Stochastic Version Research Articles

Sparse conic reformulation of structured QCQPs based on copositive optimization with applications in stochastic optimization

AbstractRecently, Bomze et al. introduced a sparse conic relaxation of the scenario problem of a two stage stochastic version of the standard quadratic optimization problem. When compared numerically to Burer’s classical reformulation, the authors showed that there seems to be almost no difference in terms of solution quality, whereas the solution time can differ by orders of magnitudes. While the authors did find a very limited special case, for which Burer’s reformulation and their relaxation are equivalent, no satisfying explanation for the high quality of their bound was given. This article aims at shedding more light on this phenomenon and give a more thorough theoretical account of its inner workings. We argue that the quality of the outer approximation cannot be explained by traditional results on sparse conic relaxations based on positive semidenifnite or completely positive matrix completion, which require certain sparsity patterns characterized by chordal and block clique graphs respectively, and put certain restrictions on the type of conic constraint they seek to sparsify. In an effort to develop an alternative approach, we will provide a new type of convex reformulation of a large class of stochastic quadratically constrained quadratic optimization problems that is similar to Burer’s reformulation, but lifts the variables into a comparatively lower dimensional space. The reformulation rests on a generalization of the set-completely positive matrix cone. This cone can then be approximated via inner and outer approximations in order to obtain upper and lower bounds, which potentially close the optimality gap, and hence can give a certificate of exactness for these sparse reformulations outside of traditional, known sufficient conditions. Finally, we provide some numerical experiments, where we asses the quality of the inner and outer approximations, thereby showing that the approximations may indeed close the optimality gap in interesting cases.

Random permutations and queues

Given a growth rule which sequentially constructs random permutations of increasing degree, the stochastic process version of the rencontre problem asks what is the limiting proportion of time that the permutation has no fixed points (singleton cycles). We show that the discrete-time Chinese Restaurant Process (CRP) does not exhibit this limit. We then consider the related embedding of the CRP in continuous time and thereby show that it does have this and other limits of the time averages. By this embedding the cycle structure of the permutation can be represented as a tandem of infinite-server queues. We use this connection to show how results from the queuing theory can be interpreted in terms of the evolution of the cycle counts of permutations.

The spread of COVID-19 in London: Network effects and optimal lockdowns

We generalise a stochastic version of the workhorse SIR (Susceptible-Infectious-Removed) epidemiological model to account for spatial dynamics generated by network interactions. Using the London metropolitan area as a salient case study, we show that commuter network externalities account for about 42% of the propagation of COVID-19. We find that the UK lockdown measure reduced total propagation by 44%, with more than one third of the effect coming from the reduction in network externalities. Counterfactual analyses suggest that: (i) the lockdown was somehow late, but further delay would have had more extreme consequences; (ii) a targeted lockdown of a small number of highly connected geographic regions would have been equally effective, arguably with significantly lower economic costs; (iii) targeted lockdowns based on threshold number of cases are not effective, since they fail to account for network externalities.

How noise induces multi-stage transformations of oscillatory regimes in a thermochemical model

A thermochemical model in the parametric zone of hysteresis with coexisting equilibrium and oscillatory attractors is considered. For the stochastic version of the model, we study probabilistic mechanisms of multi-stage noise-induced transformations of oscillatory modes. These transformations are associated with stochastic P-bifurcations. Constructive abilities of the analytical method based on confidence domains are demonstrated. It is discussed which of the coexisting regimes under increasing noise is more dangerous in the context of temperature blow-ups.

Adaptive proximal SGD based on new estimating sequences for sparser ERM

Estimating sequences introduced by Nesterov is an efficient trick to accelerate gradient descent (GD). The stochastic version of estimating sequences is also successfully used to speed up stochastic gradient descent (SGD). In solving the non-smooth convex optimization problems, the convergence rate of SGD with stochastic estimating sequences is O(1/k). Here k is the number of iterations. In this paper, we present a new way of constructing estimating sequences. The characteristic of the new estimating sequences is to replace the subgradient with the proximal stochastic gradient. The novelty of the new estimating sequences is to replace the fixed learning rate with the adaptive learning rate. The adaptive learning rate is calculated by the exponential moving average of past squared stochastic gradients. Based on the new estimating sequences, we propose an adaptive proximal SGD algorithm, called ES-APSGD, for solving the large-scale ℓ1-norm regularized empirical risk minimization (ERM). The proposed ES-APSGD simplifies the calculation and can obtain a convergence rate of O(1/k2). The significant advantage of ES-APSGD is to strengthen the sparsity of solution by adaptively adjusting the threshold magnitude. Experimental results on Lasso and ℓ1-norm regularized logistic regression show that ES-APSGD speeds up convergence and obtains the sparser optimal solutions.

