute ranges, and the limit of the ranges of iterations are described for some stochastic integral mappings. For Lévy processes, the concepts of weak mean and weak drift are helpful in giving necessary and sufficient conditions for the weak law of large numbers and for the weak version of Shtatland’s theorem on the behavior near t=0; those conditions are obtained from each other through inversion.