Random Field Distribution Research Articles

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A continuous spectral algorithm and computer routines in the R programming environment that enable the simulation of second-order stationary and intrinsic (i.e., with second-order stationary increments or generalized increments) vector Gaussian random fields in Euclidean spaces are presented. The simulation is obtained by computing a weighted sum of cosine and sine waves, with weights that depend on the matrix-valued spectral density associated with the spatial correlation structure of the random field to simulate. The computational cost is proportional to the number of locations targeted for simulation, below that of sequential, matrix decomposition and discrete spectral algorithms. Also, the implementation is versatile, as there is no restriction on the number of vector components, workspace dimension, number and geometrical configuration of the target locations. The computer routines are illustrated with synthetic examples and statistical testing is proposed to check the normality of the distribution of the simulated random field or of its generalized increments. A by-product of this work is a spectral representation of spherical, cubic, penta, Askey, J-Bessel, Cauchy, Laguerre, hypergeometric, iterated exponential, gamma, and stable covariance models in the d -dimensional Euclidean space.

Bayesian decision analysis for the optimization of inspection and repair of spatially degrading concrete structures

Pre-posterior analyses can be used to determine the value of information (VoI) of inspection strategies. As such, decisions can be made on whether or not it is relevant to perform inspections and to what extent this is the case, before these are implemented. In this article, a pre-posterior analysis framework is developed for degrading (concrete) structures, accounting for time-dependent behaviour and spatial variability. To account for the time-dependent degradation, a deterioration model is implemented and failure probabilities and costs are evaluated at different points in time. In relation to the spatial nature of the degradation, the structure is discretized in zones and elements and spatial correlation is modelled by random fields and hyperparameters. Although the framework is generally applicable, it is applied to concrete structures subjected to corrosion. First, the framework is illustrated by a simple analytical example to demonstrate the calculation procedure and show how the distributions of hyperparameters and random fields in the corrosion model can be updated based on obtained inspection outcomes. More specifically, the information on non-inspected elements which can be extracted from measurements performed elsewhere in the structure is quantified. Finally, the framework is also applied to a case study in order to prove the feasibility to optimize the timing and location of inspections in relation to the corrosion state of a bridge girder, considering the VoI as a decision variable.

Effects of random fields on the reversal of magnetisation of Ising ferromagnet

We have studied the reversal time of the magnetisation in two dimensional Ising ferromagnet in the presence of externally applied uniform magnetic field using Monte Carlo simulation based on Metropolis single spin flip algorithm. Then we have investigated the change in reversal time due to the presence of quenched random field in addition to the uniform magnetic field. We report the results of statistical distribution of reversal times in the presence of three different types of the distributions (namely, uniform, bimodal and normal) of random fields and compared the results with those obtained for uniform field only. We have observed that the reversal time decreases due to the presence of any kind (of distribution) of the random fields. The metastable volume fraction is observed to follow the Avrami’s law. Dependence of reversal times on temperature and different widths of the distributions of random fields are also reported. We have also checked whether the system obeyed Becker–Döring theory of classical nucleation in presence of additional random field and tried to investigate the range of the width of the distribution of random field. For larger width of the distribution of random field, the system fails to show the reversal via the nucleation of a single droplet (for small values of uniform field only). The possible reason is analysed.

Long-range Order Induced by Random Fields in Two-dimensional O(n) Models, and the Imry–Ma State

The influence of defects of the "random local field" type with an anisotropic distribution of random fields on two-dimensional models with continuous symmetry of the vector order parameter is considered. In the case of weak anisotropy of random fields, with decreasing temperature there takes place a smooth transition from the paramagnetic phase with dynamic fluctuations of the order parameter to the Imry-Ma phase with static fluctuations caused by fluctuations of the random field of defects. In the case of strong anisotropy of random fields, defects lead to an effective decrease in the number of components of the order parameter and the appearance of a phase transition to an ordered state at finite temperature. It is shown that in the case of the defect-free two-dimensional X-Y model, the appearance of an arbitrarily weak anisotropy in the two-dimensional space of the order parameter completely eliminates the appearance of the Berezinsky-Kosterlitz-Thouless phase and gives rise to the phase with the long-range order.

