Stochastic Domains Research Articles

Enzyme sequestration by the substrate: An analysis in the deterministic and stochastic domains.

This paper is concerned with the potential multistability of protein concentrations in the cell. That is, situations where one, or a family of, proteins may sit at one of two or more different steady state concentrations in otherwise identical cells, and in spite of them being in the same environment. For models of multisite protein phosphorylation for example, in the presence of excess substrate, it has been shown that the achievable number of stable steady states can increase linearly with the number of phosphosites available. In this paper, we analyse the consequences of adding enzyme docking to these and similar models, with the resultant sequestration of phosphatase and kinase by the fully unphosphorylated and by the fully phosphorylated substrates respectively. In the large molecule numbers limit, where deterministic analysis is applicable, we prove that there are always values for these rates of sequestration which, when exceeded, limit the extent of multistability. For the models considered here, these numbers are much smaller than the affinity of the enzymes to the substrate when it is in a modifiable state. As substrate enzyme-sequestration is increased, we further prove that the number of steady states will inevitably be reduced to one. For smaller molecule numbers a stochastic analysis is more appropriate, where multistability in the large molecule numbers limit can manifest itself as multimodality of the probability distribution; the system spending periods of time in the vicinity of one mode before jumping to another. Here, we find that substrate enzyme sequestration can induce bimodality even in systems where only a single steady state can exist at large numbers. To facilitate this analysis, we develop a weakly chained diagonally dominant M-matrix formulation of the Chemical Master Equation, allowing greater insights in the way particular mechanisms, like enzyme sequestration, can shape probability distributions and therefore exhibit different behaviour across different regimes.

Norm Conflict Resolution in Stochastic Domains

Artificial agents will need to be aware of human moral and social norms, and able to use them in decision-making. In particular, artificial agents will need a principled approach to managing conflicting norms, which are common in human social interactions. Existing logic-based approaches suffer from normative explosion and are typically designed for deterministic environments; reward-based approaches lack principled ways of determining which normative alternatives exist in a given environment. We propose a hybrid approach, using Linear Temporal Logic (LTL) representations in Markov Decision Processes (MDPs), that manages norm conflicts in a systematic manner while accommodating domain stochasticity. We provide a proof-of-concept implementation in a simulated vacuum cleaning domain.

Heat Treatment Effect on Magnetic Microstructure of Fe73.9Cu1Nb3Si13.2B8.9 Thin Films

Fe73.9Cu1Nb3Si13.2B8.9 (Finemet) thin films were deposited on the glass substrates by means of radio frequency sputtering. The films thickness was varied from 10 to 200 nm. Heat treatment at temperatures of 350, 400 and 450 °C were performed for 30 minutes in order to control thin film structural state. The X-ray powder diffractometry revealed that the crystallization of α-FeSi nanograins took place only at 450 °C whilst the other samples stayed in the amorphous state. Relation between the structure and magnetic properties of the films was discussed in the framework of random magnetic anisotropy model and the concept of stochastic magnetic domains. The latter was investigated using magnetic force microscopy (MFM). MFM data showed formation of such magnetic domains only in samples thermally treated at 450 °C. There was a tendency of the magnetic domain size reduction with the thickness decrease.

Variation of magnetic anisotropy and temperature-dependent FORC probing of compositionally tuned Co-Ni alloy nanowires

The magnetic microstructure of Co-Ni binary alloy nanowires electrodeposited with controlled composition into nanoporous aluminum oxide templates can be represented as an ensemble of stochastic magnetic domains, whose size is determined by the magnetic correlation length. Using a method based on the approaching of magnetization to saturation, we defined the dimension of regions with magnetic orientation coherency as the stochastic domain size. Based on the experimental measurements of magnetization curves near saturation and first order reversal curves (FORC), we described a relationship between the macroscopic and microscopic parameters of the nanowires depending on the crystal structure observed by high-resolution transmission electron microscopy in terms of the random anisotropy model. The alloy composition strongly determines the crystal structure, in particular, the grain size and hcp/fcc phase distribution, and influences the effective magnetic anisotropy energy providing the direction of easy magnetization in the nanowire arrays. The characterization of Co-Ni arrays by FORC method at room and low temperatures revealed the transformation of magnetic behavior and certain contributions to the energy of the effective magnetic anisotropy.

