Deterministic Rate Research Articles

Deterministic computation of quantiles in a Lipschitz framework

In this article, we focus on computing the quantiles of a random variable f(X), where X is a [0,1]d-valued random variable, d∈N∗, and f:[0,1]d→R is a deterministic Lipschitz function. We are particularly interested in scenarios where the cost of a single function evaluation is high, while the law of X is assumed to be known. In this context, we propose a deterministic algorithm to compute deterministic lower and upper bounds for the quantile of f(X) at a given level α∈(0,1). With a fixed budget of N function calls, we demonstrate that our algorithm achieves an exponential deterministic convergence rate for d=1 (O(ρN) with ρ∈(0,1)) and a polynomial deterministic convergence rate for d>1 (O(N−1d−1)) and show the optimality of those rates. Furthermore, we design two algorithms, depending on whether the Lipschitz constant of f is known or unknown.

Deterministic Multi-Objective Optimization of Analog Circuits

Stochastic optimization approaches benefit from random variance to produce a solution in a reasonable time frame that is good enough for solving the problem. Compared with them, deterministic optimization methods feature faster convergence rates and better reproducibility but may get stuck at a local optimum that is insufficient to solve the problem. In this paper, we propose a group-based deterministic optimization method, which can efficiently achieve comparable performance to heuristic optimization algorithms, such as particle swarm optimization. Moreover, the weighted sum method (WSM) is employed to further improve our deterministic optimization method to be multi-objective optimization, making it able to seek a balance among multiple conflicting circuit performance metrics. With a case study of three common analog circuits tested for our optimization methodology, the experimental results demonstrate that our proposed method can more efficiently reach a better estimation of the Pareto front compared to NSGA-II, a well-known multi-objective optimization approach.

Rolling bearing fault diagnosis based on multi-domain features and whale optimized support vector machine

Rolling bearing is an important rotating support component in mechanical equipment. It is very prone to wear, defects, and other faults, which directly affect the reliable operation of mechanical equipment. Its running condition monitoring and fault diagnosis have always been a matter of concern to engineers and researchers. A rolling bearing fault diagnosis technique based on multi-domain feature and whale optimization algorithm-support vector machine (MDF-WOA-SVM) is proposed. Firstly, recursive analysis is performed on vibration signal and the recursive features are employed as nonlinear recursive feature vector including recursive rate (RR), deterministic rate (DET), recursive entropy (RE), and diagonal average length (DAL). Then, a comprehensive multi-domain feature vector is constructed by combining three time-domain features including root mean square, variance, and peak to peak. Finally, whale optimization algorithm (WOA) is introduced to optimize the penalty factor C and kernel function parameter g to construct the optimal WOA-SVM model. The rolling bearing datasets of Jiangnan University is employed for instance analysis, and the results show that the 10-CV accuracy of the technique proposed is good with an accuracy of 99%. Compared with recursive features or time-domain features, multi-domain features are more accurate and comprehensive in describing characters of the signal. Some popular supervised learning models are also introduced for comparison including K-nearest neighbor (KNN) and decision tree (DT), and the result shows that the proposed method has a higher accuracy and certain advantages.

Pre-electoral coalition agreement from the Black–Scholes point of view

A political party can be considered as a company whose value depends on the voters support i.e. on the percentage of population supporting the party. Dynamics of the support is thus as a stochastic process with a deterministic growth rate perturbed by a white noise modeled through the Wiener process. This is in an analogy with the option modeling where the stock price behaves similarly as the voters’ support. While in the option theory we have the question of fair price of an option, the question that we ask here is what is a reasonable level of support that the coalition of a “major” party (safely above the election threshold) and a “minor” party (under or around the election threshold) should achieve in order for the “minor” party to get one more representative. We shall elaborate some of the conclusions in the case of recent elections in Montenegro (June, 2023) which are particularly interesting due to lots of political subjects entering the race.