Дальний порядок в двумерных O(n)-моделях, индуцированный случайными полями, и состояние Имри-Ма

A.A. Berzin, A.I. Morosov, and A.S. Sigov The influence of defects of the “random local field” type with an anisotropic distribution of random fields on two-dimensional models with continuous symmetry of the vector order parameter is considered. In the case of weak anisotropy of random fields, with decreasing temperature there takes place a smooth transition from the paramagnetic phase with dynamic fluctuations of the order parameter to the Imry-Ma phase with static fluctuations caused by fluctuations of the random field of defects. In the case of strong anisotropy of random fields, defects lead to an effective decrease in the number of components of the order parameter and the appearance of a phase transition to an ordered state at finite temperature.

Effects of the double Gaussian random field distribution on the phase diagrams and hysteresis characteristics of the magnetic binary alloys

The effect of the double Gaussian random magnetic field distribution on the binary alloy which represented by the formula AcB1−c has been investigated within the framework of effective field theory. By examining the behavior of the order parameter, the evolution of the critical phase boundaries has been obtained under given concentration and randomness. It has been observed that the binary Ising model under the effect of double Gaussian random field distribution could display first and second order transitions. On the other hand, double hysteresis behavior is observed for a specific set of Hamiltonian parameters. The transition between different hysteresis regimes with the level of the randomness has been determined. Besides, the quantities of hysteresis characteristics such as hysteresis loop area, remanent magnetization and coercive field, have been investigated.

Loop Corrections in Spin Models through Density Consistency.

Computing marginal distributions of discrete or semidiscrete Markov random fields (MRFs) is a fundamental, generally intractable problem with a vast number of applications in virtually all fields of science. We present a new family of computational schemes to approximately calculate the marginals of discrete MRFs. This method shares some desirable properties with belief propagation, in particular, providing exact marginals on acyclic graphs, but it differs with the latter in that it includes some loop corrections; i.e., it takes into account correlations coming from all cycles in the factor graph. It is also similar to the adaptive Thouless-Anderson-Palmer method, but it differs with the latter in that the consistency is not on the first two moments of the distribution but rather on the value of its density on a subset of values. The results on finite-dimensional Isinglike models show a significant improvement with respect to the Bethe-Peierls (tree) approximation in all cases and with respect to the plaquette cluster variational method approximation in many cases. In particular, for the critical inverse temperature β_{c} of the homogeneous hypercubic lattice, the expansion of (dβ_{c})^{-1} around d=∞ of the proposed scheme is exact up to d^{-4} order, whereas the latter two are exact only up to d^{-2} order.

On the spin – S random field Ising model

Abstract General Spin-S Ising model under the effect of the random field has been investigated within the effective field approximation. Effect of the bimodal distribution as a discrete, and Gaussian distribution as a continuous random field distribution, on the phase diagrams and order parameter of the system, have been determined. Detailed investigations display many interesting phenomena such as reentrant behavior and small ordered phase which can survive any degree of randomness. Underlying physical mechanisms have been given for the effect of the random field distributions. Especially role of the crystal field has been clarified.

Random field generation of the material properties in the lattice discrete element method

The lattice discrete element method has been successfully used to simulate the evolution of damage in structural mechanics. The approach has led to new perspectives in the solution of fracture problems in nonhomogeneous materials. For such purpose, it is necessary to introduce correctly the parameters that characterize the random nature of the material. In this article, the fracture toughness of the material is considered a three-dimensional random field, characterized by a probability density and the spatial distribution of the simulated random field which is governed by the correlation length. The methodology used to separate the random field simulated from the discretization level used is depicted in detail. Examples are shown that verify the objectivity of the results obtained respecting the discretization levels. Finally, the article concludes by emphasizing the relevance of this implementation in the damage simulation process in the so-called heterogeneous materials.