Scalable domain decomposition solvers for stochastic PDEs in high performance computing

Stochastic spectral finite element models of practical engineering systems may involve solutions of linear systems or linearized systems for non-linear problems with billions of unknowns. For stochastic modeling, it is therefore essential to design robust, parallel and scalable algorithms that can efficiently utilize high-performance computing to tackle such large-scale systems. Domain decomposition based iterative solvers can handle such systems. Although these algorithms exhibit excellent scalabilities, significant algorithmic and implementational challenges exist to extend them to solve extreme-scale stochastic systems using emerging computing platforms. Intrusive polynomial chaos expansion based domain decomposition algorithms are extended here to concurrently handle high resolution in both spatial and stochastic domains using an in-house implementation. Sparse iterative solvers with efficient preconditioners are employed to solve the resulting global and subdomain level local systems through multi-level iterative solvers. Parallel sparse matrix–vector operations are used to reduce the floating-point operations and memory requirements. Numerical and parallel scalabilities of these algorithms are presented for the diffusion equation having spatially varying diffusion coefficient modeled by a non-Gaussian stochastic process. Scalability of the solvers with respect to the number of random variables is also investigated.

Method of confidence domains in the analysis of noise-induced extinction for tritrophic population system

A problem of the analysis of the noise-induced extinction in multidimensional population systems is considered. For the investigation of conditions of the extinction caused by random disturbances, a new approach based on the stochastic sensitivity function technique and confidence domains is suggested, and applied to tritrophic population model of interacting prey, predator and top predator. This approach allows us to analyze constructively the probabilistic mechanisms of the transition to the noise-induced extinction from both equilibrium and oscillatory regimes of coexistence. In this analysis, a method of principal directions for the reducing of the dimension of confidence domains is suggested. In the dispersion of random states, the principal subspace is defined by the ratio of eigenvalues of the stochastic sensitivity matrix. A detailed analysis of two scenarios of the noise-induced extinction in dependence on parameters of considered tritrophic system is carried out.

Micromagnetism in a planar system with a random magnetic anisotropy and two-dimensional magnetic correlations

The hysteresis loops and the micromagnetic structure of a ferromagnetic nanolayer with a randomly oriented local easy magnetization axis and two-dimensional magnetization correlations are studied using a micromagnetic simulation. The properties and the micromagnetic structure of the nanolayer are determined by the competition between the anisotropy and exchange energies and by the dipole–dipole interaction energy. The magnetic microstructure can be described as an ensemble of stochastic magnetic domains and topological magnetization defects. Dipole–dipole interaction suppresses the formation of topological magnetization defects. The topological defects in the magnetic microstructure can cause a sharper change in the coercive force with the crystallite size than that predicted by the random magnetic anisotropy model.

Stochastic variational multiscale analysis of the advection–diffusion equation: Advective–diffusive regime and multi-dimensional problems

We present the variational multiscale (VMS) formulation for the stochastic advection–diffusion equation in which the source of uncertainty may arise from random advective and diffusive fields (or material parameters), a random source term, or a combination among them. In this formulation, a stabilization parameter arises that, for the advection–diffusion equation, is a complicated function of the uncertain advection and diffusion parameters. To efficiently incorporate the stochastic stabilization parameter in the numerical method, we develop a projection-based approach to obtain an approximation of the stabilization parameter in the form of a generalized polynomial chaos (gPC) expansion. We demonstrate the current approach for problems which span the advective and diffusive regimes in the stochastic domain. Specifically, four cases are considered with multi-dimensional physical and stochastic domains. These cases include an advection-dominated case with an uncertain advection parameter, a case spanning both the advective and diffusive regimes with uncertain advection and diffusion parameters, a case with an uncertain source term, and a case with scalar transport under an uncertain advection field in a channel with multiple branches. For the latter case uniform and non-uniform meshes are employed. The expectation and variance from the stochastic Galerkin and VMS formulations are compared against their counterparts from analytical or Monte Carlo sampling methods where appropriate. In summary, an accurate and stable numerical solution is obtained using the VMS formulation with projection-based approximation of the stochastic stabilization parameter for multi-dimensional problems including non-trivial physical discretization.

