A Comparison of Stochastic Optimization Methods for Multi-Reservoir Systems

Optimal Control of Stochastic Systems

This work focuses on modeling of film porosity in thin film deposition using stochastic differential equations. A deposition process is modeled via kinetic Monte Carlo (kMC) simulation on a triangular lattice. The microscopic process events involve atom adsorption and migration and the film growth allows for vacancies and overhangs to develop inside the film. Appropriate definitions of film site occupancy ratio (SOR), i.e., fraction of film sites occupied by particles over total number of film sites, and its fluctuation are introduced to describe film porosity. Deterministic and stochastic ordinary differential equation (ODE) models are also derived to describe the time evolution of film SOR and its fluctuation. The coefficients of the ODE models are estimated on the basis of data obtained from the kMC simulator of the deposition process using least-square methods. Simulation results demonstrate the applicability and effectiveness of the proposed film porosity modeling methods in the context of the deposition process under consideration.

Stochastic differential equation models play an increasingly prominent role in a wide variety of application areas, particularly those characterized by complexity and random behavior. Examples of such areas include biology, epidemiology, mechanics, economics, and finance. This issue's Education section contains a paper, "An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations," by Desmond Higham, that makes this important material accessible to a wide range of readers. Higham's paper is written from the perspective that the best way to gain insight quickly into the topics of stochastic differential equations (SDEs) is to see examples of numerical methods in use. The author feels that it is possible to explain and understand how simple numerical methods in this area work, without requiring the reader to be familiar with SDE theory. Indeed, the paper takes the approach that simulations of the dynamic behavior of SDEs and experimentation with numerical examples can help develop an intuitive understanding of SDEs that lays the groundwork for subsequent theoretical treatments. Thus Higham's paper expects only that the reader's background includes Euler's method for deterministic ordinary differential equations and an intuitive understanding of random variables. It provides a beautifully written treatment that is aimed at upper-division undergraduates and beginning graduate students. The topics covered in Higham's paper include investigation of stochastic integration, linear stability, and strong and weak convergence. The paper is built around ten complete downloadable MATLAB programs that are used as examples and contain ample comment lines. The programs are linked to relevant discussion points in the body of the paper. This paper is suitable as a module in a numerical methods course, such as a course on numerical methods for ordinary differential equations. It could also find application in courses in the areas where SDEs arise, such as finance, epidemiology, or general mathematical modeling. And it can serve excellently as the starting point for a self-study project in the area of SDEs for advanced undergraduates or beginning graduate students. Finally, many professionals in the SIAM community will find this article useful as an introduction to an area of numerical computation that has grown in prominence since many of us were students!

Event Abstract Back to Event Stochastic modeling of neuronal signaling Jukka Intosalmi1*, Tiina Manninen1, Keijo Ruohonen1 and Marja-Leena Linne1 1 Tampere University of Technology, Finland The time evolution of biochemical systems in neurons is traditionally modeled using deterministic ordinary differential equations (ODEs). Chemical reactions, however, are random in nature, and the deterministic approach is valid only for a restricted class of systems. Stochastic models take random fluctuations into account and are thus more realistic. Biochemical reactions can be modeled stochastically using numerous different methods. An ideal model would have the following three important properties. First, the model should be as realistic as possible, second, the mathematical method should be easily implementable as a computer algorithm, and third, the algorithm should be computationally effective. Naturally, all these three conditions cannot be fulfilled at the same time. Some realistic modeling approaches can be derived directly from chemical kinetics without making any approximations. Such approaches are called exact. A good example of an exact modeling approach is the stochastic simulation algorithm (SSA) developed by Gillespie. The SSA is applicable when the molecular populations in the system are small, but it becomes computationally inefficient when the numbers of molecules increase. In order to construct stochastic models that can be effectively simulated, new mathematical approaches have to be explored. As an approximate method also stochastic differential equations (SDEs) have been considered a promising way to model biochemical reactions stochastically. The SDE approach is attractive especially if we consider a system for which the SSA is computationally inefficient and the traditional deterministic ODE approach cannot be used as a good approximation. SDE models treat the chemical populations as real numbers and the construction of the model is based on the law of mass action, similarly as the construction of the traditional deterministic modeling approach. The SDE model, however, takes the random fluctuations into account by describing the time evolution of the system with the stochastic Itô process instead of the deterministic set of ODEs. The Itô process is basically a set of coupled stochastic differential equations. Although SDE modeling usually leads to equations that cannot be solved analytically, the solutions can be approximated numerically using different numerical integration methods. Thus, SDEs provide a mathematically rigorous way to construct models that can be simulated effectively. In order to investigate how well the SDE approach models biochemical systems, the results have to be compared with experimental data or with the simulation results from some exact simulation procedure. In this study, we use the methodology of spectral analysis to compare the simulation results of the SDE model and the SSA. As case studies, we consider e.g. calcium binding and protein kinase C (PKC) signal transduction pathway. We investigate the nature of noise in different modeling approaches and try to interpret the meaning of the noise in real biological systems. The main goal of our study is to find the most essential components and parameters in the SDE model and to adjust them so that the model is capable of giving similar results as the SSA. The simulations are carried out using the distributed computing resources (GRID) provided by Techila Technologies Ltd. Conference: Neuroinformatics 2008, Stockholm, Sweden, 7 Sep - 9 Sep, 2008. Presentation Type: Poster Presentation Topic: Computational Neuroscience Citation: Intosalmi J, Manninen T, Ruohonen K and Linne M (2008). Stochastic modeling of neuronal signaling. Front. Neuroinform. Conference Abstract: Neuroinformatics 2008. doi: 10.3389/conf.neuro.11.2008.01.023 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 28 Jul 2008; Published Online: 28 Jul 2008. * Correspondence: Jukka Intosalmi, Tampere University of Technology, Tampere, Finland, jukka.intosalmi@tut.fi Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Scholar Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne PubMed Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

