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# HW 5: GPU Programming

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CSCE 689-606
HW 5: GPU Programming

Gaussian Process Regression can be used to predict the values of a function at a point from
observations at other points in the domain. As an example, consider an m x m grid of points on a
two-dimensional unit square. Node coordinates are given by
(xi,yj) = (ih,jh), i=1,..,m, j=1,…,m, (1)
where h = 1/(m+1) is the mesh width. Observed data value at the point r=(xi,yj) is given by the
function
f(xi,yj) = 1−[(xi−0.5)2+(yi−0.5)2]+dij,
(2)
where dij is a random value between [−0.05,0.05]. The predicted value of the function at r*=(x,y) is
given by
f(x,y) = kT(tI+K)-1f. (3)
K is an n x n matrix with elements K(r,s)=exp(−||r−s||2), where ||r−s||2 is the distance between grid
points r and s, and exp(a)=ea (e is the base of the natural logarithm). Note that for an m x m grid,
n=m2. Further, k is an n x 1 vector with elements k(r) = exp(-||r−r*||2), f is the vector of observed
data values given by equation (2), and t is a noise parameter that is set to 0.01.
A Matlab file GPR.m is provided as an illustration.
1. (70 points) In this assignment, you have to develop GPU code to compute f(x,y) for a given point
(x,y). The code should use a single processor of the device but should be parallelized to exploit
all the cores within the processor. The code should initialize the grid points and observed data
values using equations (1) and (2) on the host and move these values to the GPU device. Next,
the matrix K’ = (tI+K) should be computed on the device. This should be followed by LU
factorization of K’ on the device. These factors should be used to compute the solution of the
system K’z=f using the L and U factors obtained in the previous step. Finally, the predicted value
should be computed as f(x,y) = kTz. You must develop your own code to compute the LU factors
and to solve the triangular systems. LU factorization can be replaced by Cholesky factorization,
which is a more efficient algorithm for symmetric positive definite matrices.
2. (20 points) Describe your strategy to parallelize the algorithm for a single multiprocessor of
the GPU. Discuss any design choices you made to improve the parallel performance of the code.
3. (10 points) Compute the flop rate you achieve in the factorization routine and in the solver
routine. Compare this value with the peak flop rate achievable on the processor, and estimate
the speedup obtained over one core and the corresponding efficiency/utilization of the cores on
the device. You may choose appropriate values for the grid size to study the features of your
implementation.
Submission: You need to upload the following to eCampus:
1. Submit the code you developed.
2. Submit a single PDF or MSWord document that includes the following.
• Responses to Problem 1, 2, and 3. Response to 1 should consist of a brief description of
how to compile and execute the code on the parallel computer
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