Stochastic Projection Method Research Articles

Dynamic stochastic projection method for multistage stochastic variational inequalities

Dynamic stochastic projection method for multistage stochastic variational inequalities

Stochastic Simulation Algorithm for Solving the System of Lame Equations for Two- and Three-Dimensional Domains by Combining the Slobodianskii Representation, the Method of Fundamental Solutions and a Stochastic Projection Method

Stochastic Simulation Algorithm for Solving the System of Lame Equations for Two- and Three-Dimensional Domains by Combining the Slobodianskii Representation, the Method of Fundamental Solutions and a Stochastic Projection Method

Stochastic simulation of magnetically controlled convection in porous media with random porosity via Karhunen-Loève expansion and intrusive polynomial chaos expansion

Understanding and quantifying uncertainty factors are crucial for accurately predicting thermomagnetic convection phenomena. This study presented a mathematical model and algorithm framework for uncertainty analysis of thermomagnetic convection in porous media with random porosity. The proposed method combined Karhunen-Loève expansion and intrusive polynomial chaos expansion to represent the input random parameters and output response, respectively. The Galerkin projection method was employed to decouple the stochastic governing equations of thermomagnetic convection into deterministic governing equations that can be efficiently solved using the finite volume method. By solving the polynomial chaos expansion of the output response and employing the stochastic projection method to solve the associated decoupled governing equations, the temporal evolution and statistical characteristics of the output response were obtained. The study revealed that porosity uncertainty in the porous medium affects the thermomagnetic convection of paramagnetic fluid, exhibiting significant chaos effects. Monte Carlo simulations were performed to validate the accuracy of the proposed approach and demonstrate its computational efficiency compared to traditional Monte Carlo methods. This research provides valuable insights into the uncertainty analysis of thermomagnetic convection and offers a promising methodology for analyzing heat transfer in various porous media applications.

Mini-batch stochastic subgradient for functional constrained optimization

In this paper, we consider finite sum composite optimization problems with many functional constraints. The objective function is expressed as a finite sum of two terms, one of which admits easy computation of (sub)gradients while the other is amenable to proximal evaluations. We assume a generalized bounded gradient condition on the objective which allows us to simultaneously tackle both smooth and nonsmooth problems. We also consider the cases of both with and without a quadratic functional growth property. Further, we assume that each constraint set is given as the level set of a convex but not necessarily a differentiable function. We reformulate the constrained finite sum problem into a stochastic optimization problem for which the stochastic subgradient projection method from Necoara and Singh [Stochastic subgradient projection methods for composite optimization with functional constraints; 2022 Journal of Machine Learning Research, 23, 1–35] specializes in a collection of mini-batch variants, with different mini-batch sizes for the objective function and functional constraints, respectively. More specifically, at each iteration, our algorithm takes a mini-batch stochastic proximal subgradient step aimed at minimizing the objective function and then a subsequent mini-batch subgradient projection step minimizing the feasibility violation. By specializing in different mini-batching strategies, we derive exact expressions for the stepsizes as a function of the mini-batch size and in some cases we also derive insightful stepsize-switching rules which describe when one should switch from a constant to a decreasing stepsize regime. We also prove sublinear convergence rates for the mini-batch subgradient projection algorithm which depend explicitly on the mini-batch sizes and on the properties of the objective function. Numerical results also show a better performance of our mini-batch scheme over its single-batch counterpart.

A stochastic selective projection method for solving a constrained stochastic variational inequality problem

A stochastic selective projection method for solving a constrained stochastic variational inequality problem

Delamination analysis of 3D-printed nylon reinforced with continuous carbon fibers

The inconsistent interfacial bonding of 3D-printed materials is accompanied by increasing scattering of the measured fracture toughness in delamination tests. This article focuses on the analysis of the delamination tests using theories developed for conventional composites. To this end, nylon 3D-printed specimens reinforced with carbon fibers are subjected to the mode I Double Cantilever Beam (DCB) and the mode II End-Loaded Split (ELS) tests. Several analytical methods and numerical models are employed to measure the fracture toughness and simulate the behavior of the specimen during delamination. The uncertainty of the data is considered by expanding the deterministic character of the models utilizing the non-intrusive stochastic pseudo-spectral projection method. The results of the simulations are well in agreement with the experimental results and support the use of stochastic techniques in combination with linear fracture mechanics for the delamination analysis of 3D-printed composites.