In this research, application of some real-time operation methods on a single-reservoir system, Karoon IV, with decreasing sum of drinking, industry and agriculture deficit has been considered. These methods include standard operation policy (SOP), stochastic dynamic programming (SDP) and some types of operational rule curves with different orders of inflow and reservoir storage volume. To rank aforesaid, a multiattribute decision method, ELECTRE I, with combination of indices, objective function and reservoir performance criteria (reliability, resiliency and vulnerability) has been used. This ranking is accomplished in two different states including: (1) performances criteria and (2) objective function and performance criteria using the same weights for all criteria. Results show that rule curves were selected as the suitable policy in real-time operation and models validation has been presented by testing nominated curve for dry, normal and wet years.

Advances in carbon capture techniques and demands in alternative fuel sources have increased over the past couple of decades. The Fischer-Tropsch Synthesis (FTS) provides a viable way to produce hydrocarbons from natural gas, coal, CO2, or biomass. However, current comprehensive models for FTS encompass large number of reacting species, readsorption and conversion of primary products, surface intermediates, and coverage-dependent reaction rates. To accurately predict the products obtained through the process a reduced order model has been developed. By reducing the number of parameters of an existing comprehensive model, uncertainty is introduced. The uncertainty can be quantified by using discrepancy functions within the chemical rate equations, there by representing the reduced order model as a set of stochastic differential equations. Representing the uncertainty as model discrepancy functions, a Bayesian approach is used to calibrate the reduced order model to data obtained from literature. Through a Bayesian Smoothing Splines (BSS-ANOVA) framework, the stochastic differential equations are decoupled into deterministic differential equations and stochastic coefficients. The parameters are solved for using a Sequential Monte Carlo approach with importance sampling. Through the use of these stochastic coefficients, fidelity is restored to the reduced order model. Thus, the model can be fully described by fewer parameters than initially needed, as well as a reduction in the computational complexity.

A data-driven approach for discovering stochastic dynamical systems with non-Gaussian Lévy noise

We are concerned with the fuzzy stochastic differential equations driven by multidimensional Brownian motion viewed as a tool used to describe the behavior of dynamic systems operating in fuzzy environments with stochastic noises. Under the uniform Lipschitz condition, we prove the local uniqueness theorem for the solutions of fuzzy stochastic differential equations. Next we show, assuming the Lipschitz condition is satisfied only locally, that these equations have a unique solution. The fact that the solution is bounded is also proved. We conclude the paper with a number of corresponding results holding for the deterministic fuzzy differential equations and set-valued stochastic differential equations with local Lipschitz condition.

A Lax equivalence theorem for stochastic differential equations

Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and the natural sciences. In particular, SDEs and Kolmogorov PDEs, respectively, are highly employed in models for the approximative pricing of financial derivatives. Kolmogorov PDEs and SDEs, respectively, can typically not be solved explicitly and it has been and still is an active topic of research to design and analyze numerical methods which are able to approximately solve Kolmogorov PDEs and SDEs, respectively. Nearly all approximation methods for Kolmogorov PDEs in the literature suffer under the curse of dimensionality or only provide approximations of the solution of the PDE at a single fixed space-time point. In this paper we derive and propose a numerical approximation method which aims to overcome both of the above mentioned drawbacks and intends to deliver a numerical approximation of the Kolmogorov PDE on an entire region \([a,b]^d\) without suffering from the curse of dimensionality. Numerical results on examples including the heat equation, the Black–Scholes model, the stochastic Lorenz equation, and the Heston model suggest that the proposed approximation algorithm is quite effective in high dimensions in terms of both accuracy and speed.

