Stochastic Solver Research Articles

Fast initialization methods for the nonconvex economic dispatch problem

This paper develops a set of fast initialization methods to generate candidate preliminary points in the search space of the non-convex economic dispatch problem. These initial points are either the global optimal solution or close enough from this solution to clearly facilitate and accelerate the process of solving the problem while increasing the probability of attaining the global optimal solution. The proposed methods can approach the global optimal solution in minimal time irrespective of the size of the system. In addition, a two-stage framework is also proposed to accommodate the proposed initialization methods. In the first stage, initial solutions are generated by the proposed initialization methods and in the second stage, any powerful stochastic solver can be utilized to confirm obtaining the global optimal solution. The proposed framework is flexible with respect to treating the physical constraints and the practical features of the problem such as the valve point effects, prohibited operating zones, and multiple fuel options. To generate initial solutions for specific variants of the problem, three fast initialization methods are proposed, and to generate initial solutions considering several practical features simultaneously, an integrating strategy is developed. The interior-point method implemented in MATLAB is employed to solve the approximated convex economic dispatch problems incorporated within the proposed initialization techniques. Several powerful metaheuristic algorithms and benchmark problems have been simulated to demonstrate the effectiveness of the proposed initialization methods, the generic applicability feature of them, and to evaluate the closeness degree from the global optimal solution. The results demonstrate that the proposed initialization methods are capable of generating high-quality solutions in a highly computational efficient manner.

Efficient and stochastic multireference perturbation theory for large active spaces within a full configuration interaction quantum Monte Carlo framework.

Full Configuration Interaction Quantum Monte Carlo (FCIQMC) has been effectively applied to very large configuration interaction (CI) problems and was recently adapted for use as an active space solver and combined with orbital optimization. In this work, we detail an approach within FCIQMC to allow for efficient sampling of fully internally contracted multireference perturbation theories within the same stochastic framework. Schemes are described to allow for the close control over the resolution of stochastic sampling of the effective higher-body intermediates within the active space. It is found that while complete active space second-order perturbation theory seems less amenable to a stochastic reformulation, strongly contracted N-Electron Valence second-order Perturbation Theory (NEVPT2) is far more stable, requiring a similar number of walkers to converge the sc-NEVPT2 expectation values as to converge the underlying CI problem. We demonstrate the application of the stochastic approach to the computation of sc-NEVPT2 within a (24, 24) active space in a biologically relevant system and show that small numbers of walkers are sufficient for a faithful sampling of the sc-NEVPT2 energy to chemical accuracy, despite the active space already exceeding the limits of practicality for traditional approaches. This raises prospects of an efficient stochastic solver for multireference chemical problems requiring large active spaces, with an accurate treatment of external orbitals.

Uncertainty propagation and numerical evaluation of viscoelastic sandwich plates having nonlinear behavior

The dynamic analysis of nonlinear viscoelastic systems in the frequency domain is not an easy task. In most cases, it is due to the frequency- and temperature-dependent properties of the viscoelastic part. Additionally, due to the inherent uncertainties affecting the viscoelastic efficiency in practical situations, their handling in the nonlinear modeling methodology becomes essential nowadays. However, it is still an issue. Thus, this paper presents a numerical modeling methodology intended to perform dynamic analyses in the frequency domain of thin sandwich plates under large displacements. The uncertainties characterizing the nonlinear dynamics of the viscoelastic system are introduced on the random linear and nonlinear finite element matrices by performing the Karhunen–Loève expansion technique. The Latin hypercube sampling method is used herein as the stochastic solver, and the nonlinear frequency responses are computed using the harmonic balance method combined with the Galerkin bases. To overcome the difficulty in solving the resulting complex nonlinear eigenproblem with a frequency-dependent viscoelastic stiffness, making the stochastic nonlinear analyses in the frequency domain very costly, sometimes unfeasible, an efficient and accurate iterative reduction method is proposed to approximate the complex eigenmodes. The envelopes of nonlinear frequency responses demonstrate clearly the relevance of considering the uncertainties in design variables of viscoelastic systems having nonlinear behavior to deal with more realistic situations.

