Computational Science Models

Week 5 - Computational Science

General Concepts

1. A model is an abstraction of reality. How does abstraction help us understand the object/system being modeled? An abstraction is the process of removing physical, spatial, or temporal details or attributes in the study of objects or systems to focus attention on details of greater importance. It is an important tool for simplifying a complex phenomenon to a level where analysis, experimentation, or understanding can take place.

   For example, in attempting to understand the mechanics of the solar system, early mathematicians and astronomers applied abstraction to a “planet”, treating the planet as a body all of whose mass is concentrated at a single point. Such an abstraction ignores a wealth of details about each planet - its actual diameter, its atmospheric content, its average temperature, etc. However, these details are not relevant to understanding and modeling the basic orbital mechanics of the solar system.

2. What does a model allow us to do? Models could be, a lab animal, a mathematical equation, a simplified abstraction of reality, and a representation of a phenomenon in a mathematical or computer-based language. This kind of models allows us to describe and understand problems. In this way, we can predict outcomes and control certain scenarios.

3. Explain: “Everything should be made as simple as possible, but not simpler.” This quote from Einstein shows that abstraction is done to make things simple, but making it more simple will defeat its purpose. Just break down a complex problem to a level that a computer or human can analyze it and can answer your specific problem.

4. Exemplify: “The same system can be modeled at different scales.” A system is composed of different elements, and you can create different models with different scales based on what elements you’re trying to understand. For example, in dermatology the skin can be breakdown to tissue, cells, skin, and etc. If you would like to study a skin disease, you need to define the scale (tissue level, cellular level, and etc.) you would like to examine, to formulate the best model for your problem.

5. Explain: “One considers a discrete universe as an abstraction of the real world” Real world is arguably could be a continuous or discrete.¹ In quantum theory, the real world is seen as discrete universe wherein we could describe every thing to its smallest possible particle.

Discrete Universe in Quantum Theory

In Quantum Theory, there’s the smallest possible particle of gold, the gold atom. There’s the smallest possible particle of electricity, the electron. There’s the smallest possible particle of light, the photon. In quantum theory, we have this idea of discreteness, that there is the smallest possible thing from which everything else is built.

This shows us that a discrete universe can be an abstraction of a real world wherein we can break down the complex systems around us to its smallest possible unit.

Continuous Universe in General Relativity

In general relativity, the idea is the opposite. It says things can continuously vary, and the mathematics requires that things be continuously variable so they can be differentiated and so on. The idea is that you can keep on dividing up space and you can keep on dividing up time.

General relativity also suggest that the real world is infinitely continuous and it involves different complex interconnected systems that follow foundational theories rooted in a continuous universe.

6. Explain: “Simulation is a numerical experiment in a computer-based virtual universe.” A simulation is from a model that is programmed and run its simulation many times to study a certain problem. Simulations is done through use of computer programs, software engineering, algorithms, data-structures, hardware(parallel machines, GPUs), code optimization, and data analysis. This is why it is called a computer-based virtual universe. It is a task commonly run in a virtual simulations thus creating its own interpretation of an abstraction of real world entities.

7. How do we validate/benchmark a model? To validate/benchmark a model, it is required to have an initial knowledge of the field, known phenomena, previous model to validate the model performance. A newly created model in a field that has no known phenomena is hard to validate thus, physical and traditional experimentation is prerequisite before the creation of models.

8. Exemplify: “Natural processes occurs in space and evolve over time.” Different natural processes that is occupying a space and will eventually change its state over time. This indicates that every natural processes could be different base on its spatial variation, and it state is different in past, present, and future state due to temporal variation. For an example, a human will undergo in aging which eventually lead to its end life and create another natural process which is fossilization, this is a manifestation of both spatial and temporal variation since a human being is in different spatial variation and affected by several temporal variation. Another example of both manifestation of spatial and temporal variation is the time dilation.

9. Consider a study on the growth of population. Which is more relevant its evolution in time or its spatial variations? Growth of population changes evolves over time which means it is more relevant with its temporal variations.