Indian summer monsoon governs agricultural productivity, water resources, and economic activity affecting 1.4 billion people, yet accurate prediction remains challenging due to complex non-linear dynamics involving atmosphere-ocean interactions, land surface processes, and stochastic forcing. Traditional numerical weather prediction models solve deterministic partial differential equations but struggle capturing uncertainty quantification and sub-grid scale processes. Machine learning approaches demonstrate skill learning patterns from historical data but lack physical constraints and interpretability. This research develops hybrid framework integrating stochastic differential equations representing atmospheric dynamics with deep learning capturing non-linear relationships, enabling probabilistic monsoon forecasting with improved accuracy and uncertainty quantification. The research formulated Indian monsoon system as coupled stochastic differential equation system: dX_t = μ(X_t, θ)dt + σ(X_t, θ)dW_t, where X_t represents state variables (rainfall, temperature, humidity, pressure), μ denotes drift coefficient encoding deterministic dynamics, σ represents diffusion coefficient capturing stochastic fluctuations, W_t denotes Wiener process, and θ parameters learned from data. Long Short-Term Memory neural networks parameterized drift and diffusion coefficients enabling flexible representation of complex non-linear relationships while preserving stochastic process structure. Training utilized 40 years of daily rainfall data (1980-2019) from 64 India Meteorological Department stations across eight monsoon regions, supplemented by atmospheric reanalysis including temperature, humidity, wind, and sea surface temperature. Numerical integration employed Euler-Maruyama scheme with adaptive time stepping ensuring stability and accuracy. Model evaluation compared hybrid stochastic-machine learning approach against traditional numerical prediction, pure stochastic models, and pure machine learning on 2017-2019 validation period. Performance metrics included root mean square error, mean absolute error, R-squared, continuous ranked probability score, and skill scores assessing probabilistic forecast quality. Results demonstrated hybrid model achieved RMSE of 2.1 mm daily rainfall compared to 3.2 mm for pure stochastic models and 5.4 mm for deterministic numerical predictions, representing 34% accuracy improvement. R-squared reached 0.967 indicating excellent predictive skill. Probabilistic forecasts provided well-calibrated uncertainty estimates with continuous ranked probability score of 1.8 mm outperforming deterministic predictions. Spatial analysis revealed consistent performance across regions with coastal areas (R²=0.967) and northeast India (R²=0.956) achieving highest accuracy, while northwest plains showed moderate performance (R²=0.912) due to continental influences. Temporal validation demonstrated robust monsoon onset prediction with 89.3% accuracy detecting rainfall commencement within ±3 days. Computational efficiency analysis revealed hybrid model required 2.3 hours training on GPU hardware compared to 48 hours for traditional numerical models, enabling operational forecasting. The research establishes mathematical framework for physics-informed machine learning combining stochastic calculus with deep learning, providing interpretable probabilistic forecasts supporting agricultural planning, water resource management, and disaster preparedness across India's monsoon-dependent economy.

A stochastic dynamic programming approach for delay management of a single train line

Reservoir operation rules are logical or mathematical equations that take into account system variables to calculate water release from a reservoir based on inflow and storage volume values. In fact, previous experiences of the system are used to balance reservoir system parameters in each operational period. Commonly, reservoir operation rules have been considered to be linear decision rules (LDRs) and constant coefficients developed by using various optimization procedures. This paper addresses the application of real-time operation rules on a reservoir system whose purpose is to supply total downstream demand. Those rules include standard operation policy (SOP), stochastic dynamic programming (SDP), LDR, and nonlinear decision rule (NLDR) with various orders of inflow and reservoir storage volume. Also, a multi-attribute decision method, elimination and choice expressing reality (ELECTRE)-I, with a combination of indices, objective functions, and reservoir performance criteria (reliability, resiliency, and vulnerability) are used to rank the aforementioned rules. The ranking method employs two combinations of indices: (1) performance criteria and (2) objective function and performance criteria by using the same weights for all criteria. Results show that the NLDR gives an appropriate rule for real-time operation. Moreover, NLDR validation is presented by testing predefined curves for dry, normal, and wet years.

Moment boundedness of linear stochastic delay differential equations with distributed delay

Pathwise mild solutions for quasilinear stochastic partial differential equations