Backtracking Search Optimization Paradigm for Pattern Correction of Faulty Antenna Array in Wireless Mobile Communications

The demand for wireless mobile communication is growing exponentially with expectations that in the near future mobile device or user will be available in every corner of the globe. Alternatively, it increases the importance of antenna arrays which are responsible for transmission and reception of information. Every antenna array is projected to generate a desired pattern and, hence, failure of any antenna causes misrepresentation of the overall pattern in terms of increased side lobe levels and displacement of nulls from their original position. The aim of the study is to present viable, simple, and accurate stochastic solver based on backtracking search optimization algorithm (BSA) for the pattern correction of faulty antenna array in mobile communication systems. A fitness function is developed to optimize the weights of the remaining healthy antenna elements in the array. The fitness function consists of two parts: the first part is based on mean square error approach for the reduction of sidelobes level, while, in the second part, steering vectors are used for the repositioning of nulls. Simulation results establish the validity of the BSA from its counterparts based on genetic algorithm and its memetic combination with pattern search technique.

Quantification of Uncertainties on the Critical Buckling Load of Columns under Axial Compression with Uncertain Random Materials.

This study is devoted to the modeling and simulation of uncertainties in the constitutive elastic properties of material constituting a circular column under axial compression. To this aim, a probabilistic model dedicated to the construction of positive-definite random elasticity matrices was first used, involving two stochastic parameters: the mean value and a dispersion parameter. In order to compute the nonlinear effects between load and lateral deflection for the buckling problem of the column, a finite element framework combining a Newton-Raphson solver was developed. The finite element tool was validated by comparing the as-obtained critical buckling loads with those from Euler’s formula at zero-fluctuation of the elasticity matrix. Three levels of fluctuations of material uncertainties were then propagated through the validated finite element tool using the probabilistic method as a stochastic solver. Results showed that uncertain material properties considerably influenced the buckling behavior of columns under axial loading. The coefficient of variation of a critical buckling load over 500 realizations were 15.477%, 26.713% and 41.555% when applying dispersion parameters of 0.3, 0.5 and 0.7, respectively. The 95% confidence intervals of column buckling response were finally given. The methodology of modeling presented in this paper is a potential candidate for accounting material uncertainties with some instabilities of structural elements under compression.

Design of computational intelligent procedure for thermal analysis of porous fin model

The importance of rectangular porous fins for the transformation of heat through the system is well-recognized to analyze the physical characteristics of material in practical applications. In this study, a neuro-computing based stochastic numerical paradigm has been designed to study the dynamics of temperature distribution in porous fin model by exploiting the strength of artificial neural network (ANN) modeling integrated with global search exploration with genetic algorithms (GAs) and efficient local search with interior-point technique (IPT). The governing porous fin equation is transformed into an equivalent nonlinear second order ordinary differential equation. Effect of heat on rectangular type fin with thermal conductivity and temperature dependent internal heat generation is measured through stochastic solver based on ANN optimized with GA-IPT in case of two different materials, Silicon nitride Si3N4 and Aluminium Al. The proposed technique ANN-GA-IPT has been applied on transformed equation for multi-times and the accuracy, convergence and robustness of designed model has been validated by analysis of variance test.

Flight control of a hexa-rotor airship: Uncertainty quantification for a range of temperature and pressure conditions

The present paper is concerned with the dynamic modeling and design of control laws for a small non-rigid multi-rotor airship constituted of an oblate-spheroid helium balloon coupled with an electric-powered hexa-rotor airframe. The vehicle is assumed to operate in windless and low-speed conditions. A six-degree-of-freedom nonlinear dynamic model is derived for it using the Newton–Euler approach and considering, among other efforts, a restoring torque due to the displacement of the balloon’s center of buoyancy above the vehicle’s center of mass and the added-mass effect resulting from the air–structure interaction. Using the derived model and assuming a time-scale separation between the translational and rotational dynamics, the attitude and position control laws are designed separately from each other. Both laws are formulated using feedback linearization combined with control input saturation within appropriate parallelepipedal sets, which are carefully chosen to respect pre-defined bounds on the control torque, control force and maximum inclination angle. The effect of temperature and pressure fluctuations is taken into account through a parametric probabilistic approach, where Maximum Entropy Principle is used to construct a physically consistent stochastic model and Monte Carlo method is used as the stochastic solver to propagate the uncertainties through the system. Extensive simulation results show the effectiveness of the proposed control system and quantify the uncertainty of its performance over a wide range of local temperature and pressure.

Design of cascade artificial neural networks optimized with the memetic computing paradigm for solving the nonlinear Bratu system

A nature-inspired, integrated computational heuristic paradigm is developed for the piecewise solution of the nonlinear Bratu problem arising in fuel ignition model, electrically conducting solids and related fields, by exploiting the strength of Cascade Artificial Neural Networks (CANN) modeling, optimized with the memetic computing procedure based on global search efficacy of genetic algorithms (GAs), aided with the efficient local search of teaching learning based optimization (TLBO). The proposed technique incorporates the log-sigmoid activation function in the CANN model, trained by GAs hybridized with TLBO, i.e., CANN-GA-TLBO. As a first application of CANN-GA-TLBO, 1D nonlinear Bratu's system represented with a boundary value problem of the second-order ordinary different equation has been solved, which is a benchmark for testing new algorithms. Comparison of the results with exact solution and previously reported solutions, including Adomian decomposition method, Laplace transformed decomposition method, B-Spline method and artificial neural network solutions, confirms the superiority of the designed stochastic solver CANN-GA-TLBO in terms of accuracy and convergence measures.