Optimal microwave imaging with random field illuminations

In the recent years, the theory and technologies of electromagnetic computational imaging have been well developed and several novel imaging methods have been proposed, one of which is known as the microwave imaging under random field illumination. In order to solve the matrix equation of imaging model, the key of such an imaging system is to generate the random electromagnetic radiation field distribution, implementing the independent measurements under random field illuminations. In this work, an optimal microwave imaging system for the desired imaging region and resolution is theoretically analyzed and experimentally implemented. In the randomness analysis, the correlation between different measurements is evaluated by the singular value decomposition, which is also adopted as a criterion for choosing the optimal parameters of the imaging system. Based on random field illuminations generated by the least number of antenna elements, a full-rank matrix equation can be used to reconstruct the object by direct matrix inversion, which can be completed in nearly real-time once the system calibration is implemented in advance. The numerical simulation and experimental investigation are performed, and the results prove the effectiveness of the proposed optimal imaging system. By using the traditional array theory, it is found that for an <i>N</i>-element phase array, <i>N</i> illuminations with each element excited by a single frequency, equal amplitude and randomized 0 or <inline-formula><tex-math id="M5000">\begin{document}${\text{π}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20182122_M5000.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20182122_M5000.png"/></alternatives></inline-formula> phase signal will result in <i>N</i> independent measurements. Theoretically, any additional measurement under random illumination will be correlated with the previous <i>N</i> measurements. Since the random field illumination is obtained by array antennas with 1-bit random phase modulation, the power radiated by each transmitting element is not sacrificed, resulting in an optimal power efficiency of the imaging system compared with those of earlier metasurface-based imaging systems. Besides, a single frequency signal source is used in the system, which also realizes the optimal spectrum efficiency. In conclusion, there are two major innovations of the proposed imaging system: 1) the completely random field illuminations based on 1-bit phase modulation; 2) the approach to optimizing the system on desired demand. The compact and low-cost imaging system promises to have various imaging applications, such as public security and indoor localization.

Binary alloy in random field: Phase diagrams

Critical behavior of the binary alloy represented by the formula AcB1−c under the effect of bimodal random field distribution has been investigated within the framework of effective field theory. The system consists of type A atoms (spin-1∕2) with concentration c and type B atoms (spin-1) with concentration 1−c. Special attention has been paid on the evolution of the phase diagrams with concentration. Besides, by inspecting the behavior of the order parameter, some ideas have been given about the first and second order transitions under several concentrations and randomness.

Analysis of Travel Time Distributions for Uncertainty Propagation in Channelized Porous Systems

In the context of stochastic two-phase flow in porous media, one is often interested in estimating the statistics of fluid saturations in the reservoir. In this work, we show how we can efficiently compute the probability distribution of water saturation in channelized porous systems by pursuing the methodology proposed in Ibrahima et al. (Transp Porous Media 109:(1):81–107, 2015; Comput Geosci 22:(1):389–412, 2018). One of the key metrics of the developed distribution method, called the frozen streamline (FROST) method, is the logarithm of time-of-flight (log-TOF). Therefore, we dedicate a significant portion of the paper to study the cumulative distribution function and probability density function of the log-TOF. We also demonstrate that a Gaussian mixture model can be applied for parametrization of the probability distribution of the log-TOF random field, which is expected to lead to significant computational speedup. We compare the results of the saturation statistics obtained from both the FROST method and exhaustive Monte Carlo (MC) simulations of coupled two-phase flow and transport (i.e., streamlines may change with time), and we show that the agreement between the two approaches is good and improves with grid refinement. Finally, we compare FROST with MC simulations that correspond to the exact assumptions used in FROST (i.e., the streamlines are time independent), and we demonstrate excellent agreement.