Canards Oscillations, Noise-Induced Splitting of Cycles and Transition to Chaos in Thermochemical Kinetics

We study how noise generates complex oscillatory regimes in the nonlinear thermochemical kinetics. In this study, the basic mathematical Zeldovich–Semenov model is used as a deterministic skeleton. We investigate the stochastic version of this model that takes into account multiplicative random fluctuations of temperature. In our study, we use direct numerical simulation of stochastic solutions with the subsequent statistical analysis of probability densities and Lyapunov exponents. In the parametric zone of Canard cycles, qualitative effects caused by random noise are identified and investigated. Stochastic P-bifurcations corresponding to noise-induced splitting of Canard oscillations are parametrically described. It is shown that such P-bifurcations are associated with splitting of both amplitudes and frequencies. Studying stochastic D-bifurcations, we localized the rather narrow parameter zone where transitions from order to chaos occur.

Strong solutions for the stochastic Allen-Cahn-Navier-Stokes system

We study in this article a stochastic version of a coupled Allen-Cahn-Navier-Stokes model on a bounded domain of . The model consists of the Navier-Stokes equations for the velocity, coupled with an Allen-Cahn model for the order (phase) parameter. We prove the existence and uniqueness of a local maximal strong solution when the initial data takes values in . Moreover in the two-dimensional case, we prove that our solution is global.

Sex, ducks, and rock "n" roll: Mathematical model of sexual response.

In this paper, we derive and analyze a mathematical model of a sexual response. As a starting point, we discuss two studies that proposed a connection between a sexual response cycle and a cusp catastrophe and explain why that connection is incorrect but suggests an analogy with excitable systems. This then serves as a basis for derivation of a phenomenological mathematical model of a sexual response, in which the variables represent levels of physiological and psychological arousal. Bifurcation analysis is performed to identify stability properties of the model's steady state, and numerical simulations are performed to illustrate different types of behavior that can be observed in the model. Solutions corresponding to the dynamics associated with the Masters-Johnson sexual response cycle are represented by "canard"-like trajectories that follow an unstable slow manifold before making a large excursion in the phase space. We also consider a stochastic version of the model, for which spectrum, variance, and coherence of stochastic oscillations around a deterministically stable steady state are found analytically, and confidence regions are computed. Large deviation theory is used to explore the possibility of stochastic escape from the neighborhood of the deterministically stable steady state, and the methods of an action plot and quasi-potential are employed to compute most probable escape paths. We discuss implications of the results for facilitating better quantitative understanding of the dynamics of a human sexual response and for improving clinical practice.

Effect of Transmission and Vaccination on Time to Dominance of Emerging Viral Strains: A Simulation-Based Study.

We studied the effect of transmissibility and vaccination on the time required for an emerging strain of an existing virus to dominate in the infected population using a simulation-based experiment. The emergent strain is assumed to be completely resistant to the available vaccine. A stochastic version of a modified SIR model for emerging viral strains was developed to simulate surveillance data for infections. The proportion of emergent viral strain infections among the infected was modeled using a logistic curve and the time to dominance (TTD) was recorded for each simulation. A factorial experiment was implemented to compare the TTD values for different transmissibility coefficients, vaccination rates, and initial vaccination coverage. We discovered a non-linear relationship between TTD and the relative transmissibility of the emergent strain for populations with low vaccination coverage. Furthermore, higher vaccination coverage and high vaccination rates in the population yielded significantly lower TTD values. Vaccinating susceptible individuals against the current strain increases the susceptible pool of the emergent virus, which leads to the emergent strain spreading faster and requiring less time to dominate the infected population.

Lagrangian stochastic model for the orientation of inertialess spheroidal particles in turbulent flows: An efficient numerical method for CFD approach

In this work, we propose a model for the orientation of inertialess spheroidal particles suspended in turbulent flows. This model consists in a stochastic version of the Jeffery equation that can be included in a statistical Lagrangian description of particles suspended in a flow. It is compatible and coherent with turbulence models that are widely used in CFD codes for the simulation of the flow field in practical large-scale applications. In this context, we propose and analyze a numerical scheme based on a splitting scheme algorithm that decouples the orientation dynamics into its main contributions: stretching and rotation. We detail its implementation in an open-source CFD software. We analyze the weak and strong convergence of both the global scheme and of each sub-part. Subsequently, the splitting technique yields to a highly efficient hybrid algorithm coupling pure probabilistic and deterministic numerical schemes. Various numerical experiments were implemented and the results were compared with analytical predictions of the model to assess the algorithm efficiency and accuracy.

