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// Licensed to the .NET Foundation under one or more agreements.
// The .NET Foundation licenses this file to you under the MIT license.
// See the LICENSE file in the project root for more information.
//
// .NET SIMD to solve Burgers' equation
//
// Benchmark based on
// https://github.com/taumuon/SIMD-Vectorisation-Burgers-Equation-CSharp
// http://www.taumuon.co.uk/2014/10/net-simd-to-solve-burgers-equation.html
using Microsoft.Xunit.Performance;
using System;
using System.Linq;
using System.Numerics;
using System.Runtime.CompilerServices;
[assembly: OptimizeForBenchmarks]
public class Burgers
{
private static double BurgersAnalytical(double t, double x, double nu)
{
return -2 * nu * (-(-8 * t + 2 * x) * Math.Exp(-Math.Pow((-4 * t + x), 2) / (4 * nu * (t + 1))) / (4 * nu * (t + 1)) - (-8 * t + 2 * x - 12.5663706143592) * Math.Exp(-Math.Pow(-4 * t + x - 6.28318530717959, 2) / (4 * nu * (t + 1))) / (4 * nu * (t + 1))) / (Math.Exp(-Math.Pow(-4 * t + x - 6.28318530717959, 2) / (4 * nu * (t + 1))) + Math.Exp(-Math.Pow(-4 * t + x, 2) / (4 * nu * (t + 1)))) + 4;
}
private static double[] linspace(double first, double last, int num)
{
var step = (last - first) / (double)num;
return Enumerable.Range(0, num).Select(v => (v * step) + first).ToArray();
}
private static double[] GetAnalytical(double[] x, double t, double nu)
{
double[] u = new double[x.Length];
for (int i = 0; i < x.Length; ++i)
{
u[i] = BurgersAnalytical(t, x[i], nu);
}
return u;
}
private static double[] GetCalculated0(int nt, int nx, double dx, double dt, double nu, double[] initial)
{
double[] u = new double[nx];
Array.Copy(initial, u, u.Length);
for (int tStep = 0; tStep < nt; tStep++)
{
double[] un = new double[nx];
Array.Copy(u, un, u.Length);
for (int i = 1; i < nx - 1; i++)
{
u[i] = un[i] - un[i] * dt / dx * (un[i] - un[i - 1]) + Math.Pow(nu * dt / dx, 2.0) *
(un[i + 1] - 2 * un[i] + un[i - 1]);
}
u[0] = un[0] - un[0] * dt / dx * (un[0] - un[nx - 1]) + Math.Pow(nu * dt / dx, 2.0) *
(un[1] - 2 * un[0] + un[nx - 1]);
u[nx - 1] = un[nx - 1] - un[nx - 1] * dt / dx * (un[nx - 1] - un[nx - 2]) + Math.Pow(nu * dt / dx, 2.0) *
(un[0] - 2 * un[nx - 1] + un[nx - 2]);
}
return u;
}
// Reduce new array allocation and copying, ping-pong between them
private static double[] GetCalculated1(int nt, int nx, double dx, double dt, double nu, double[] initial)
{
double[] u = new double[nx];
double[] un = new double[nx];
Array.Copy(initial, un, un.Length);
for (int tStep = 0; tStep < nt; tStep++)
{
for (int i = 1; i < nx - 1; i++)
{
u[i] = un[i] - un[i] * dt / dx * (un[i] - un[i - 1]) + Math.Pow(nu * dt / dx, 2.0) *
(un[i + 1] - 2 * un[i] + un[i - 1]);
}
u[0] = un[0] - un[0] * dt / dx * (un[0] - un[nx - 1]) + Math.Pow(nu * dt / dx, 2.0) *
(un[1] - 2 * un[0] + un[nx - 1]);
u[nx - 1] = un[nx - 1] - un[nx - 1] * dt / dx * (un[nx - 1] - un[nx - 2]) + Math.Pow(nu * dt / dx, 2.0) *
(un[0] - 2 * un[nx - 1] + un[nx - 2]);
double[] swap = u;
u = un;
un = swap;
}
return un;
}
// Pull calculation of (nu * dt / dx)^2 out into a variable
private static double[] GetCalculated2(int nt, int nx, double dx, double dt, double nu, double[] initial)
{
double[] u = new double[nx];
double[] un = new double[nx];
Array.Copy(initial, un, un.Length);
double factor = Math.Pow(nu * dt / dx, 2.0);
for (int tStep = 0; tStep < nt; tStep++)
{
for (int i = 1; i < nx - 1; i++)
{
u[i] = un[i] - un[i] * dt / dx * (un[i] - un[i - 1]) + factor *
(un[i + 1] - 2 * un[i] + un[i - 1]);
}
u[0] = un[0] - un[0] * dt / dx * (un[0] - un[nx - 1]) + factor *
(un[1] - 2 * un[0] + un[nx - 1]);
u[nx - 1] = un[nx - 1] - un[nx - 1] * dt / dx * (un[nx - 1] - un[nx - 2]) + factor *
(un[0] - 2 * un[nx - 1] + un[nx - 2]);
double[] swap = u;
u = un;
un = swap;
}
return un;
}
// SIMD
private static double[] GetCalculated3(int nt, int nx, double dx, double dt, double nu, double[] initial)
{
var nx2 = nx + (Vector<double>.Count - (nx % Vector<double>.Count));
double[] u = new double[nx2];
double[] un = new double[nx2];
Array.Copy(initial, un, initial.Length);
double factor = Math.Pow(nu * dt / dx, 2.0);
for (int tStep = 0; tStep < nt; tStep++)
{