Exchange bias mechanism in FM/FM/AF spin valve systems in the presence of random unidirectional anisotropy field at the AF interface: The role played by the interface roughness due to randomness

We propose an atomistic model and present Monte Carlo simulation results regarding the influence of FM/AF interface structure on the hysteresis mechanism and exchange bias behavior for a spin valve type FM/FM/AF magnetic junction. We simulate perfectly flat and roughened interface structures both with uncompensated interfacial AF moments. In order to simulate rough interface effect, we introduce the concept of random exchange anisotropy field induced at the interface, and acting on the interface AF spins. Our results yield that different types of the random field distributions of anisotropy field may lead to different behavior of exchange bias.

Mixed-spin ferrimagnetic bilayer films with a random crystal field distribution

The magnetic properties (phase diagrams and magnetizations) of two mixed spin-1/2 and spin-1 Ising ferrimagnetic bilayer films with a random crystal field distribution are investigated by the use of the effective field theory with correlations. The single-ion anisotropy of spin-1 atoms is randomly distributed by the use of a bimodal distribution function. It is examined how unique phenomena can be realized for the magnetic properties in the two systems, due to the effects of the random crystal field distribution on the spin-1 atoms. The compensation point phenomena are found in the both systems, depending on the selections of physical parameters. But, the reentrant phenomena are found only for one of the two systems.

Dynamical implications of sample shape for avalanches in 2-dimensional random-field Ising model with saw-tooth domain wall

We study dynamics of a built-in domain wall (DW) in 2-dimensional disordered ferromagnets with different sample shapes using random-field Ising model on a square lattice rotated by 45 degrees. The saw-tooth DW of the length Lx is created along one side and swept through the sample by slow ramping of the external field until the complete magnetisation reversal and the wall annihilation at the open top boundary at a distance Ly. By fixing the number of spins N=Lx×Ly=106 and the random-field distribution at a value above the critical disorder, we vary the ratio of the DW length to the annihilation distance in the range Lx∕Ly∈[1∕16,16]. The periodic boundary conditions are applied in the y-direction so that these ratios comprise different samples, i.e., surfaces of cylinders with the changing perimeter Lx and height Ly. We analyse the avalanches of the DW slips between following field updates, and the multifractal structure of the magnetisation fluctuation time series. Our main findings are that the domain-wall lengths materialised in different sample shapes have an impact on the dynamics at all scales. Moreover, the domain-wall motion at the beginning of the hysteresis loop (HLB) probes the disorder effects resulting in the fluctuations that are significantly different from the large avalanches in the central part of the loop (HLC), where the strong fields dominate. Specifically, the fluctuations in HLB exhibit a wide multi-fractal spectrum, which shifts towards higher values of the exponents when the DW length is reduced. The distributions of the avalanches in this segments of the loops obey power-law decay and the exponential cutoffs with the exponents firmly in the mean-field universality class for long DW. In contrast, the avalanches in the HLC obey Tsallis density distribution with the power-law tails which indicate the new categories of the scale invariant behaviour for different ratios Lx∕Ly. The large fluctuations in the HLC, on the other hand, have a rather narrow spectrum which is less sensitive to the length of the wall. These findings shed light to the dynamical criticality of the random-field Ising model at its lower critical dimension; they can be relevant to applications of the dynamics of injected domain walls in two-dimensional nanowires and ferromagnetic films.

Critical behavior of the two-dimensional Coulomb glass at zero temperature

The lattice model of Coulomb Glass in two dimensions with box-type random field distribution is studied at zero temperature for system size upto $96^{2}$. To obtain the minimum energy state we annealed the system using Monte Carlo simulation followed by further minimization using cluster-flipping. The values of the critical exponents are determined using the standard finite size scaling. We found that the correlation length $\xi$ diverges with an exponent $\nu=1.0$ at the critical disorder $W_{c} = 0.2253$ and that $\chi_{dis} \approx \xi^{4-\bar{\eta}}$ with $\bar{\eta}=2$ for the disconnected susceptibility. The staggered magnetization behaves discontinuously around the transition and the critical exponent of magnetization $\beta=0$. The probability distribution of the staggered magnetization shows a three peak structure which is a characteristic feature for the phase coexistence at first-order phase transition. In addition to this, at the critical disorder we have also studied the properties of the domain for different system sizes. In contradiction with the Imry-Ma arguments, we found pinned and non-compact domains where most of the random field energy was contained in the domain wall. Our results are also inconsistent with Binder's roughening picture.