10. Consider a study on the migration of population. Which is more relevant its evolution in time or its spatial variations? Migration of population occurs mainly due to several events such as war, economy, climate, food supply, and etc. which is mostly relevant to spatial variations. Good example of this is the migration of several individuals from China to Taiwan and rest of the world due to the overtaking of Mao Zedong which set up a communist state in China. ²

11. Explain: “To capture the temporal dimension in a model, time is discretized but the process is followed continuously over it’s duration.” As stated in Special and General Relativity, time is continuously flowing. But in the perspective of model and simulations this is hard to utilize since abstraction considers discrete variable to break it down and create a suitable model. This can be done through discretization of time using time intervals. Furthermore, this could be also done using quantum mechanics which defines the time it takes for a photon to travel a distance equal to the Planck length, this is also called as Planck Time. By doing this, we can make use of this smaller units to create a simulated temporal dimension.

12. Exemplify: “In a Discrete Event Simulation (DES) approach, the evolution of a system is broken down into events.” The discrete event simulation focuses on considering the time at which significant and remarkable events occurs. This could be contextualized in a daily life scenario wherein special events such as entering a school, doing a recitation is considered as a remarkable event and can be a basis of analysis and models.

13. Explain/Interpret the diagram: For the first line the time is interpreted as a continuous (this could be also seen as an interpretation of Special and General Relativity), while in the second one it is interpreted as an uniform interval of smaller units (this could be also seen as an interpretation of Quantum Mechanics), and lastly the third one is interpreted as a division of remarkable events (which is also an interpretation of DES).

14. Exemplify: “To model space, one can take the Eulerian approach, which attach a property of the system at each spatial locations. Take the point of view (POV) of an observer who sits at a fixed position (x’) in space and record what you see.” Eulerian approach is an approach wherein the observer focuses on specific locations in the space through which the system flow/move as time passes. A good example of this is an individual from the earth which observes a moving spaceship in the space. (This is a part of Twin Paradox).³

Twin Paradox

Suppose that one of two identical twin sisters flies off into space at nearly the speed of light. According to relativity, time runs more slowly on her spacecraft than it does on Earth; therefore, when she returns to Earth, she will be younger than her Earth-bound sister. But in relativity, what one observer sees as happening to a second one, the second one sees as happening to the first one. To the space-going sister, time moves more slowly on Earth than it does in her spacecraft; when she returns, her Earth-bound sister is the one who is younger.

15. Explain: “Space can be discretized, forming a mesh covering the region of interest.” Space can be continuous and discrete. In a continuous space it is interpreted using mathematical models that break downs continuous problems, while in discrete space it is interpreted through a cell or cartesian plane wherein the observations, or states will eventually cover a certain region of interest which is also seen in a mesh or points.

16. Exemplify: “In the Lagrangian approach to modeling space, one takes the positions of all objects of interest as a function of time. The observer takes the point of view (POV) of the moving objects.” Lagrangian approach is an approach wherein the observer follows an individual unit of the systems as it flow/move through space and time. A good example of this is an individual that travelling in space in a spaceship relative to an observer to the earth. (This is a part of Twin Paradox).³

17. Interpret: In Eulerian point of view, the regions observed is put in a cell which is fixed due to a specific observed position. The regions differs in intensity of colors based on the whole time of observation. While in a Langrangian point of view the regions observed is plotted in a cartesian plane and is not fixed since the observer is moving together with the systems. The regions specifies the space and time of the observed phenomena.

18. “In some systems, it is not so much the spatial positions of the components that matter but whether these can interact with each other. We use graphs.” This specific graph is an example of community network based on the video. Wherein the circle pertains to a person, and the bigger circle means that the it has a bigger network of individuals, while the color represents their opinion. In this case you can understand their opinion and its changes over time.

19. Using Monte-Carlo approach, how can you programmatically simulate this problem? “When tossing a coin 4 times, what is the probability of getting 3 tail, 1 head?”

    Code Output

20. Consider: Markov-Chain Monte-Carlo (MCMC) MCMC can be simulated as a “random walk” in the state space. How can this be done Programmatically?

Simulation of Markov-Chain Monte-Carlo

Code

Output

Footnotes

1. Is the Universe Discrete or Continuous? (nav.al) ↩

2. 8 of the Greatest Migrations in History - Owlcation ↩

3. twin paradox | physics | Britannica ↩ ↩²