Stochastic Galerkin approximation of the Reynolds equation with irregular film thickness

We consider the approximation of the Reynolds equation with an uncertain film thickness. The resulting stochastic partial differential equation is solved numerically by the stochastic Galerkin finite element method with high-order discretizations both in spatial and stochastic domains. We compute the pressure field of a journal bearing in various numerical examples that demonstrate the effectiveness and versatility of the approach. The results suggest that the stochastic Galerkin method is capable of supporting design when manufacturing imperfections are the main sources of uncertainty.

On features of magnetization self-organization in 1D stochastic ferromagnetic systems

The magnetic structure of a polycrystalline nanowire at the weak or missing magnetostatic interaction exhibits the special self-organization of magnetization. As is known, the magnetization structure forming in a random crystallographic anisotropy field has a characteristic length range, which involves tens and hundreds of crystallites. This leads to the occurrence of stochastic domains. The induced uniform anisotropy of magnetostatic nature or the texture co-directed with the crystallite anisotropy axes masks the picture of stochastic domains. Nevertheless, as we show, the information on stochastic domains remains in the magnetization structure. The experimental techniques for obtaining information on the magnetic properties of stochastic domains are proposed.

Creating Affective Autonomous Characters Using Planning in Partially Observable Stochastic Domains

The ability to reason about and respond to their own emotional states can enhance the believability of Non-Player Characters (NPCs). In this paper, we use a Partially Observable Markov Decision Process (POMDP)-based framework to model emotion over time. A two-level appraisal model, involving quick and reactive vs. slow and deliberate appraisals, is proposed for the creation of affective autonomous characters based on POMDPs, wherein the probability of goal satisfaction is used in an appraisal and reappraisal process for emotion generation. We not only extend Probabilistic Computation Tree Logic (PCTL) for reasoning about the properties of emotional states based on POMDPs but also illustrate how four reactive (primary) emotions and nine deliberate (secondary) emotions can be derived by combining PCTL with the belief-desire theory of emotion. The results of an empirical study suggest that the proposed model can be used to create characters that appear to be more believable and more intelligent.

Stochastic sensitivity analysis of noise-induced order-chaos transitions in discrete-time systems with tangent and crisis bifurcations

Abstract We study noise-induced order-chaos transitions in discrete-time systems with tangent and crisis bifurcations. To study these transitions parametrically, we suggest a generalized mathematical technique using stochastic sensitivity functions and confidence domains for randomly forced equilibria, cycles, and chaotic attractors. This technique is demonstrated in detail for the simple one-dimensional stochastic system, in which points of crisis and tangent bifurcations are borders of the order window lying between two chaotic parametric zones. A stochastic phenomenon of the extension and shift of this window towards crisis bifurcation point, under increasing noise, is presented and analyzed. Shifts of borders of this order window are found as functions of the noise intensity. By our analytical approach based on stochastic sensitivity functions, we construct a parametric diagram of chaotic and regular regimes for the stochastically forced system.

Learning Relational Dynamics of Stochastic Domains for Planning

Probabilistic planners are very flexible tools that can provide good solutions for difficult tasks. However, they rely on a model of the domain, which may be costly to either hand code or automatically learn for complex tasks. We propose a new learning approach that (a) requires only a set of state transitions to learn the model; (b) can cope with uncertainty in the effects; (c) uses a relational representation to generalize over different objects; and (d) in addition to action effects, it can also learn exogenous effects that are not related to any action, e.g., moving objects, endogenous growth and natural development. The proposed learning approach combines a multi-valued variant of inductive logic programming for the generation of candidate models, with an optimization method to select the best set of planning operators to model a problem. Finally, experimental validation is provided that shows improvements over previous work.

