- When modeling population growth, describing it in terms of birth rate B(t) and death rate D(t) alone is simplistic. Why is this model not realistic? Explain.
- Other factors/parameters, such as accessible natural resources, diseases, and so on, are not taken into account in this model, which could have a significant impact on population increase. Thus making it not realistic.
- A more realistic model for population growth which considers a caring “capacity”. Assuming that the caring capacity is 1,000. Describe/differentiate the growth/decay of the population of two population samples where one started with 100 versus 1,500 when observed for an extended period.
- Because the starting population of 100 persons is smaller than the 1,000-person care capacity, there will almost certainly be an increase in population. The second population of 1,500, on the other hand, will experience population decline because it is larger than the caring capacity of 1,000 people.
- This model assimilates the fact that when there is an existential dependence between 2 population, the growth/decay of population of the prey and predator species behaves in a certain way. Describe the dynamics of this scenario. How does the growth/decay of the prey population affect the predators?
- Prey population growth/decay is proportional to predator population growth/decay. Due to increasing food sources, if the number of prey grows, the population of predators will likely grow as well. This is also true when the prey population declines.
- The same system can be described/modeled at different scales, and different methods apply depending on the scale one is interested in. Take humans as the object of study/observation. Enumerate 5 different scales by which humans can be modeled/simulated. Proceed from micro to macro level.
- The human body can be modeled from the smallest building block of (1) cell which is composed of atoms, molecules, and organelles, up to (2) tissues, (3) organs, (4) organ systems, and lastly (5) human organism
- “Once considers a discrete universe as an abstraction of the real world.” One has to discretize a continuous phenomenon in order to model/simulate it in a computer. What are the advantages and disadvantages of such approach?
- Real world is arguably could be a continuous or discrete. In quantum theory, the real world is seen as discrete universe wherein we could describe every thing to its smallest possible particle. The advantage of this approach is that having a discretize world will make it possible for computer to break down complex ideas into smaller chunks of discrete units. The disadvantage of this is that breaking down into smallest unit will affect the model efficiency in simulating real world.
- In a natural setup, space takes a real value. Only mathematical models can deal with this (using differential equations. Any attempt to model/simulate using a computer requires space discretization. Differentiate the 2 major ways that this can be achieved.
- 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. 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.
- The goal of Monte-Carlo methods is the sampling of a process in order to determine some statistical properties some dynamical systems. How is the Monte-Carlo Markov-Chain (MCMC) different? Explain in simple terms.
- Regular Monte-Carlo simulation relies on a repeated random sampling method which you will created a model based on continuous randomness, while Markov-Chain Monte-Carlo simulation uses the temporal and spatial dimension as well as the previous iteration of random sampling to create a model.
- Dynamical systems are by nature continuous but can be modeled discretely. The discrete model is of course just an approximation of a continuous one in this case. This approximate discrete model has an inherent error in it by virtue of its approximate nature. The error in the discrete version of the model varies depending on one critical factor. What is this factor and how do we tweak its value to minimize the modeling error?
- The time increment is the critical factor; the greater the time increment, the greater the cumulative inaccuracy. As a result, in order to reduce the cumulative mistake, we must reduce the time increment.
- Explain the major areas of application of computational science and how the computational science cycle work?
- Computational Science involves different areas but it is mostly focused on Applied Mathematics and Numerical Methods, Science Disciplines and Computer Science. Computational Science cycle starts at Selecting a real world problems and start to simplify it to create an abstraction for models. Abstract ideas is converted to computational model by representing it with mathematics and algorithms which leads to computer models Simulations is done through the help of computer models in an experimental testbed. Through the simulations we could get some data which could generate fractions of real world situation. Lastly, the data will be interpreted if it could be apply on a real world scenarios (imitation, creation of systems and etc.)
- How are the three types of simulations different?
- Live Simulation uses real people that uses physical and real systems to simulate real situation (Army trainings is one of the example).
- Virtual Simulation uses real people that use and operate simulated system to mimic and perform a real situation (Pilots uses this kind of simulation, AR is one of the use case of this scenarios)
- Constructive Simulation uses simulated environment and users (people) to test different scenarios that is not capable or dangerous for people to test. This kind of simulations is facilitated by real people to show different scenarios that could be a result of different environments. (One example of this is VR)