The UWHAM and SWHAM Software Package

We introduce the UWHAM (binless weighted histogram analysis method) and SWHAM (stochastic UWHAM) software package that can be used to estimate the density of states and free energy differences based on the data generated by multi-state simulations. The programs used to solve the UWHAM equations are written in the C++ language and operated via the command line interface. In this paper, first we review the theoretical bases of UWHAM, its stochastic solver RE-SWHAM (replica exchange-like SWHAM)and ST-SWHAM (serial tempering-like SWHAM). Then we provide a tutorial with examples that explains how to apply the UWHAM program package to analyze the data generated by different types of multi-state simulations: umbrella sampling, replica exchange, free energy perturbation simulations, etc. The tutorial examples also show that the UWHAM equations can be solved stochastically by applying the RE-SWHAM and ST-SWHAM programs when the data ensemble is large. If the simulations at some states are far from equilibrium, the Stratified RE-SWHAM program can be applied to obtain the equilibrium distribution of the state of interest. All the source codes and the tutorial examples are available from our group’s web page: https://ronlevygroup.cst.temple.edu/software/UWHAM_and_SWHAM_webpage/index.html.

Cyclic virtual test on wood furniture by Monte Carlo simulation: from compression behavior to connection modeling

Prediction of durability of wood product is a major challenge and an important goal for furniture industry. Numerical simulation based on approximation methods such as the finite element method (FEM) is an efficient and powerful tool to address this challenge while avoiding expensive experimental testing campaigns. Nevertheless, the strong heterogeneity of wood-based materials, the specific geometrical characteristics of wood-based structures (such as furniture that can often be represented as an assembly of beams, plates and/or shells) and the complex nonlinear 3D local behavior near the connections between structural parts may induce some difficulties in the numerical modeling and virtual testing of furniture for robust design purposes. Especially, when cyclic loading occurs, the behavior of junctions in furniture involves a local permanent strain that increases with the number of cycles and that can lead to an important gap potentially affecting the structural integrity of furniture. In this paper, we present an experimental campaign of cyclic compression tests carried out on spruce specimens. Theses specimens are cut out from a bunk bed and loaded under cyclic compression. The cyclic compression loading applied to the specimens leads to an evolution of the permanent strain during cycles that is modeled using a simple law describing the displacement gap as a function of the number of cycles. Considering the strong dispersion in the mechanical properties of wood-based materials and the variabilities induced by the experimental configuration, a stochastic modeling of the gap is proposed by having recourse to the maximum entropy (MaxEnt) principle in order to take into account the random uncertainties on the experimental setup and between the test specimens. The random mechanical response of a complex corner junction in a bunk bed under cyclic loading is then numerically simulated by using a Monte Carlo numerical simulation method as stochastic solver. This provides independent realizations of the random gap evolution (with respect to the number of cycles) in the bunk bed corner, allowing probabilistic quantities of interest related to the random gap, such as first- and second-order statistical moments (mean value, standard deviation) as well as confidence regions (with a given probability level), to be estimated.

Intelligent computing to analyze the dynamics of Magnetohydrodynamic flow over stretchable rotating disk model

In this study a novel application of neurocomputing technique is presented for nonlinear fluid mechanics problem arising in the model of the flow over stretchable rotating disk in the presence of strong magnetic field. The scheme comprises of the power of effective modelling of neural networks supported with integrated optimization strength of genetic algorithm and interior-point method. The governing partial differential equation of the system is converted to nonlinear systems of simultaneous ordinary differential equations by incorporating the similarity variables. Neural network based approximate differential equation models are formulated for the transformed system that are used to construct the merit function in mean squared error sense. The networks are trained initially by genetic algorithm for the global search and rapid local refinements is attained through efficient interior point method. The given scheme is applied for dynamical analysis of the system model in terms of radial, tangential, axial velocities and heat effects by varying magnetic interaction parameters, unsteadiness factors, disk stretchable magnitudes, and Prandtl numbers. The statistical performance indices based on error from standard numerical solutions are used to validate the correctness, consistency, robustness and stability of the proposed stochastic solver.

