Final Exam Questions Flashcards

Question

Answer 1

2 types: Task parallelism & data parallelism Task parallelism: used to parallelise the independent algorithms A, B and C by running them concurrently on 3 different processes. Each process performs a different task, and the tasks are executed in parallel, thus reducing the overall execution time. Data Parallelism: Used to parallelise the matrix operation in D. The matrix is partitioned into smaller sub-matrices, and each sub-matrix is processed by a different process in parallel. This approach allowed multiple processes to work on different parts of the matrix simultaneously, which can significantly reduce the overall execution time.

Answer 2

1. Process of identifying and prioritising the most important parameters in a simulation model that have the greatest impact on the output. 2. An important tool for uncertainty quantification and sensitivity analysis in computational physics. 3. The ranking of uncertain parameters helps to reduce the dimensionality of the problem by focussing on a subset of parameters that have the greatest impact. 4. This approach can simplify the calibration, validation, and optimisation of the simulation model and improve its accuracy and reliability.

Answer 3

1. This method economises on the number of model evaluations, and guarantees to generate an elementary effect for each uncertain parameter 2. helps to identify the most important parameters in a simulation model, guiding researches to focus their efforts and resources on the parameters that have the highest impact on the output of the model. 3. Increases the transparency and credibility of the Simulation model, allowing researchers to explain and justify the choices and assumptions made in the model calibration and validation process. 4. Can help to identify areas of the model that need improvement or further investigation, such as parameters that have high uncertainty or that are highly sensitive to changes in the model. 5. Can help to improve the accuracy and reliability of the simulation model, as the focus is on the parameters that have the greatest impact on the output.

Answer 4

1. Morris method identifies important parameters in a simulation model. 2. It requires defining ranges of uncertain parameters and generating parameter sets. 3. Model output is evaluated for each parameter set and elementary effects are calculated. 4. Morris indices are calculated from the average and standard deviation of elementary effects to rank parameters. 5. Robustness of rankings is checked by performing the method on multiple independent sets of parameter samples.

Answer 5

May not be the most effective approach because algorithm A, B and C cannot themselves be parallelised. In a functional decomposition strategy, the problem is decomposed into a set of functions or subroutines that can be executed in parallel. In this case, the independent algorithms A, B, and C cannot be further decomposed into smaller functions that can be executed independently.

Answer 6

-- Fraction of the code that can be parallelised is now 1/4, since only D is left to be parallelised. -- Assuming perfect parallelism for D, this means that the parallel portion of the code can be run in 1/8th of the original time (since there are 8 processes available) -- Therefore, the maximum theoretical parallel speedup can be calculated using Ahmdals law as: Maximum speedup = 1/[(1-1/4) + (1/4)/8] = 1/(3/4 + 1/32) = 1/(25/32) = 1.28 So the maximum theoretical parallel speedup for the new application is 1.28

Answer 7

1. it is symplectic - preserves the Hamiltonian structure, and hence the energy conservation property of the system. 2. It is time-reversible - can be run forwards / backwards in time with equal accuracy. 3. It is second-order accurate in time - error in integration decreases quadratically with decreasing time step size. 4. It is computationally efficient, requiring only one function evaluation per time step size.

Answer 8

1. the system is conservative - energy is conserved over time. 2. The system has a periodic / oscillatory behaviour. 3. The system has a well-defined Hamiltonian structure (can be described in terms of its position and momentum) Compared to Runge-Kutta methods, Leapfrog is advantageous for long-term problems / those that require high accuracy. It is therefore more computationally efficient better with long term stability properties

Answer 9

If the constraint ω_0Δt > 2 is violated, then the numerical solution behaves in the following ways: 1. The numerical solution becomes unstable and oscillates with an amplitude that grows with time. 2. The solution may diverge to infinity or oscillate with an amplitude that is not physically realistic. 3. The energy of the system may increase with time, which is not possible for a conservative system. 4. The numerical solution may fail to converge or converge slowly, leading to inaccurate results.

Answer 10

1. Stiffness is a problem because the time step required to accurately resolve the stiff dynamics is often much smaller than the tilmestep requires to resolve other, non-stiff dynamics in the system. 2. This means that using a fixed time step for the entire simulation can result in very slow simulations, since the time-set must be set to the size required by the stiff dynamics. 3. Alternatively, if a larger time step is used, the numerical solution may become useable and diverge from the true solution due to numerical error. 4. Stiffness can also make the simulation more difficult to parallelise, since different processes may meed to use different time steps to accurately resolve the dynamics.

Answer 11

1. Implicit methods - treat the stiff terms implicitly and the non-stop terms explicitly. They are generally more stable and can handle larger time steps than explicit methods. 2. Operator splitting - splits the system of equations into stiff and non-stiff parts and treats each part separately using appropriate numerical methods. 3. Extrapolation and deferred correction - use multiple evaluations of a less accurate method to approximate a higher-order method, allowing for larger time steps to be taken, especially useful in cases where the stiffness varies over time

Answer 12

1. In the context of solving stiff differential equations, an implicit integration scheme can be beneficial as it can handle stiffness better than explicitly schemes. 2. However the stability of this specific implicit leapfrog scheme is limited by the value of ω_0Δt. 3. if the stiffness of the problem is such that ω_0Δt > 2, this method would not be suitable, and an alternative implicit scheme should be considered for better stability

Answer 13

1. The backward Euler method for integrating the ODE dy/dx = -y is given by the equation y_{n+1} = y_n - Δx y_{n+1}, where y_{n+1} is the value of y at the next time step and y_n is the value of y at the current time step. 2. The backward Euler method is an implicit method, meaning that y_{n+1} appears on both sides of the equation and needs to be solved for iteratively. 3. The backward Euler method is unconditionally stable, meaning that it is always stable for any choice of Δx. 4. Unlike the forward Euler method, which becomes increasingly unstable as Δx is increased, the backward Euler method remains stable for any Δx. However, it may become computationally expensive as Δx is decreased, as more iterations are required to solve for y_{n+1}.

Answer 14

1. refers to a situation where the solution of a system of differential equations changes very rapidly in some variables but very slowly in others. 2. It occurs when the system has widely different characteristic time scales. 3. The can cause numerical integration methods to become unstable or require very small time steps to maintain stability, leading to slow convergence and computational inefficiency.

Answer 15

Forward Euler method: 1. An explicit integration method that can be unstable for stiff systems. 2. The algorithms behaviour for various step sizes is dependent on the size of the time step, with the solution becoming more unstable as the step size increases. Backward Euler method: 1. An implicit integration method that can be more stable for still systems than explicit methods. 2. The algorithm is unconditionally stable for any step size, but its accuracy decreases as the step size increases. Desirability of simplicity integration: 1. They are unconditionally stable and can handle large step sizes, unlike explicit methods. 2. Use the current time step and the next time step to compute the solution, making them less sensitive to the system's stiffness.

Answer 16

1. Require solving a system of nonlinear equations at each time step, which can be computationally expensive. 2. The convergence of the iterative solution of the nonlinear system can be slow, particularly when the stiffness is severe. 3. The choice of the initial guess for the iterative solver can be critical for achieving convergence. 4. The stiffness can vary over time, making it challenging to select the best algorithm and time step sizes.