Explicit cost bounds of stochastic Galerkin approximations for parameterized PDEs with random coefficients

This work analyzes the overall computational complexity of the stochastic Galerkin finite element method (SGFEM) for approximating the solution of parameterized elliptic partial differential equations with both affine and non-affine random coefficients. To compute the fully discrete solution, such approaches employ a Galerkin projection in both the deterministic and stochastic domains, produced here by a combination of finite elements and a global orthogonal basis, defined on an isotopic total degree index set, respectively. To account for the sparsity of the resulting system, we present a rigorous cost analysis that considers the total number of coupled finite element systems that must be simultaneously solved in the SGFEM. However, to maintain sparsity as the coefficient becomes increasingly nonlinear in the parameterization, it is necessary to also approximate the coefficient by an additional orthogonal expansion. In this case we prove a rigorous complexity estimate for the number of floating point operations (FLOPs) required per matrix–vector multiplication of the coupled system. Based on such complexity estimates we also develop explicit cost bounds in terms of FLOPs to solve the stochastic Galerkin (SG) systems to a prescribed tolerance, which are used to compare with the minimal complexity estimates of a stochastic collocation finite element method (SCFEM), shown in our previous work (Galindo et al., 2015). Finally, computational evidence complements the theoretical estimates and supports our conclusion that, in the case that the coefficient is affine, the coupled SG system can be solved more efficiently than the decoupled SC systems. However, as the coefficient becomes more nonlinear, it becomes prohibitively expensive to obtain an approximation with the SGFEM.

Heaviside enriched extended stochastic FEM for problems with uncertain material interfaces

This paper is concerned with the modeling of heterogeneous materials with uncertain inclusion geometry. The eXtended stochastic finite element method (X-SFEM) is a recently proposed approach for modeling stochastic partial differential equations defined on random domains. The X-SFEM combines the deterministic eXtended finite element method (XFEM) with a polynomial chaos expansion (PCE) in the stochastic domain. The X-SFEM has been studied for random inclusion problems with a $$C^0$$C0-continuous solution at the inclusion interface. This work proposes a new formulation of the X-SFEM using the Heaviside enrichment for modeling problems with either continuous or discontinuous solutions at the uncertain inclusion interface. The Heaviside enrichment formulation employs multiple enrichment levels for each material subdomain which allows more complex inclusion geometry to be accurately modeled. A PCE is applied in the stochastic domain, and a random level set function implicitly defines the uncertain interface geometry. The Heaviside enrichment leads to a discontinuous solution in the spatial and stochastic domains. Adjusting the support of the stochastic approximation according to the active stochastic subdomain for each degree of freedom is proposed. Numerical examples for heat diffusion and linear elasticity are studied to illustrate convergence and accuracy of the scheme under spatial and stochastic refinements. In addition to problems with discontinuous solutions, the Heaviside enrichment is applicable to problems with $$C^0$$C0-continuous solutions by enforcing continuity at the interface. A higher convergence rate is achieved using the proposed Heaviside enriched X-SFEM for $$C^0$$C0-continuous problems when compared to using a $$C^0$$C0-continuous enrichment.

Hybrid approaches for multiple-species stochastic reaction–diffusion models

Reaction–diffusion models are used to describe systems in fields as diverse as physics, chemistry, ecology and biology. The fundamental quantities in such models are individual entities such as atoms and molecules, bacteria, cells or animals, which move and/or react in a stochastic manner. If the number of entities is large, accounting for each individual is inefficient, and often partial differential equation (PDE) models are used in which the stochastic behaviour of individuals is replaced by a description of the averaged, or mean behaviour of the system. In some situations the number of individuals is large in certain regions and small in others. In such cases, a stochastic model may be inefficient in one region, and a PDE model inaccurate in another. To overcome this problem, we develop a scheme which couples a stochastic reaction–diffusion system in one part of the domain with its mean field analogue, i.e. a discretised PDE model, in the other part of the domain. The interface in between the two domains occupies exactly one lattice site and is chosen such that the mean field description is still accurate there. In this way errors due to the flux between the domains are small. Our scheme can account for multiple dynamic interfaces separating multiple stochastic and deterministic domains, and the coupling between the domains conserves the total number of particles. The method preserves stochastic features such as extinction not observable in the mean field description, and is significantly faster to simulate on a computer than the pure stochastic model.

Components of Magnetic Anisotropy of Soft Magnetic Nanocrystalline Fe-Based Films

We present quantitative evaluation results of micromagnetic structure parameters of nanocrystalline Fe, Fe95Zr5, Fe90N10, and Fe85Zr5N10films obtained by magnetron sputtering. It is shown that quantities of magnetocrystallineK1, magnetoelasticKME, magnetostaticKMS, and surfaceKa,Sanisotropies are components of experimentally measured effective local anisotropyKeff. The shape of the hysteresis loops of the films is determined by the presence of two main macroscopic effective magnetic anisotropies, one of which is the anisotropy field of stochastic domains, and the other is the magnetoelastic anisotropy field due to residual macrostresses.