Average decayed dynamics of one-step transformation process with randomly varying return transition rate

The problem of stochastic averaging of the decayed dynamics of the output state population of a one-step transformation process with constant deterministic forward transition rate and randomly varying return transition rate is solved in approximation where random variation is modeled as a dichotomous stochastic process. The form of the obtained solution represented as a product between bimodal sigmoid rise of average population and its unimodal exponential decay is shown to largely be dependent on the stochastic frequency and amplitude parameters. For example, at high stochastic frequency, the behavior of population is reduced to that of a decayed one-step deterministic system. However, for resonance stochastic amplitude at low stochastic frequency, such behavior coincides with that of three-exponential rise-decay kinetics typical rather of a three-step deterministic slowly decaying process. Thus, there is an equivalence between using a more complex deterministic kinetic model and a less complex stochastic kinetic model for describing the decayed dynamics of different irreversible systems.

Universally optimal staffing of Erlang-A queues facing uncertain arrival rates

In many service systems, the staffing decisions must be made before the arrival rate is known with certainty. Thus, it is more appropriate to consider the arrival rate as a random variable at the time of the staffing decision. Motivated by this observation, we study the staffing problem in a service system modeled as an Erlang-A queue facing a random arrival rate. For linear staffing costs, linear waiting costs, and a cost per customer abandonment, we propose a policy that is based on modifying the well-known square-root safety staffing policy to explicitly account for the randomness in the arrival rate. Our primary contribution is to show that our proposed policy is “universally optimal”, i.e., irrespective of the magnitude of randomness in the arrival rate, the optimality gap between our proposed policy and the exact optimal policy remains bounded as the system size grows large. This is important because earlier performance guarantees for Erlang-A queues either (1) are not universal and offer performance guarantees that depend on the magnitude of uncertainty in the arrival rate or (2) are universal but assume a deterministic arrival rate. The practical relevance of this provable robustness is that our proposed policy is a “one-size-fits-all” as it is guaranteed to perform well for all levels of arrival rate uncertainty.

Stochastic Chain-Ladder Reserving with Modeled General Inflation

We consider two possible approaches to the problem of incorporating explicit general (i.e., economic) inflation in the non-life claims reserve estimates and the corresponding reserve SCR, defined—as in Solvency II—under the one-year view. What we call the actuarial approach provides a simplified solution to the problem, obtained under the assumption of deterministic interest rates and absence of inflation risk premia. The market approach seeks to eliminate these shortcomings by combining a stochastic claims reserving model with a stochastic market model for nominal and real interest rates. The problem is studied in details referring to the stochastic chain-ladder provided by the Over-dispersed Poisson model. The application of the two approaches is illustrated by a worked example based on market data.

Age-Optimal Scheduling Over Hybrid Channels

We consider the problem of minimizing the age of information when a source can transmit status updates over two heterogeneous channels. Our work is motivated by recent developments in 5 G mmWave technology, where transmissions may occur over an unreliable but fast (e.g., mmWave) channel or a slow reliable (e.g., sub-6 GHz) channel. The unreliable channel is modeled as a time-correlated Gilbert-Elliot channel at a high rate when the channel is in the “ON” state. The reliable channel provides a deterministic but lower data rate. The scheduling strategy determines the channel to be used for transmission in each time slot, aiming to minimize the time-average age of information (AoI). The optimal scheduling problem is formulated as a Markov Decision Process (MDP), which is challenging to solve because super-modularity does not hold in a part of the state space. We address this challenge and show that a multi-dimensional threshold-type scheduling policy is optimal for minimizing the age. By exploiting the structure of the MDP and analyzing the discrete time Markov chains (DTMCs) of the threshold-type policy, we devise a low-complexity bisection algorithm to compute the optimal thresholds. We compare different scheduling policies using numerical simulations.

Experimental investigations through modeling and optimization for fabrication of fused silica in medium-pressure plasma process