Stratified UWHAM and Its Stochastic Approximation for Multicanonical Simulations Which Are Far from Equilibrium.

We describe a new analysis tool called Stratified unbinned Weighted Histogram Analysis Method (Stratified-UWHAM), which can be used to compute free energies and expectations from a multicanonical ensemble when a subset of the parallel simulations is far from being equilibrated because of barriers between free energy basins which are only rarely (or never) crossed at some states. The Stratified-UWHAM equations can be obtained in the form of UWHAM equations but with an expanded set of states. We also provide a stochastic solver, Stratified RE-SWHAM, for Stratified-UWHAM to remove its computational bottleneck. Stratified-UWHAM and Stratified RE-SWHAM are applied to study three test topics: the free energy landscape of alanine dipeptide, the binding affinity of a host-guest binding complex, and path sampling for a two-dimensional double well potential. The examples show that when some of the parallel simulations are only locally equilibrated, the estimates of free energies and equilibrium distributions provided by the conventional UWHAM (or MBAR) solutions exhibit considerable biases, but the estimates provided by Stratified-UWHAM and Stratified RE-SWHAM agree with the benchmark very well. Lastly, we discuss features of the Stratified-UWHAM approach which is based on coarse-graining in relation to two other maximum likelihood-based methods which were proposed recently, that also coarse-grain the multicanonical data.

Techno-economic feasibility study of membrane based propane/propylene separation process

One of the most important issues for developing propylene/propane gas separation membranes is to identify the minimum required membrane performances. Herein, we reported the technical and economic feasibility of facilitated propylene transport membranes containing silver nanoparticles, which produces 99.6% purity of propylene with 97% recovery for 46,000kg/h capacity. In order to suggest minimum specification, the membrane separation process is compared with advanced distillation process employing direct vapor recompression process. The distillation process is optimized using a stochastic solver interfaced with a commercial process simulator. With the distillation process, the energy and total costs are approximately $4.40 and 18.6$ to produce per ton of propylene, respectively. The single-stage membrane processes with various stage-cut conditions were evaluated to meet the same total propylene production cost. Result indicates that required membrane permeance and selectivity ranges between 11.3 to 251.5 GPU (permeance unit, 1 GPU=1×10−6cm3 (STP)/(cm s cmHg)) and 61.9 to 1950 depending on stage-cut. Sensitivity analysis was also performed according to membrane costs, feed flow rates and composition.

Stochastic Galerkin Formulations for $$\hbox {CO}_2$$ Transport in Aquifers: Numerical Solutions with Uncertain Material Properties

Abstract A stochastic Galerkin formulation for the transport of $$\hbox {CO}_2$$ CO 2 in a tilted aquifer with uncertain heterogeneous properties is presented. We consider a simplified physics model assuming capillary pressure to be negligible compared to hydrostatic and viscous pressure. The flow is dominated by buoyancy and capillary trapping. We assume a stochastic permeability field and a stochastic model for the uncertain relative permeabilities. We prove that the proposed stochastic Galerkin formulation results in a hyperbolic system of equations, and we devise a numerical method that captures the expected solution discontinuities. The shock-capturing solver for the flux function is combined with an adaptive quadrature method for discontinuous isosurfaces that is used to compute the discontinuous stochastic accumulation coefficient. The stochastic solver is validated against Monte Carlo sampling of an analytical solution for the deterministic problem. The sharp features of the statistics of the solution are accurately captured by the numerical solver. The polynomial chaos framework admits low-cost post-processing of the output to obtain statistics of interest. By construction of an accurate polynomial chaos surrogate model of the output, fast sampling admits calculation of risk for leakage and failure probabilities.

Stochastic blind motion deblurring.

Blind motion deblurring from a single image is a highly under-constrained problem with many degenerate solutions. A good approximation of the intrinsic image can, therefore, only be obtained with the help of prior information in the form of (often nonconvex) regularization terms for both the intrinsic image and the kernel. While the best choice of image priors is still a topic of ongoing investigation, this research is made more complicated by the fact that historically each new prior requires the development of a custom optimization method. In this paper, we develop a stochastic optimization method for blind deconvolution. Since this stochastic solver does not require the explicit computation of the gradient of the objective function and uses only efficient local evaluation of the objective, new priors can be implemented and tested very quickly. We demonstrate that this framework, in combination with different image priors produces results with Peak Signal-to-Noise Ratio (PSNR) values that match or exceed the results obtained by much more complex state-of-the-art blind motion deblurring algorithms.

