Laplace Equation Inversion Introduction – Maths Assignment Help

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Laplace equation inversion Introduction In this assignment, we will build on top of our work from the previous assignment. Previously, we built a Laplace equation solver that can go from initial conditions to the steady-state solution. For details on that, please refer to the instruction document on CA2 and your own submission for it. In many areas of science and engineering, we have equations to describe the behaviour of a system, where, given a set of initial conditions and system parameters, the equation can describe the behaviour of the system at steady-state or as a function of time. These equations are usually partial differential equations (PDEs), and PDE-based simulations of such systems are a primary driver of high-performance computing research. In CA2 previously, we saw an example where we used the Laplace equation (a PDE), to calculate the steady-state temperature distribution of a room. Inverse Problems In real-life problems, it is often the case that the parameters of the system are harder to measure than the final observations. For example, when studying wave propagation through a medium, it is easier to record the received signal than it is to measure the speed of sound at every point in the medium. Sometimes our interest is in finding out what initial conditions would lead to a desired final state. These are both examples of inverse problems – where we are solving for the inputs to a PDE, already knowing the output, essentially solving the PDE in reverse (although, as you will see, we are not literally reversing the PDE). As computer systems get faster and bigger, the utility of inverse problems is rapidly rising, especially in the last decade. It is common to see inverse problems in Astrophysics (predicting gravitational waves), Geophysics (understanding earthquakes, tsunamis), Computational imaging (making images of the human brain, or of inside the earth), and also in computational design (what shape of a car produces the maximum amount of downforce while meeting a prescribed weight target?) In this assessment, we will solve an inverse version of the problem from CA2.

Instead of solving the Laplace equation for a given set of initial conditions, we will invert for the initial conditions, knowing the steady-state distribution. Most realistic inverse problems are solved using gradient-based methods, which is out of scope for our module (but feel free to read about it in your time). Here we will carry out the inversion using a modified version of the bisection method. 2021-2022 Problem Statement Solve for this temperature 10m 4m 10m 10?C 37?C (5.0, 5.0) Figure 1: 10m × 10m room with a radiator As before, we have a two-dimensional room with four walls that are 10m long each. The four walls are held at a constant temperature of 10?C. One of the walls has a 4m long radiator attached to the middle of the wall. The temperature of this radiator is also constant but that constant setting is unknown. This is a radiator that can be set to any temperature using a thermostat and our objective is to find out what temperature to set it to, such that the middle of the room (5m, 5m) is at exactly 37?C (within floating-point precision) in the steady-state distribution. Bisection method The Bisection method (also called bisection search) is a very old and well-studied method for finding roots of a smooth, continuous function when given an interval within which at least one root lies. 2021-2022 2 Suppose the function f(x) describes the temperature at the middle of the room (in the steadystate), for a given radiator temperature, for example, f(100) is the temperature of the middle of the room if the radiator is set to 100?C. The equation we are interested in solving here is: f(x) = 37 (1) , where the function f is a solution of the Laplace equation (refer to CA2 for more information).

Purely from intuition, we can tell that the function f is: • Smooth: It doesn’t exhibit jumpy behaviour. From intuition of our physical world, we know that the temperature of the middle of the room will change in predictable ways when changing the temperature of the radiator. • Continuous: Again from intuition, we know that changing the temperature of the radiator by 10?C won’t suddenly change the temperature of the room by 100?C. • Monotonous: We know that increasing the temperature of the radiator will only increase the temperature of the room, and vice-versa. These properties make the function f an excellent target for the bisection method. The bisection method involves choosing two starting points, a0 and b0, where we know that the solution x lies in the interval (a0, b0). In other words: a0 < x xss=removed> 37, that means x lies somewhere in the interval (a0, c), so we set a1 = a0, b1 = c. However, if we see that f(c) < 37 xss=removed xss=removed>

If we can evaluate the function n times per iteration, we can reduce the interval by a factor of n + 1, speeding up convergence. Running over MPI, we can set n to be the number of ranks. At every iteration, every rank runs an instance of your Laplace equation solver from CA2, with a slightly different value of the radiator temperature. At the end of the Laplace equation solve, you get the final temperatures for each of the different radiator temperatures. Use this to decide the interval to be explored in the next iteration, continuing until the interval is as small as machine epsilon or until a maximum of 1000 iterations. Having done this, you would have written a program that uses both MPI and OpenMP together. I hope you are proud of yourself. Problem specifications N = 1024, K = 100000. Wall temperatures at 10?C. a0 = 10, b0 = 5000. Target temperature at the middle (5.0, 5.0) of the room should be 37?C. Your final submission should include a slurm submission script that works with the following command sba tch ?c ?p run?mpi . sh to call your program on the Barkla HPC system in batch mode. Note that the ?c setting above sets the number of cores per process, which affects the speed of your Laplace solver, and 2021-2022 4 the ?p affects the number of ranks and hence the convergence speed of your search method. Try a few different values for these and report solution times for them. Note that the product of the two numbers should always be less than 40, i.e. calling sbatch ?c 20 ?p 6 run?mpi.sh is expected to never complete, since it asks for 20*6=120 cores, when the nodes of Barkla only have 40 cores each. All computations should be in double precision. Only use sbatch to run your programs in batch mode. Do not run these programs on the login nodes of Barkla.