A mathematical model with nonlinear relapse: conditions for a forward-backward bifurcation

ABSTRACT We constructed a Susceptible-Addicted-Reformed model and explored the dynamics of nonlinear relapse in the Reformed population. The transition from susceptible considered at-risk is modeled using a strictly decreasing general function, mimicking an influential factor that reduces the flow into the addicted class. The basic reproductive number is computed, which determines the local asymptotically stability of the addicted-free equilibrium. Conditions for a forward-backward bifurcation were established using the basic reproductive number and other threshold quantities. A stochastic version of the model is presented, and some numerical examples are shown. Results showed that the influence of the temporarily reformed individuals is highly sensitive to the initial addicted population.

Bistability and criticality in the stochastic Wilson-Cowan model.

We study a stochastic version of the Wilson-Cowan model of neural dynamics, where the response function of neurons grows faster than linearly above the threshold. The model shows a region of parameters where two attractive fixed points of the dynamics exist simultaneously. One fixed point is characterized by lower activity and scale-free critical behavior, while the second fixed point corresponds to a higher (supercritical) persistent activity, with small fluctuations around a mean value. When the number of neurons is not too large, the system can switch between these two different states with a probability depending on the parameters of the network. Along with alternation of states, the model displays a bimodal distribution of the avalanches of activity, with a power-law behavior corresponding to the critical state, and a bump of very large avalanches due to the high-activity supercritical state. The bistability is due to the presence of a first-order (discontinuous) transition in the phase diagram, and the observed critical behavior is connected with the line where the low-activity state becomes unstable (spinodal line).

Deterministic and stochastic phase-field modeling of anisotropic brittle fracture

We investigate variational phase-field formulations of anisotropic brittle fracture to model zigzag crack patterns in cubic materials. Our objective is twofold: (i) to analytically derive and numerically test the fundamental behavioral aspects predicted by the two main available fourth-order models, and to guide the calibration of their unknown parameters; (ii) motivated by the pronounced non-uniqueness of the phase-field solution in the anisotropic case and uncertainties in the material model, to transition from a deterministic to a stochastic model by introducing a material-related random field in the anisotropic phase-field energy functional. We employ Monte Carlo, randomized quasi-Monte Carlo and stochastic spectral methods to estimate statistical moments of the phase-field variable. The two fourth-order phase-field models are shown to predict a very different response both in their deterministic and stochastic versions. For either modeling choice, we believe that the stochastic approach, which captures several possible zigzag crack paths, holds significant promise to enable meaningful predictions of anisotropic fracture with phase-field models.

Calibration of quantum decision theory: aversion to large losses and predictability of probabilistic choices

We present the first calibration of quantum decision theory (QDT) to a dataset of binary risky choice. We quantitatively account for the fraction of choice reversals between two repetitions of the experiment, using a probabilistic choice formulation in the simplest form without model assumption or adjustable parameters. The prediction of choice reversal is then refined by introducing heterogeneity between decision makers through their differentiation into two groups: ‘majoritarian’ and ‘contrarian’ (in proportion 3:1). This supports the first fundamental tenet of QDT, which models choice as an inherent probabilistic process, where the probability of a prospect can be expressed as the sum of its utility and attraction factors. We propose to parameterize the utility factor with a stochastic version of cumulative prospect theory (logit-CPT), and the attraction factor with a constant absolute risk aversion function. For this dataset, and penalising the larger number of QDT parameters via the Wilks test of nested hypotheses, the QDT model is found to perform significantly better than logit-CPT at both the aggregate and individual levels, and for all considered fit criteria for the first experiment iteration and for predictions (second ‘out-of-sample’ iteration). The distinctive QDT effect captured by the attraction factor is mostly appreciable (i.e. most relevant and strongest in amplitude) for prospects with big losses. Our quantitative analysis of the experimental results supports the existence of an intrinsic limit of predictability, which is associated with the inherent probabilistic nature of choice. The results of the paper can find applications both in the prediction of choice of human decision makers as well as for organizing the operation of artificial intelligence.

Deep capsule encoder–decoder network for surrogate modeling and uncertainty quantification

AbstractWe propose a novel capsule‐based deep encoder–decoder model for surrogate modeling and uncertainty quantification of systems in mechanics from sparse data. The proposed framework is developed by adapting Capsule Network (CapsNet) architecture into an image‐to‐image regression encoder–decoder network. Specifically, the aim is to exploit the benefits of CapsNet over convolution neural network (CNN) – retaining pose and position information related to an entity to name a few. The performance of the proposed approach is illustrated by solving two different variants of elliptic partial differential equations (PDE): a stochastic version without a source term having an input dimensionality of 1024 and a deterministic pressure Poisson equation for flow past a cylinder. The first PDE, that is, the stochastic PDE (SPDE), also governs systems in mechanics such as steady heat conduction, groundwater flow, or other diffusion processes. In this article, the problem definition for this SPDE is such that it does not restrict the random diffusion field to a particular covariance structure, and the more strenuous task of response prediction for an arbitrary diffusion field is solved. Finally, we also evaluate the performance of our model on the uncertainty propagation problem based on this equation. The obtained results from the performance evaluation of the developed model on the mentioned problems indicate that the proposed approach is accurate, data efficient, and robust.