Inexact stochastic subgradient projection method for stochastic equilibrium problems with nonmonotone bifunctions: application to expected risk minimization in machine learning

This paper discusses a stochastic equilibrium problem for which the function is in the form of the expectation of nonmonotone bifunctions and the constraint set is closed and convex. This problem includes various applications such as stochastic variational inequalities, stochastic Nash equilibrium problems, and nonconvex stochastic optimization problems. For solving this stochastic equilibrium problem, we propose an inexact stochastic subgradient projection method. The proposed method sets a random realization of the bifunction and then updates its approximation by using both its stochastic subgradient and the projection onto the constraint set. The main contribution of this paper is to present a convergence analysis showing that, under certain assumptions, any accumulation point of the sequence generated by the proposed method using a constant step size almost surely belongs to the solution set of the stochastic equilibrium problem. A convergence rate analysis of the method is also provided to illustrate the method’s efficiency. Another contribution of this paper is to show that a machine learning algorithm based on the proposed method achieves the expected risk minimization for a class of least absolute selection and shrinkage operator (lasso) problems in statistical learning with sparsity. Numerical comparisons of the proposed machine learning algorithm with existing machine learning algorithms for the expected risk minimization using LIBSVM datasets demonstrate the effectiveness and superior classification accuracy of the proposed algorithm.

Моделі оптимізації завантаження мережі розподілених ситуаційних центрів

The article substantiates the relevance of the creation and formulates a mathematical statement of the problem of technical interoperability maintenance in a network of distributed situational centers based on convergent technological solutions. Technical interoperability is considered in the context of optimizing the distribution of situational management tasks between individual situation centers in a distributed network and minimizing information processing time. For this purpose the model and algorithms of optimization of performance of tasks by participants of process of situational management in a network of the distributed situational centers are offered. The model of technical interoperability optimization is presented in the form of a zero-one integer programming problem with the search for the local extremum of the objective function. The information-analytical process is formed as a request that determines the need for information resources, analytical and expert capabilities and service time in each local situation center. The task of optimizing the load of the network of distributed situational centers is to redistribute the load between network nodes by equalizing the use of organizational, human, technical and information resources for timely and quality decision-making in situational centers that serve participants in situational management at different levels. The problem of minimizing service time in a network of distributed situational centers is dual to the primal problem of load redistribution and is solved using a stochastic quasigradient projection method. Algorithms for solving the formulated problems are developed and a test example of application is given. The potential for further improvement of situational systems of strategic planning management based on capability management for target systems is shown

Krahn's sales territory for PPG precipitated silica extended to cover Benelux

A well designed territory enhances customer coverage, increases sales, fosters fair performance and rewards systems and lower travel cost. This paper considers a real life case study to design a sales territory for a business sales plan. The business plan consists in assigning the optimal quantity of sellers to a territory including the scheduling and routing plans for each seller. The problem is formulated as a combination of assignment, scheduling and routing optimization problems. The solution approach considers a meta-heuristic using stochastic iterative projection method for large systems. Several real life instances of different sizes were tested with stochastic data to represent raise/fall in the customers demand as well as the appearance/loss of customers.

Effects of coastal acidification on North Atlantic bivalves: interpreting laboratory responses in the context of in situ populations.

Experimental exposure of early life stage bivalves has documented negative effects of elevated pCO2 on survival and growth, but the population consequences of these effects are unknown. Following standard practices from population viability analysis and wildlife risk assessment, we substituted laboratory-derived stress-response relationships into baseline population models of Mercenaria mercenaria and Argopecten irradians. The models were constructed using inverse demographic analyses with time series of size-structured field data in NY, USA, whereas the stress-response relationships were developed using data from a series of previously published laboratory studies. We used stochastic projection methods and diffusion approximations of extinction probability to estimate cumulative risk of 50% population decline during ten-year population projections at 1, 1.5 and 2 times ambient pCO2 levels. Although the A. irradians population exhibited higher growth in the field data (12% per year) than the declining M. mercenaria population (-8% per year), cumulative risk was high for A. irradians in the first ten years due to high variance in the stochastic growth rate estimate (log λs = -0.02, σ2 = 0.24). This ten-year cumulative risk increased from 69% to 94% and >99% at 1.5 and 2 times ambient scenarios. For M. mercenaria (log λs = -0.09, σ2 = 0.01), ten-year risk was 81%, 96% and >99% at 1, 1.5 and 2 times ambient pCO2, respectively. These estimates of risk could be improved with detailed consideration of harvest effects, disease, restocking, compensatory responses, other ecological complexities, and the nature of interactions between these and other effects that are beyond the scope of available data. However, results clearly indicate that early life stage responses to plausible levels of pCO2 enrichment have the potential to cause significant increases in risk to these marine bivalve populations.

