159.735 Assignment 4

Modelling Gravitational Lensing on a GPU

Gravitational (micro)lensing occurs when a foreground star (the lens) passes

along the line-of-sight of a background star (the source). A consequence of

Einstein’s theory of general relativity is that light rays do not always travel in

straight lines. A light ray will follow the natural ”curvature” of space-time if

it passes by an object that has mass. The result of this is that the foreground

star acts as a lens on the background star. The lens star will produce two

arc shaped images of the source star. More complicated image patterns are

produced if there are several lens stars. In this exercise, you will see what

they look like.

In gravitational lensing, the mapping between the source and the lens plane

is given by the lens equation. If the two dimensional source, ζ and image

positions, z, are expressed in complex form, the lens equation for NL lenses

with positions zm,

In principle, given the position of the source star, one can solve this equation

to derive the image positions on the source plane. In practice, this is very

difficult to do, as it requires one to solve a high order complex polynomial. A

conceptually simple way is to use the technique of “inverse ray tracing”. the

lens plane is divided into pixels and a ray is shot from each pixel position.

The corresponding position on the source plane is then calculated directly

from the lens equation. If the ray shot from a particular pixel happens to

land on the source star, then that pixel forms part of the gravitationally

lensed image of the star. This works for simple lens configurations and very

complex patterns resulting from many lenses.

The Assignment

On the Stream site, download a copy of lensing.tar and unpack this.

This package includes lenses.cpp which provides an implementation of the

lens equation and a startup program lens demo.cpp You need to complete

the section of the code that generates the lens image. Try the sequential

version for different lens configurations. You can use ds9 to view the images.

Note that stars are not generally of uniform brightness across their discs.

They appear darker towards the edge. You can include this ”limb darkening”

by weighting the lens image pixel according to where the ray lands within

the source star disc.

and r is the distance from the center of the star where the ray lands, rstar

is the radius of the star, and λ is the limb darkening coefficient (typically

λ = 0.5).

Now write a CUDA implementation that runs on the GPU. You

may use the GPU cards installed in the lab workstations, or any other card

on which you have access.

Submission

Please submit your C or C++ source code together with a report where you

should address at least the following.

? State the GPU card that use implement your assignment. Give the

number of processing units on that card and any other information

that is relevant.

? Describe how you implemented your solution to the inverse ray tracing

problem on the GPUs. Marks will be awarded for thoughtful answers

that demonstrate your understanding of the GPU architecture.

? Try running your program for different problem sizes, ie try generating

lens images of different resolutions and for different sizes of the lensing

system. Comment on the performances you see.

Due date: TBD

This assignment is worth 20% of your final grade

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