A Bayesian screening approach for hepatocellular carcinoma using multiple longitudinal biomarkers.

Advanced hepatocellular carcinoma (HCC) has limited treatment options and poor survival, therefore early detection is critical to improving the survival of patients with HCC. Current guidelines for high-risk patients include ultrasound screenings every six months, but ultrasounds are operator dependent and not sensitive for early HCC. Serum α-Fetoprotein (AFP) is a widely used diagnostic biomarker, but it has limited sensitivity and is not elevated in all HCC cases so, we incorporate a second blood-based biomarker, des'γ carboxy-prothrombin (DCP), that has shown potential as a screening marker for HCC. The data from the Hepatitis C Antiviral Long-term Treatment against Cirrhosis (HALT-C) Trial is a valuable source of data to study biomarker screening for HCC. We assume the trajectories of AFP and DCP follow a joint hierarchical mixture model with random changepoints that allows for distinct changepoint times and subsequent trajectories of each biomarker. The changepoint indicators are jointly modeled with a Markov Random Field distribution to help detect borderline changepoints. Markov chain Monte Carlo methods are used to calculate posterior distributions, which are used in risk calculations among future patients and determine whether a patient has a positive screen. The screening algorithm was compared to alternatives in simulations studies under a range of possible scenarios and in the HALT-C Trial using cross-validation.

Influence of the interaction of polymeric chains on thermal transitions of Jahn—Teller exchange clusters in compounds of "breathing" crystal family

Based on the method of random field distribution functions in Ising magnets, for the first time a theoretical description of thermal transitions in Jahn—Teller exchange clusters is suggested, which takes into account the interactions of polymeric chains in "breathing" crystal compounds. It was shown that the inclusion of this interaction can lead to both an intensification of spin-crossover type cooperative transitions in exchange clusters and their deceleration. The approach in question is generalized to include non-Jahn—Teller metal impurities in "breathing" crystals, which suppress cooperative effects in thermal transitions of Jahn—Teller exchange clusters. The obtained results qualitatively reproduce the temperature dependencies of magnetic properties of solid solutions Cu1—x Ni x (hfac)2LR (hfac is hexafluoroacetylacetonate anion, ligand LR is nitronyl nitroxide radical with pyrazole substituent at the second position of the imidazoline ring) when R = Et.

Temporally coherent disparity maps using CRFs with fast 4D filtering

Abstract State-of-the-art methods for disparity estimation achieve good results for single stereo frames, but temporal coherence in stereo videos is often neglected. In this paper, we present a method to compute temporally coherent disparity maps. We define an energy over whole stereo sequences and optimize their conditional random field (CRF) distributions using the mean-field approximation. In addition, we introduce novel terms for smoothness and consistency between the left and right views. We perform CRF optimization by fast, iterative spatio-temporal filtering with linear complexity in the total number of pixels. We propose two CRF optimization techniques, using parallel and sequential updates, and compare them in detail. While parallel updates are not guaranteed to converge, we show that, in practice with appropriate initialization, they provide the same quality as sequential updates and they also lead to faster implementations. Finally, we demonstrate that the results of our approach rank among the state of the art while having significantly less flickering artifacts in stereo sequences.

Phase Transitions in Disordered Systems: The Example of the Random-Field Ising Model in Four Dimensions.

By performing a high-statistics simulation of the D=4 random-field Ising model at zero temperature for different shapes of the random-field distribution, we show that the model is ruled by a single universality class. We compute to a high accuracy the complete set of critical exponents for this class, including the correction-to-scaling exponent. Our results indicate that in four dimensions (i)dimensional reduction as predicted by the perturbative renormalization group does not hold and (ii)three independent critical exponents are needed to describe the transition.