UNCERTAINTY QUANTIFICATION FOR NATURAL CONVECTION IN RANDOM POROUS MEDIA WITH INTRUSIVE POLYNOMIAL CHAOS EXPANSION

The paper presents a framework for the construction of a stochastic projection method for the natural convection in random porous media under the local thermal nonequilibrium conditions. In this approach, the input stochastic field (porosity) is represented by the Karhunen−Loeve expansion and the output fields (e.g., velocity, pressure, fluid phase temperature, and solid phase temperature) are expressed by the polynomial chaos expansion. Using the spectral decomposition, the stochastic problem in random porous media is reformulated to a set of deterministic problems to be solved for each polynomial chaos. A stochastic projection method is implemented to obtain the chaos coefficients in the corresponding deterministic governing equations and the statistics of the numerical solution is obtained. The prediction results are compared against results obtained using a Monte Carlo method. Excellent agreement between these results indicates the efficiency and accuracy of the proposed method.

A Computation-Efficient Decentralized Algorithm for Composite Constrained Optimization

This paper focuses on solving the problem of composite constrained convex optimization with a sum of smooth convex functions and non-smooth regularization terms (ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm) subject to locally general constraints. Motivated by the modern large-scale information processing problems in machine learning (the samples of a training dataset are randomly decentralized across multiple computing nodes), each of the smooth objective functions is further considered as the average of several constituent functions. To address the problem in a decentralized fashion, we propose a novel computation-efficient decentralized stochastic gradient algorithm, which leverages the variance reduction technique and the decentralized stochastic gradient projection method with constant step-size. Theoretical analysis indicates that if the constant step-size is less than an explicitly estimated upper bound, the proposed algorithm can find the exact optimal solution in expectation when each constituent function (smooth) is strongly convex. Concerning the existing decentralized schemes, the proposed algorithm not only is suitable for solving the general constrained optimization problems but also possesses low computation cost in terms of the total number of local gradient evaluations. Furthermore, the proposed algorithm via differential privacy strategy can effectively mask the privacy of each constituent function, which is more practical in applications involving sensitive messages, such as military affairs or medical treatment. Finally, numerical evidence is provided to demonstrate the appealing performance of the proposed algorithm.

Numerical simulations for stochastic differential equations on manifolds by stochastic symmetric projection method

Stochastic standard projection technique, as an efficient approach to simulate stochastic differential equations on manifolds, is widely used in practical applications. However, stochastic standard projection methods usually destroy the geometric properties (such as symplecticity or reversibility), even though the underlying methods are symplectic or symmetric, which seriously affect long-time behavior of the numerical solutions. In this paper, a modification of stochastic standard projection methods for stochastic differential equations on manifolds is presented. The modified methods, called the stochastic symmetric projection methods, remain the symmetry and the ρ-reversibility of the underlying methods and maintain the numerical solutions on the correct manifolds. The mean square convergence order of these methods are proved to be the same as the underlying methods’. Numerical experiments are implemented to verify the theoretical results and show the superiority of the stochastic symmetric projection methods over the stochastic standard projection methods.

Territorial design optimization for business sales plan

A well designed territory enhances customer coverage, increases sales, fosters fair performance and rewards systems and lower travel cost. This paper considers a real life case study to design the sales territory for a business sales plan. The business plan consist of assigning the optimal quantity of sellers to a territory including the scheduling and routing plans for each seller. The problem is formulated as a combination of assignment, scheduling and routing optimization models. The solution approach considers a meta-heuristic using stochastic iterative projection method for large systems. Several real life instances of different sizes were tested with stochastic data to represent raise/fall in the customers demand as well as the appearance/loss of customers.

