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mathematics

Simulation

Simulation is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Simulation rather than just read about it. In short: A simulation is an imitative representation of a process or system that could exist in the real world. In this broad sense, simulation can often be used interchangeably with model.

Simulation — main illustration
Simulation — illustration

Key takeaways

  • Simulation belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Simulation to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Simulation from memory before moving on to harder problems.

Reference excerpt

A simulation is an imitative representation of a process or system that could exist in the real world. In this broad sense, simulation can often be used interchangeably with model. Sometimes a clear distinction between the two terms is made, in which simulations require the use of models; the model represents the key characteristics or behaviors of the selected system or process, whereas the simulation represents the evolution of the model over time. Another way to distinguish between the terms is to define simulation as experimentation with the help of a model. This definition includes time-independent simulations. Often, computers are used to execute the simulation. Simulation is used in many contexts, such as simulation of technology for performance tuning or optimizing, safety engineering, testing, training, education, and video games. Simulation is also used with scientific modelling of natural systems or human systems to gain insight into their functioning, as in economics. Simulation can be used to show the eventual real effects of alternative conditions and courses of action. Simulation is also used when the real system cannot be engaged, because it may not be accessible, or it may be dangerous or unacceptable to engage, or it is being designed but not yet built, or it may simply not exist. Key issues in modeling and simulation include the acquisition of valid sources of information about the relevant selection of key characteristics and behaviors used to build the model, the use of simplifying approximations and assumptions within the model, and the fidelity and validity of the simulation outcomes. Procedures and protocols for model verification and validation are an ongoing field of academic study, refinement, research and development in simulations technology or practice, particularly in the work of computer simulation.

Classification and terminology

Historically, simulations used in different fields developed largely independently, but 20th-century studies of systems theory and cybernetics combined with the spreading use of computers across all those fields have led to some unification and a more systematic view of the concept. Physical simulation refers to simulation in which physical objects are substituted for the real thing. These physical objects are often chosen because they are smaller or cheaper than the actual object or system. (See also: physical model and scale model.) Alternatively, physical simulation may refer to computer simulations considering selected laws of physics, as in multiphysics simulation. (See also: Physics engine.) Interactive simulation is a special kind of physical simulation, often referred to as a human-in-the-loop simulation, in which physical simulations include human operators, such as in a flight simulator, sailing simulator, or driving simulator. Continuous simulation is a simulation based on continuous-time rather than discrete-time steps, using numerical integration of differential equations. Discrete-event simulation studies systems whose states change their values only at discrete times. For example, a simulation of an epidemic could change the number of infected people at time instants when susceptible individuals get infected or when infected individuals recover. Stochastic simulation is a simulation where some variable or process is subject to random variations and is projected using Monte Carlo techniques using pseudo-random numbers. Thus replicated runs with the same boundary conditions will each produce different results within a specific confidence band. Deterministic simulation is a simulation which is not stochastic: thus the variables are regulated by deterministic algorithms. So replicated runs from the same boundary conditions always produce identical results. Hybrid simulation (or combined simulation) corresponds to a mix between continuous and discrete event simulation and results in integrating numerically the differential equations between two sequential events numerically to reduce the number of discontinuities. A stand-alone simulation is a simulation running on a single workstation by itself. A distributed simulation is one which uses more than one computer simultaneously, to guarantee access from/to different resources (e.g. multi-users operating different systems, or distributed data sets); a classical example is Distributed Interactive Simulation (DIS). Parallel simulation speeds up a simulation's execution by concurrently distributing its workload over multiple processors, as in high-performance computing. Interoperable simulation is where multiple models, simulators (often defined as federates) interoperate locally, distributed over a network; a classical example is High-Level Architecture. Modeling and simulation as a service is where simulation is accessed as a service over the web. Modeling, interoperable simulation and serious games is where serious game approaches (e.g. game engines and engagement methods) are integrated with interoperable simulation. Simulation fidelity is used to describe the accuracy of a simulation and how closely it imitates the real-life counterpart. Fidelity is broadly classified as one of three categories: low, medium, and high. Specific descriptions of fidelity levels are subject to interpretation, but the following generalizations can be made:

Low – the minimum simulation required for a system to respond to accept inputs and provide outputs Medium – responds automatically to stimuli, with limited accuracy High – nearly indistinguishable or as close as possible to the real system A synthetic environment is a computer simulation that can be included in human-in-the-loop simulations. Simulation in failure analysis refers to simulation in which we create environment/conditions to identify the cause of equipment failure. This can be the best and fastest method to identify the failure cause.

Data simulation Data simulation creates artificial data to replicate the behavior of real-world data. The artificial data is generated using computer models or programs based on specific rules, patterns, or characteristics. This type of data is designed to imitate the operations of real systems, processes, or behaviors, enabling researchers and analysts to study and test scenarios without directly observing or collecting information from the real world.

In computers

… excerpt ends here. Continue reading the full article.

Illustrations

Simulation: Human-in-the-loop simulation of outer space
Human-in-the-loop simulation of outer space
Simulation: Visualization of a direct numerical simulation model
Visualization of a direct numerical simulation model
Simulation: Example of computer simulation modeling a two body gravitational system
Example of computer simulation modeling a two body gravitational system
Simulation: Military simulators pdf
Military simulators pdf
Simulation: Motorcycle simulator of Bienal do Automóvel exhibition, in Belo Horizonte, Brazil
Motorcycle simulator of Bienal do Automóvel exhibition, in Belo Horizonte, Brazil

Worked examples

Example 1 — a first encounter with Simulation

Start with the simplest possible case. Write down what Simulation claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Simulation before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Simulation ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Simulation

In research
Simulation appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Simulation in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Simulation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Mathematical and quantitative methods (economics), Simulation, so understanding it makes those chapters shorter.
In everyday life
Look for Simulation outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Simulation in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Simulation means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Simulation out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Simulation in simple terms?

A simulation is an imitative representation of a process or system that could exist in the real world. In this broad sense, simulation can often be used interchangeably with model.

Why does Simulation matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Simulation?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Simulation.

Tags

  • Mathematical and quantitative methods (economics)
  • Simulation

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