Due to its deterministic high material removal rate (MRR) and exceptional surface finish, the medium-pressure plasma process, which is based solely on chemical etching, has been developed as a promising fabrication method for fused silica. In the present investigation, the input parameters, such as radio-frequency power, total pressure, and processing time, are investigated and optimized for MRR and % change in surface roughness (%ΔRa) using response surface methodology. The process parameters, i.e., pressure ratio (SF6/O2) of 1:1 and gas composition (He: (SF6+O2)) of 90:10, have been kept constant during all experiments. The optimized value achieved for MRR and %ΔRa is 0.29 mm3/min and 3.4%, at RF power of 80 W, total pressure of 11.6 mbar, and processing time of 60 min. The result depicts that surface roughness marginally changes from 2.79 to 2.82 μm after plasma processing at optimum parametric conditions. Moreover, FESEM result shows that the cracks and irregularity have been reduced. EDX results reveal that elements silicon, oxygen, fluorine, and carbon appear on the plasma processed surface, showing the reaction occurring during plasma processing. Moreover, XPS results show the peaks of silicon (Si2p, Si2s), carbon (C1s), oxygen (O1s), and fluorine (F2p, F2s, F1s) on the processed surface of fused silica.

Determinism of indicators of child disability due to mental disorders in the conditions of its registered and calculated level

The purpose of the study was to identify the presence of deterministic rates of childhood disability due to mental disorders in the context of its registered (recorded) and calculated levels.
 Material and methods. We examined associations between registered, counted and uncounted disability and integral measures of children with mental disorders in 0–14, 15–17 and 0–17years groups. In addition, health resource and activity measures, mental disorder morbidity rates, demographic indicators and their association with child disability rates were examined. Period of analysis was from 2010 to 2020. The study is analytic, non-personalized, retrospective, not randomised. 
 Results. In the Arkhangelsk region, in the 0–14 years group the rate of calculated total disability was on average 55.9% higher than the rate of registered disability. The underreporting of total disability was higher than in the 15–17 years group. We found no significant links between demographic, socio-economic, health resource indicators and the level of official childhood disability.
 Limitations of the study. To avoid errors of extrapolation, caution should be exercised when attempting to scale the results obtained in the study to the population of other regions or the country as a whole, other types of disability, separate age and gender groups, and individual cases of disability. 
 Conclusions. In the Arkhangelsk region there is underreporting of childhood disability caused by mental disorders. The levels of children’s disability due to mental disorders recorded by official statistics have no significant relationship with indicators reflecting the demographic, socio-economic situation, resources and health care activities. The estimated and unrecorded levels of disability are found to be significantly associated with indicators of socioeconomic status, health care resources, and the overall incidence of mental disorders.

Modified Smoluchowski Rate Equations for Aggregation and Fragmentation in Finite Systems.

Protein self-assembly into supramolecular structures is important for cell biology. Theoretical methods employed to investigate protein aggregation and analogous processes include molecular dynamics simulations, stochastic models, and deterministic rate equations based on the mass-action law. In molecular dynamics simulations, the computation cost limits the system size, simulation length, and number of simulation repeats. Therefore, it is of practical interest to develop new methods for the kinetic analysis of simulations. In this work we consider the Smoluchowski rate equations modified to account for reversible aggregation in finite systems. We present several examples and argue that the modified Smoluchowski equations combined with Monte Carlo simulations of the corresponding master equation provide an effective tool for developing kinetic models of peptide aggregation in molecular dynamics simulations.

Workload analysis of a two-queue fluid polling model

AbstractIn this paper, we analyze a two-queue random time-limited Markov-modulated polling model. In the first part of the paper, we investigate the fluid version: fluid arrives at the two queues as two independent flows with deterministic rate. There is a single server that serves both queues at constant speeds. The server spends an exponentially distributed amount of time in each queue. After the completion of such a visit time to one queue, the server instantly switches to the other queue, i.e., there is no switch-over time.For this model, we first derive the Laplace–Stieltjes transform (LST) of the stationary marginal fluid content/workload at each queue. Subsequently, we derive a functional equation for the LST of the two-dimensional workload distribution that leads to a Riemann–Hilbert boundary value problem (BVP). After taking a heavy-traffic limit, and restricting ourselves to the symmetric case, the BVP simplifies and can be solved explicitly.In the second part of the paper, allowing for more general (Lévy) input processes and server switching policies, we investigate the transient process limit of the joint workload in heavy traffic. Again solving a BVP, we determine the stationary distribution of the limiting process. We show that, in the symmetric case, this distribution coincides with our earlier solution of the BVP, implying that in this case the two limits (stationarity and heavy traffic) commute.

Is It Feasible to Preserve a Self-Sustaining Population of Yangtze Finless Porpoise in the Highest Density Section of Yangtze River?