Claims Reserving with a Stochastic Vector Projection

In the last three decades, a variety of stochastic reserving models have been proposed in the general insurance literature mainly using (or reproducing) the well-known Chain-Ladder claims-reserving estimates. In practice, when the data do not satisfy the Chain-Ladder assumptions, high prediction errors might occur. Thus, in this article, a combined methodology is proposed based on the stochastic vector projection method and uses the regression through the origin approach of Murphy, but with heteroscedastic errors instead, and different from those that used by Mack. Furthermore, the Mack distribution-free model appears to have higher prediction errors when compared with the proposed one, particularly, for data sets with increasing (regular) trends. Finally, three empirical examples with irregular and regular data sets illustrate the theoretical findings, and the concepts of best estimate and risk margin are reported.

Claims Reserving with a Stochastic Vector Projection

In the last three decades, a variety of stochastic reserving models has been proposed in the general insurance literature mainly using (or reproducing) the eminent Chain-Ladder claims reserving estimates. In practice, when the data doesn’t satisfy the Chain-Ladder assumptions, high prediction errors might occur. Thus, in this paper, a combined methodology is proposed which is based on the stochastic vector projection method and uses the regression through the origin approach of Murphy (1994), but with heteroscedastic errors instead, and different to those that used by Mack (1993, 1994). Furthermore, the Mack (1993) distribution-free model appears to have higher prediction errors when it is compared with the pro-posed one, particularly, for data sets with increasing (regular) trends. Finally, three empirical examples with irregular and regular data sets illustrate the theoretical findings, and the concepts of best estimate and risk margin are reported.

Robust scheduling for flexible processing networks

Abstract Modern processing networks often consist of heterogeneous servers with widely varying capabilities, and process job flows with complex structure and requirements. A major challenge in designing efficient scheduling policies in these networks is the lack of reliable estimates of system parameters, and an attractive approach for addressing this challenge is to design robust policies, i.e. policies that do not use system parameters such as arrival and/or service rates for making scheduling decisions. In this paper we propose a general framework for the design of robust policies. The main technical novelty is the use of a stochastic gradient projection method that reacts to queue-length changes in order to find a balanced allocation of service resources to incoming tasks. We illustrate our approach on two broad classes of processing systems, namely the flexible fork-join networks and the flexible queueing networks, and prove the rate stability of our proposed policies for these networks under nonrestrictive assumptions.

Nonintrusive Polynomial Chaos Expansions for Sensitivity Analysis in Stochastic Differential Equations

A Galerkin polynomial chaos (PC) method was recently proposed to perform variance decomposition and sensitivity analysis in stochastic differential equations (SDEs), driven by Wiener noise and involving uncertain parameters. The present paper extends the PC method to nonintrusive approaches enabling its application to more complex systems hardly amenable to stochastic Galerkin projection methods. We also discuss parallel implementations and the variance decomposition of the derived quantity of interest within the framework of nonintrusive approaches. In particular, a novel hybrid PC-sampling-based strategy is proposed in the case of nonsmooth quantities of interest (QoIs) but smooth SDE solution. Numerical examples are provided that illustrate the decomposition of the variance of QoIs into contributions arising from the uncertain parameters, the inherent stochastic forcing, and joint effects. The simulations are also used to support a brief analysis of the computational complexity of the method, providing...

Iterative Projection approach for solving the Territorial Business Sales optimization problem

A well designed territory enhances customer coverage, increases sales, fosters fair performance and rewards systems and lower travel cost. This paper considers a real life case study to design a sales territory for a business sales plan. The business plan consists in assigning the optimal quantity of sellers to a territory including the scheduling and routing plans for each seller. The problem is formulated as a combination of assignment, scheduling and routing optimization problems. The solution approach considers a meta-heuristic using stochastic iterative projection method for large systems. Several real life instances of different sizes were tested with stochastic data to represent raise/fall in the customers demand as well as the appearance/loss of customers.

Stochastic projection methods and applications to some nonlinear inverse problems of phase retrieving

In this short paper we present a stochastic projection based Monte Carlo algorithm for solving a nonlinear ill-posed inverse problem of recovering the phase of a complex-valued function provided its absolute value is known, under some additional information. The method is developed here for retrieving the step structure of the epitaxial films from the X-ray diffraction analysis. We suggest to extract some additional information from the measurements which makes the problem well-posed, and with this information, the method suggested works well even for noisy measurements. Results of simulations for a layer structure recovering problem with 26 sublayers are presented.