Stochastic versus Deterministic Approaches

Abstract

In broad sense, modelling refers to the process of generating a simplified representation of a real system. A suitable model must be able to explain past observations, integrate present data and predict with reasonable accuracy the response of the system to planned stresses (Carrera et al., 1987). Models have evolved together with science and nowadays modelling is an essential and inseparable part of scientific activity. In environmental sciences, models are used to guarantee suitable conditions for sustainable development and are a pillar for the design of social and industrial policies.Model types include analogue models, scale models and mathematical models. Analogue models represent the target system by another, more understandable or analysable system. These models rely on Feynman's principle (Feynman et al., 1989, sec. 12-1): 'The same equations have the same solutions.' For example, the electric/hydraulic analogy (Figure 8.1a) establishes the parallelism between voltage and water-pressure difference or between electric current and flow rate of water. Scale models are representations of a system that is larger or smaller (most often) than the actual size of the system being modelled. Scale models (Figure 8.1b) are often built to analyse physical processes in the laboratory or to test the likely performance of a particular design at an early stage of development without incurring the full expense of a full-sized prototype. Notwithstanding the use of these types of models in other branches of science and engineering, the most popular models in environmental sciences are mathematical. A mathematical model describes a system by a set of state variables and a set of equations that establish relationships between those variables and the governing parameters. Mathematical models can be analytical or numerical. Analytical models often require many simplifications to render the equations amenable to solution. Instead, numerical models are more versatile and make use of computers to solve the equations.Mathematical models (either analytical or numerical) can be deterministic or stochastic (from the Greek τóχoς for 'aim' or 'guess'). A deterministic model is one in which state variables are uniquely determined by parameters in the model and by sets of previous states of these variables. Therefore, deterministic models perform the same way for a given set of parameters and initial conditions and their solution is unique. Nevertheless, deterministic models are sometimes unstable -i.e., small perturbations (often below the detection limits) of the initial conditions or the parameters governing the problem lead to large variations of the final solution (Lorenz, 1963). Thus, despite the fact that the solution is unique, one can obtain solutions that are dramatically different by perturbing slightly a single governing parameter or the initial condition at a single point of the domain.

Optimal Asset Planning for Prosumers Considering Energy Storage and Photovoltaic (PV) Units: A Stochastic Approach

“…While the calculation of the PV potential is based on solar irradiance and temperature, tools and approaches [33][34][35] can be used to obtain the PV potential data to be used as an input. To combat the uncertainty of forecasting solar PV potential, a stochastic approach was proposed and the results are to be compared with the deterministic approach [36].…”

A Network-Based Stochastic Epidemic Simulator: Controlling COVID-19 with Region-Specific Policies

“…Both approaches have their advantages and limitations [14], [15]. The main strength of the deterministic model is that it clearly shows how the initial conditions and parameters impact the model behavior, but at the expense of more realistic and nuanced simulation.…”

Appearance Modelling of Living Human Tissues

“…In the next sections, we introduce a review of skin sampling methods to capture skin appearance, then we approach models to represent skin light–material interaction. Incidentally, the terms ‘deterministic’ and ‘stochastic’ employed in the taxonomy presented in Section 5.1.3 and 5.1.4 refer to whether or not the models possess some inherent randomness [RAG13].…”