Using the VORTEX v. 10. 5.0.0, population viability analysis (PVA) was performed for Yangtze finless porpoise (YFP, Neophocaena asiaeorientalis) in the highest density section between Hukou and Meilong section (HMS) of the Yangtze River. Baseline model showed that this population was in a relatively vulnerability state; the deterministic growth rate (Det-r) was −0.0230; the stochastic growth rate (Stoch-r) was −0.0385; the probability of extinction (PE) was 0.5690; the mean population size of extant populations (N-extant) was 22; the genetic diversity (GD) was 0.7698. Under the general protection model, the Det-r was 0.0015, and the Stoch-r was −0.0092; Under the medium protection model, the Det-r was 0.0219, and the Stoch-r was 0.0144; Under the optimal protection model, the Det-r was 0.0383, and the Stoch-r was 0.0357. Sensitivity analysis found that adult females breeding rate, sex ratio at birth, and mortality rate of juvenile YFP were sensitive to maintaining population stability. The PVA showed that the conservation of YFP population in HMS depends on: neutralization of all threats affecting YFP population in the HMS; maintenance and, whenever possible, enhancement of the functional connectivity of the waterbody, increasing the food resources of YFP and reducing the risk of injury to YFP caused by human.

Deterministic generation of indistinguishable photons in a cluster state

Entanglement between particles is a basic concept of quantum sciences. The ability to produce entangled particles in a controllable manner is essential for any quantum technology. Entanglement between light particles (photons) is particularly crucial for quantum communication due to light’s non-interactive nature and long-lasting coherence. Resources producing entangled multiphoton cluster states will enable communication between remote quantum nodes, as the inbuilt redundancy of cluster photons allows for repeated local measurements—compensating for losses and probabilistic Bell measurements. For feasible applications, the cluster generation should be fast, deterministic and, most importantly, its photons indistinguishable, which will allow measurements and fusion of clusters by interfering photons. Here, using periodic excitation of a semiconductor quantum-dot-confined spin, we demonstrate a multi-indistinguishable photon cluster, featuring a continuously generated string of photons at deterministic gigahertz generation rates, and an optimized entanglement length of about ten photons. The indistinguishability of the photons opens up new possibilities for scaling up the cluster’s dimensionality by fusion, thus building graph states suited for measurement-based photonic quantum computers and all-photonic quantum repeaters.

Dynamic mean-downside risk portfolio selection with a stochastic interest rate in continuous-time

Even though it has long been agreed that the interest rate is driven by a stochastic process, most of the existing studies on dynamic mean-downside risk portfolio optimization problem focuses on deterministic interest rates. This work investigates a continuous-time mean-downside risk portfolio optimization problem with a stochastic interest rate. More specifically, we introduce the Vasicek interest rate model and choose some common downside risk measures to model our risk measures, such as, the lower-partial moments(LPM), value-at-risk(VaR) and conditional value-at-risk(CVaR). By using the martingale method and the inverse Fourier Transformation, we successfully derive the semi-analytical optimal portfolio policies and the optimal wealth processes for the mean-downside risk measures with stochastic interest rate. Finally, we provide some illustrative examples to show how the stochastic interest rate affects the investment behavior of investors with mean-downside risk preferences.

On the computation of equilibria in monotone and potential stochastic hierarchical games