for (int i = 1; i < nx2 - Vector<double>.Count + 1; i += Vector<double>.Count)
{
var vectorIn0 = new Vector<double>(un, i);
var vectorInPrev = new Vector<double>(un, i - 1);
var vectorInNext = new Vector<double>(un, i + 1);
var vectorOut = vectorIn0 - vectorIn0 * (dt / dx) * (vectorIn0 - vectorInPrev) + factor *
(vectorInNext - 2.0 * vectorIn0 + vectorInPrev);
vectorOut.CopyTo(u, i);
}
u[0] = un[0] - un[0] * dt / dx * (un[0] - un[nx - 1]) + factor *
(un[1] - 2 * un[0] + un[nx - 1]);
u[nx - 1] = un[nx - 1] - un[nx - 1] * dt / dx * (un[nx - 1] - un[nx - 2]) + factor *
(un[0] - 2 * un[nx - 1] + un[nx - 2]);
double[] swap = u;
u = un;
un = swap;
}
return un;
}
public static int Main()
{
if (!Vector.IsHardwareAccelerated)
{
Console.WriteLine("Not hardware accelerated!");
}
else
{
Console.WriteLine("Vector<double>.Length: " + Vector<double>.Count);
}
int nx = 10001;
#if DEBUG
int nt = 10;
#else
int nt = 10000;
#endif
double dx = 2.0 * Math.PI / (nx - 1.0);
double nu = 0.07;
double dt = dx * nu;
double[] x = linspace(0.0, 2.0 * Math.PI, nx);
double[] initial = GetAnalytical(x, 0.0, nu);
// Warmup
GetCalculated0(1, nx, dx, dt, nu, initial);
GetCalculated1(1, nx, dx, dt, nu, initial);
GetCalculated2(1, nx, dx, dt, nu, initial);
GetCalculated3(1, nx, dx, dt, nu, initial);
double[][] results = new double[4][];
var stopwatch = new System.Diagnostics.Stopwatch();
stopwatch.Start();
results[0] = GetCalculated0(nt, nx, dx, dt, nu, initial);
stopwatch.Stop();
Console.WriteLine("Baseline: " + stopwatch.ElapsedMilliseconds);
stopwatch.Reset();
stopwatch.Start();
results[1] = GetCalculated1(nt, nx, dx, dt, nu, initial);
stopwatch.Stop();
Console.WriteLine("Reduce copy: " + stopwatch.ElapsedMilliseconds);
stopwatch.Reset();
stopwatch.Start();
results[2] = GetCalculated2(nt, nx, dx, dt, nu, initial);
stopwatch.Stop();
Console.WriteLine("CSE of Math.Pow: " + stopwatch.ElapsedMilliseconds);
stopwatch.Reset();
stopwatch.Start();
results[3] = GetCalculated3(nt, nx, dx, dt, nu, initial);
stopwatch.Stop();
Console.WriteLine("SIMD: " + stopwatch.ElapsedMilliseconds);
stopwatch.Reset();
for (int i = 0; i < x.Length; i += 33)
{
double expected = results[0][i];
for (int j = 1; j < results.Length; j++)
{
bool valid = Math.Abs(expected - results[j][i]) < 1e-4;
if (!valid)
{
Console.WriteLine("Failed to validate");
return -1;
}
}
}
return 100;
}
static volatile object VolatileObject;
[MethodImpl(MethodImplOptions.NoInlining)]
static void Escape(object obj)
{
VolatileObject = obj;
}
[Benchmark]
public static void Test0()
{
int nx = 10001;
int nt = 10000;
double dx = 2.0 * Math.PI / (nx - 1.0);
double nu = 0.07;
double dt = dx * nu;
double[] x = linspace(0.0, 2.0 * Math.PI, nx);
double[] initial = GetAnalytical(x, 0.0, nu);
foreach (var iteration in Benchmark.Iterations)
{
using (iteration.StartMeasurement())
{
double[] results = GetCalculated0(nt, nx, dx, dt, nu, initial);
Escape(results);
}
}
}
[Benchmark]
public static void Test1()
{
int nx = 10001;
int nt = 10000;
double dx = 2.0 * Math.PI / (nx - 1.0);
double nu = 0.07;
double dt = dx * nu;
double[] x = linspace(0.0, 2.0 * Math.PI, nx);
double[] initial = GetAnalytical(x, 0.0, nu);
foreach (var iteration in Benchmark.Iterations)
{
using (iteration.StartMeasurement())
{
double[] results = GetCalculated1(nt, nx, dx, dt, nu, initial);
Escape(results);
}
}
}
[Benchmark]
public static void Test2()
{
int nx = 10001;
int nt = 10000;
double dx = 2.0 * Math.PI / (nx - 1.0);
double nu = 0.07;
double dt = dx * nu;
double[] x = linspace(0.0, 2.0 * Math.PI, nx);
double[] initial = GetAnalytical(x, 0.0, nu);
foreach (var iteration in Benchmark.Iterations)
{
using (iteration.StartMeasurement())
{
double[] results = GetCalculated2(nt, nx, dx, dt, nu, initial);
Escape(results);
}
}
}
[Benchmark]
public static void Test3()
{
// Make SIMD version work a bit harder....
int nx = 10001;
int nt = 2 * 10000;
double dx = 2.0 * Math.PI / (nx - 1.0);
double nu = 0.07;
double dt = dx * nu;
double[] x = linspace(0.0, 2.0 * Math.PI, nx);
double[] initial = GetAnalytical(x, 0.0, nu);
foreach (var iteration in Benchmark.Iterations)
{
using (iteration.StartMeasurement())
{
double[] results = GetCalculated3(nt, nx, dx, dt, nu, initial);
Escape(results);
}
}
}
}
|
