Advanced-Systems-Lab

Many problems in scientific computing include: Solving linear equations Eigenvalue computations Singular value decomposition LU/Cholesky/QR decompositions etc… And the userbase is quite large for this types of computation (number of scientists in the world is growing exponentially ) Quick History of Performance Computing Early seventies it was EISPACK and LINPACK. Then In similar years Matlab was invented, which simplified a lot compared to previous systems. LAPACK redesigned the algorithms in previous libraries to have better block-based locality. BLAS are kernel functions for each computer, while LAPACK are the higher level functions build on top of BLAS (1, 2,3). Then another innovation was ATLAS, which automatically generates the code for BLAS for each architecture. This is called autotuning because it does a search of possible enumerations and chooses the fastest one. Now autotuning has been done a lot for NN systems. ...

Locality principles Remember the two locality principles in Memoria. Temporal locality and spatial locality. Temporal Locality Some elements just are accessed many times in time. This is an example of a temporal locality. Spatial locality Some elements are accessed close to each other, this is an idea of spatial locality. In modern architectures, the a line of cache is usually 64 bytes. For example consider this snippet: sum = 0; for (i = 0; i < n; i++) sum += a[i]; return sum; Sum is an example of temporal locality as the same memory location (or register) is accessed many times, and the access of the array a is an example of spatial locality. loops cycle through the same instructions, this is an example of temporal locality. ...

On Compiler Adding compilation flags to gcc not always makes it faster, it just enables a specific set of optimization methods. It’s also good to turn on platform specific flags to turn on some specific optimization methods to that architecture. Remember that compilers are conservative, meaning they do not apply that optimization if they think it does not always apply. What are they good at Compilers are good at: mapping program to machine ▪ register allocation ▪ instruction scheduling ▪ dead code elimination ▪ eliminating minor inefficiencies ...

The algorithm has been the same, some ideas are in Fourier Series, but architectures change, which means there are new ways to make this algorithm even faster. Example of transforms We have learned in Algebra lineare numerica, Cambio di Base that linear transforms are usually a change of basis. They are matrix vector multiplications (additions and multiplications by constants). The optimizations are based on what sorts of transforms we have (e.g. Sparse Matrix Vector Multiplication, or dense versions). The same idea applies also for Fourier transforms. ...

The Skylake processor is a 2015 Intel processor. The Intel Processor In 1978 Intel made the choice to have retrocompatibility for every processor. At that time they had the 8086 processor, with some number of memory bits. For backwards compatibility intructions have usually just grown. They used geographic locations because these are not suable. If we want new code to run for old processors, we should need to put specific flags. ...

Algorithms for Sparse Matrix-Vector Multiplication Compressed Sparse Row This is an optimized way to store rows for sparse matrices: Sparse MVM using CSR void smvm(int m, const double* values, const int* col_idx, const int* row_start, double* x, double* y) { int i, j; double d; /* Loop over m rows */ for (i = 0; i < m; i++) { d = y[i]; /* Scalar replacement since reused */ /* Loop over non-zero elements in row i */ for (j = row_start[i]; j < row_start[i + 1]; j++) { d += values[j] * x[col_idx[j]]; } y[i] = d; } } Let’s analyze this code: Spatial locality: with respect to row_start, col_idx and values we have spatial locality. Temporal locality: with respect to y we have temporal locality. (Poor temporal with respect to $x$) Good storage efficiency for the sparse matrix. But it is 2x slower than the dense matrix multiplication when the matrix is dense. Block CSR But we cannot do block optimizations for the cache with this storage method. ...