We consider a class of noncooperative hierarchical $${\textbf{N}}$$ -player games where the ith player solves a parametrized stochastic mathematical program with equilibrium constraints (MPEC) with the caveat that the implicit form of the ith player’s in MPEC is convex in player strategy, given rival decisions. Few, if any, general purpose schemes exist for computing equilibria, motivating the development of computational schemes in two regimes: (a) Monotone regimes. When player-specific implicit problems are convex, then the necessary and sufficient equilibrium conditions are given by a stochastic inclusion. Under a monotonicity assumption on the operator, we develop a variance-reduced stochastic proximal-point scheme that achieves deterministic rates of convergence in terms of solving proximal-point problems in monotone/strongly monotone regimes with optimal or near-optimal sample-complexity guarantees. Finally, the generated sequences are shown to converge to an equilibrium in an almost-sure sense in both monotone and strongly monotone regimes; (b) Potentiality. When the implicit form of the game admits a potential function, we develop an asynchronous relaxed inexact smoothed proximal best-response framework, requiring the efficient computation of an approximate solution of an MPEC with a strongly convex implicit objective. To this end, we consider an $$\eta $$ -smoothed counterpart of this game where each player’s problem is smoothed via randomized smoothing. In fact, a Nash equilibrium of the smoothed counterpart is an $$\eta $$ -approximate Nash equilibrium of the original game. Our proposed scheme produces a sequence and a relaxed variant that converges almost surely to an $$\eta $$ -approximate Nash equilibrium. This scheme is reliant on resolving the proximal problem, a stochastic MPEC whose implicit form has a strongly convex objective, with increasing accuracy in finite-time. The smoothing framework allows for developing a variance-reduced zeroth-order scheme for such problems that admits a fast rate of convergence. Numerical studies on a class of multi-leader multi-follower games suggest that variance-reduced proximal schemes provide significantly better accuracy with far lower run-times. The relaxed best-response scheme scales well with problem size and generally displays more stability than its unrelaxed counterpart.

Operator splitting method for the stochastic production–inventory model equation

Stochastic optimal control of an inventory model with a deterministic rate of deteriorating items is first provided by Alshamerni (2013). The main difficulty of solving this partial differential equation is the non-linear term (ux)2 which has no exactly meaning from mathematical view. The normal method to obtain the analytic solution is to give the conjecture that its solution takes the given form (quadratic). Then we solve the ordinary differential equation with initial or terminal conditions. There are two drawbacks for such method: (1) We do not know the solution curve tendency with respect to the time t; (2) Solve the ODE system directly is complicated computing. Instead we apply the operator splitting method after Cole–Hopf transformation for the initial equation. Split the partial differential equation into two parts, each part can be solved with an analytical solution. Numerical application of the method will be presented to verify the result.

Regret Analysis of a Markov Policy Gradient Algorithm for Multiarm Bandits

We consider a policy gradient algorithm applied to a finite-arm bandit problem with Bernoulli rewards. We allow learning rates to depend on the current state of the algorithm rather than using a deterministic time-decreasing learning rate. The state of the algorithm forms a Markov chain on the probability simplex. We apply Foster–Lyapunov techniques to analyze the stability of this Markov chain. We prove that, if learning rates are well-chosen, then the policy gradient algorithm is a transient Markov chain, and the state of the chain converges on the optimal arm with logarithmic or polylogarithmic regret.

Regulation of stem cell dynamics through volume exclusion

The maintenance and regeneration of adult tissues rely on the self-renewal of stem cells. Regeneration without over-proliferation requires precise regulation of the stem cell proliferation and differentiation rates. The nature of such regulatory mechanisms in different tissues, and how to incorporate them in models of stem cell population dynamics, is incompletely understood. The critical birth-death (CBD) process is widely used to model stem cell populations, capturing key phenomena, such as scaling laws in clone size distributions. However, the CBD process neglects regulatory mechanisms. Here, we propose the birth-death process with volume exclusion (vBD), a variation of the birth-death process that considers crowding effects, such as may arise due to limited space in a stem cell niche. While the deterministic rate equations predict a single non-trivial attracting steady state, the master equation predicts extinction and transient distributions of stem cell numbers with three possible behaviours: long-lived quasi-steady state (QSS), and short-lived bimodal or unimodal distributions. In all cases, we approximate solutions to the vBD master equation using a renormalized system-size expansion, QSS approximation and the Wentzel–Kramers–Brillouin method. Our study suggests that the size distribution of a stem cell population bears signatures that are useful to detect negative feedback mediated via volume exclusion.

Determinism and invariant measures for diffusing passive scalars advected by unsteady random shear flows

We study a complicated fluid flow generated by a robotic arm moving randomly back and forth that is shown to stir a tracer in an unexpectedly simple way. One might expect that such random motions would lead to highly random observations. But it turns out that after a certain, fixed amount of time, a broad class of random motions leads to an enhanced non-random, deterministic tracer spreading rate. Looking closer, the tracer shows non-Gaussian random fluctuations which are explicitly computed.