Dissipation Properties of Transport Noise in the Two-Layer Quasi-geostrophic Model

A stochastic version of the two-layer quasi-geostrophic model (2LQG) with multiplicative transport noise is analysed. This popular intermediate complexity model describes large scale atmosphere and ocean dynamics at the mid-latitudes. The transport noise, which acts on both layers, accounts for the unresolved small scales. After establishing the well-posedness of the perturbed equations, we show that, under a suitable scaling of the noise, the solutions converge to the deterministic 2LQG model with enhanced dissipation. Moreover, these solutions converge to the deterministic stationary ones on the long time horizon.

Dynamic event-triggered adaptive control for zero-sum games of nonlinear stochastic systems

This paper considers solving the zero-sum games (ZSGs) of nonlinear stochastic systems under the architecture of dynamic event-triggered mechanism (DETM). The adaptive dynamic programming (ADP) approach with great approximate computing power is proposed to seek the saddle-point equilibrium solution of ZSGs. First, the Hamilton–Jacobi–Isaacs equation of stochastic version is constructed which is the key to solving the two-player ZSGs. Then, the ADP method is employed to search the approximated solution under the dynamic triggering mechanism. On the foundation of static event-triggered mechanism (SETM), the DETM is derived through introducing an internal dynamic invariable to further lower the update frequency for the strategies. Additionally, the stability of the closed system is demonstrated in detail via the Lyapunov theory. Finally, two representative examples are provided to present the availability and reliability of this approach from different perspectives.

Investigation of nonlinear problems governed by stochastic phi-4 type equations in nuclear and particle physics

In this article, the stochastic Phi-4 model is under consideration analytically. The stochastic Phi-4 equation has been crucial to nuclear and particle physics. The randomness of the nuclear particle at the micro level is represented by the stochastic phenomena which are shown by the white noise. The stochastic version of the Phi-4 model is investigated for the first time analytically to gain the soliton solutions by using the two techniques such as Sardar sub-equation and modified exponential rational function. The stochastic soliton solutions are successfully constructed in the form of exponential, trigonometric, and hyperbolic forms. Additionally, the stability analysis of this model is observed. Lastly, the stochastic behavior of some solutions is shown in the form of 2D, and 3D by choosing the different values of parameters.

Pattern formation in nanodomain aggregates of membrane proteins.

Transmembrane proteins, including receptors for growth factors and adhesion, play an important role in cell growth and migration; aberrant spatiotemporal signaling of these membrane proteins may underlie many diseases. Recent studies found that the accurate signaling of these membrane proteins is governed by the nanodomain formations; examples include the formation of synaptic receptor domains and receptor nanodomains encaged in flat clathrin lattice (FCL). However, the underlying physical mechanisms for the formation of such nanodomains still remain unknown. In this work, we compared three different model formulations for nanodomain formation, including 1) deterministic Turing model, 2) stochastic Turing model, and 3) particle-based model, to explore how the biophysics underlying these different models can capture different aspects of nanodomain formation. These nanodomains do not have a large occupancy but are long-term stable, and exhibit changed size and number under extracellular stimulus. Although the deterministic Turing model can generate a stable pattern, it fails to capture the property of changing pattern size and number under stimulus. The stochastic version of Turing model compensates for this shortcoming in the deterministic formulation, but the amount of new nanodomain number is still much less than that observed in the experiment. However, the particle-based model is able to generate more new nanodomains than above Turing models, indicating that the diffusion of each molecule may be driven by a more precise way than the concentration gradient. These findings may shed light on the mechanisms of nanodomain formation of membrane proteins.

Dynamics of a stochastic multi-stage sheep brucellosis model with incomplete immunity

This paper considered a multi-stage sheep brucellosis model with incomplete immunity. First, we established a deterministic model, calculated the basic reproduction number [Formula: see text], set out the conditions for the global stability of the disease-free equilibrium and endemic equilibrium. Second, considering the influence of environmental white noise on brucellosis infection, we further established the stochastic version of the model. By constructing a suitable Lyapunov function, we proved the existence and uniqueness of the global positive solution. Further, we got the sufficient conditions for disease extinction and the existence of ergodic stationary distribution. Finally, we carried out some numerical simulations to verify the theoretical results.