Robust Design in Multibody Dynamics – Application to Vehicle Ride-comfort Optimization

This research is devoted to the robust design of multibody dynamical systems, that is to say to the optimal design of a multibody system which is carried out using an uncertain computational model. The probabilistic model of uncertainties is constructed using a probabilistic approach yielding a stochastic differential equation with random initial conditions. Then the robust design of the multibody system, in presence of uncertainties, is performed using the least-square method for optimizing the cost function, and the Monte Carlo simulation method as stochastic solver. The application consists in a simple multibody model of an automotive vehicle crossing a rough road and for which the suspensions have to be designed in order to optimize the comfort of the passengers.

Application of exergy analysis for improving energy efficiency of natural gas liquids recovery processes

Thermodynamic analysis and optimization method is applied to provide design guidelines for improving energy efficiency and cost-effectiveness of natural gas liquids recovery processes. Exergy analysis is adopted in this study as a thermodynamic tool to evaluate the loss of exergy associated with irreversibility in natural gas liquids recovery processes, with which conceptual understanding on inefficient design feature or equipment can be obtained. Natural gas liquids processes are modeled and simulated within UniSim® simulator, with which detailed thermodynamic information are obtained for calculating exergy loss. The optimization framework is developed by minimizing overall exergy loss, as an objective function, subject to product specifications and engineering constraints. The optimization is carried out within MATLAB® with the aid of a stochastic solver based on genetic algorithms. The process simulator is linked and interacted with the optimization solver, in which optimal operating conditions can be determined. A case study is presented to illustrate the benefit of using exergy analysis for the design and optimization of natural gas liquids processes and to demonstrate the applicability of design method proposed in this paper.

Efficient simulation of stochastic chemical kinetics with the Stochastic Bulirsch-Stoer extrapolation method.

BackgroundBiochemical systems with relatively low numbers of components must be simulated stochastically in order to capture their inherent noise. Although there has recently been considerable work on discrete stochastic solvers, there is still a need for numerical methods that are both fast and accurate. The Bulirsch-Stoer method is an established method for solving ordinary differential equations that possesses both of these qualities.ResultsIn this paper, we present the Stochastic Bulirsch-Stoer method, a new numerical method for simulating discrete chemical reaction systems, inspired by its deterministic counterpart. It is able to achieve an excellent efficiency due to the fact that it is based on an approach with high deterministic order, allowing for larger stepsizes and leading to fast simulations. We compare it to the Euler τ-leap, as well as two more recent τ-leap methods, on a number of example problems, and find that as well as being very accurate, our method is the most robust, in terms of efficiency, of all the methods considered in this paper. The problems it is most suited for are those with increased populations that would be too slow to simulate using Gillespie’s stochastic simulation algorithm. For such problems, it is likely to achieve higher weak order in the moments.ConclusionsThe Stochastic Bulirsch-Stoer method is a novel stochastic solver that can be used for fast and accurate simulations. Crucially, compared to other similar methods, it better retains its high accuracy when the timesteps are increased. Thus the Stochastic Bulirsch-Stoer method is both computationally efficient and robust. These are key properties for any stochastic numerical method, as they must typically run many thousands of simulations.

A Stochastic Solver of the Generalized Born Model

Abstract A stochastic generalized Born (GB) solver is presented which can give predictions of energies arbitrarily close to those that would be given by exact effective GB radii, and, unlike analytical GB solvers, these errors are Gaussian with estimates that can be easily obtained from the algorithm. This method was tested by computing the electrostatic solvation energies (ΔGsolv ) and the electrostatic binding energies (ΔGbind ) of a set of DNA-drug complexes, a set of protein-drug complexes, a set of protein-protein complexes, and a set of RNA-peptide complexes. Its predictions of ΔGsolv agree with those of the linearized Poisson-Boltzmann equation, but it does not predict ΔGbind well, although these predictions of ΔGbind may be marginally better than those of traditional analytical GB solvers. Apparently, the GB model itself must be improved before accurate estimates of ΔGbind can be obtained.

Stochastic 2D Incompressible Navier-Stokes Solver Using the Vorticity-Stream Function Formulation

A two-dimensional stochastic solver for the incompressible Navier-Stokes equations is developed. The vorticity-stream function formulation is considered. The polynomial chaos expansion was integrated with an unstructured node-centered finite-volume solver. A second-order upwind scheme is used in the convection term for numerical stability and higher-order discretization. The resulting sparse linear system is solved efficiently by a direct parallel solver. The mean and variance simulations of the cavity flow are done for random variation of the viscosity and the lid velocity. The solver was tested and compared with the Monte-Carlo simulations and with previous research works. The developed solver is proved to be efficient in simulating the stochastic two-dimensional incompressible flows.