A Bayesian mixed shrinkage prior procedure for spatial–stochastic basis selection and evaluation of gPC expansions: Applications to elliptic SPDEs

We propose a new fully Bayesian method to efficiently obtain the spectral representation of a spatial random field, which can conduct spatial–stochastic basis selection and evaluation of generalized Polynomial Chaos (gPC) expansions when the number of the available basis functions is significantly larger than the size of the training data-set. We develop a fully Bayesian stochastic procedure, called mixed shrinkage prior (MSP), which performs both basis selection and coefficient evaluation simultaneously. MSP involves assigning a prior probability to the gPC structure and assigning conjugate priors to the expansion coefficients that can be thought of as mixtures of Ridge–LASSO shrinkage priors, in augmented form. The method offers a number of advantages over existing compressive sensing methods in gPC literature, such that it recovers possible sparse structures in both stochastic and spatial domains while the resulting expansion can be re-used directly to economically obtain results at any spatial input values. Yet, it inherits all the advantages of Bayesian model uncertainty methods, e.g. accounts for uncertainty about basis significance and provides interval estimation through posterior distributions. A unique highlight of the MSP procedure is that it can address heterogeneous sparsity in the spatial domain for different random dimensions. Furthermore, it yields a compromise between Ridge and LASSO regressions, and hence combines a weak (l2-norm) and strong (l1-norm) shrinkage, in an adaptive, data-driven manner. We demonstrate the good performance of the proposed method, and compare it against other existing compressive sensing ones on elliptic stochastic partial differential equations.

HIGH DIMENSIONAL SENSITIVITY ANALYSIS USING SURROGATE MODELING AND HIGH DIMENSIONAL MODEL REPRESENTATION

In this paper, a new non-intrusive method for the propagation of uncertainty and sensitivity analysis is presented. The method is based on the cut-HDMR approach, which is here derived in a different way and new conclusions are presented. The cut-HDMR approach decomposes the stochastic space into sub-domains, which are separately interpolated via a selected interpolation technique. This leads to a dramatic reduction of necessary samples for high dimensional spaces and decreases the influence of the Curse of Dimensionality. The proposed non-intrusive method is based on the coupling of an interpolation technique with the cut-HDMR (high dimension model representation) approach. The new conclusions obtained from the new derivation of the cut-HDMR approach allow one to interpolate each stochastic domain separately, including all stochastic variables and interactions between variables. Moreover, the same conclusions allow one to neglect non-important stochastic domains and therefore, drastically reduce the number of samples to detect and interpolate the higher order interactions. A new sampling strategy is introduced, which is based on a tensor product, but it uses the idea of Smoylak sparse grid for higher domains. For this work, the multi-dimensional Lagrange interpolation technique is selected and is applied for all parts of the cut-HDMR approach. However, the nature of the method allows one to use a combination of various interpolation techniques. The sensitivity analysis is performed on the surrogate model using the Monte Carlo sampling. In this work, the Sobol's approach is followed and sensitivity indices are established for each variable and interaction. Moreover, due to the obtained conclusions, the separate surrogate models allow one to visualize the uncertainty in the high dimensional space via histograms. The usage of a histogram for each stochastic domain allows one to establish full statistical properties of a given stochastic domain. This helps the user to better understand the stochastic propagation for the model of interest. The proposed interpolation technique and sensitivity analysis approach are tested on a simple example and applied on the well-known Borehole problem. Results of the proposed method are compared to the Monte Carlo sampling using the mean value and the standard deviation. Results of the sensitivity analysis of the Borehole case are compared to the literature results and the statistical visualization of each variable is provided.

The manifestations of the two-dimensional magnetic correlations in the nanocrystalline ribbons Fe64Co21B15

Abstract We report and discuss the grain size dependence of the coercivity and approach magnetization to saturation in the FeCoB ribbons. Instead of expected three-dimensional magnetic correlations for 20 μm thick ribbon with grain size about 10 nm, the observed behavior could be attributed to the two-dimensional stochastic magnetic domains and two dimensional magnetization ripple.