# (Simple) Multigrid solver

This example a multigrid solver that uses the grids $$\Omega_{h}$$, $$\Omega_{2h}$$, $$\Omega_{4h}$$, $$\Omega_{8h}$$

## One iteration a multigrid V-cycle

### Code

#include <cmath>

#include <hpc/matvec/axpy.hpp>
#include <hpc/matvec/copy.hpp>
#include <hpc/matvec/densevector.hpp>
#include <hpc/matvec/iterators.hpp>
#include <hpc/matvec/traits.hpp>
#include <hpc/matvec/print.hpp>

//------------------------------------------------------------------------------

using namespace hpc;
using namespace hpc::matvec;

template <typename T>
const T &
assertEqual(const T &a, const T &b)
{
assert(a==b);
return a;
}

//------------------------------------------------------------------------------

template <
typename F, typename G,
typename T, template<typename> class VectorF,
template<typename> class VectorU,
Require< Dense<VectorF<T>>,
Dense<VectorU<T>> > = true
>
void
discretizeProblem(const F &f, const G &g, VectorF<T> &f_h, VectorU<T> &u_h)
{
auto N  = assertEqual(f_h.length(), u_h.length()) - 2;
auto h  = T(1)/(N+1);

u_h(0)   = g(0);
u_h(N+1) = g(1);

for (std::size_t i=1; i<N; ++i) {
u_h(i) = 0;
f_h(i) = f(i*h);
}
// actually we don't have to initialize f_h(0), f_h(N+1) so we store some
// good and evil numbers there:
f_h(0)   = 42;
f_h(N+1) = 666;
}

//------------------------------------------------------------------------------

template <
typename T, template<typename> class Vector,
Require< Dense<Vector<T>> > = true
>
T
lInfNorm(const Vector<T> &x_h)
{
T res = 0;

for (auto [i, xi] : x_h) {
i=i;
if (std::abs(xi) > res) {
res = std::abs(xi);
}
}
return res;
}

//------------------------------------------------------------------------------

template <
typename U,
typename T, template<typename> class VectorU,
template<typename> class VectorE,
Require< Dense<VectorU<T>>,
Dense<VectorE<T>> > = true
>
void
evalError(const U &u, const VectorU<T> &u_h, VectorE<T> &e_h)
{
auto N  = assertEqual(u_h.length(), e_h.length()) - 2;
auto h  = T(1)/(N+1);

for (auto [i, ei] : e_h) {
ei = u(i*h) - u_h(i);
}
}

//------------------------------------------------------------------------------

template <
typename T, template<typename> class VectorU,
template<typename> class VectorF,
template<typename> class VectorR,
Require< Dense<VectorF<T>>,
Dense<VectorU<T>>,
Dense<VectorR<T>> > = true
>
void
evalResidual(const VectorU<T> &u_h, const VectorF<T> &f_h, VectorR<T> &r_h)
{
auto N  = assertEqual(u_h.length(), r_h.length()) - 2;
auto h  = T(1)/(N+1);
auto h2 = h*h;

r_h(0) = r_h(N+1) = 0;
for (std::size_t i=1; i<=N; ++i) {
r_h(i) = f_h(i) - 1./h2 * (2*u_h(i) - u_h(i-1) - u_h(i+1));
}
}

//------------------------------------------------------------------------------

template <
typename T, template<typename> class VectorF,
template<typename> class VectorU,
Require< Dense<VectorF<T>>,
Dense<VectorU<T>> > = true
>
void
direct_solver(const VectorF<T> &f_h, VectorU<T> &u_h)
{
auto N  = assertEqual(f_h.length(), u_h.length()) - 2;
auto h  = T(1)/(N+1);
auto h2 = h*h;

// forward substitution
u_h(1) = h2*f_h(1) + u_h(0);
for (std::size_t k=2; k<N; ++k) {
u_h(k) = h2*f_h(k) + u_h(k-1)*(k-1)/k;
}
u_h(N) = h2*f_h(N) + u_h(N+1) + u_h(N-1)*(N-1)/N;

// backward substitution
u_h(N) = u_h(N)*N/(N+1);
for (std::size_t k=N-1; k>=1; --k) {
u_h(k) = (u_h(k)+u_h(k+1))*k/(k+1);
}
}

//==============================================================================

template <
typename T, template<typename> class Vector,
Require< Dense<Vector<T>> > = true
>
void
{
auto N  = x_h.length()-2;
auto h  = T(1)/(N+1);

for (std::size_t i=1; i<=N; ++i) {
x_h(i) += sin(M_PI*k*i*h);
}
}

//------------------------------------------------------------------------------

template <
typename T, template<typename> class VectorF,
template<typename> class VectorU,
template<typename> class VectorS,
Require< Dense<VectorF<T>>,
Dense<VectorU<T>>,
Dense<VectorS<T>> > = true
>
void
jacobi_step(double omega, const VectorF<T> &f_h, const VectorU<T> &u_h,
VectorS<T> &s_h)
{
auto N  = assertEqual(f_h.length(), u_h.length()) - 2;
auto h  = T(1)/(N+1);
auto h2 = h*h;

s_h(0)   = u_h(0);
s_h(N+1) = u_h(N+1);
}

//------------------------------------------------------------------------------

int
main()
{
/*
auto f = [](double x) { return -2; };
auto g = [](double x) { return 1 + 3*x; };
auto u = [](double x) { return (x+1)*(x+1); };
*/
auto f = [](double x) { return 0; };
auto g = [](double x) { return 0; };
auto u = [](double x) { return 0; };

std::size_t N = 100;

DenseVector<double>  f_h(N+2), u_h(N+2), tmp_h(N+2), r_h(N+2), e_h(N+2);

discretizeProblem(f, g, f_h, u_h);

// First initialize u_h with exact solution
direct_solver(f_h, u_h);

double omega = 0.5;     // smoothing parameter

for (std::size_t it=0; it<=20; ++it) {
jacobi_step(omega, f_h, u_h, tmp_h);
jacobi_step(omega, f_h, tmp_h, u_h);

evalResidual(u_h, f_h, r_h);
evalError(u, u_h, e_h);

fmt::printf("#lInfNorm(r_h) = %.5e\n", lInfNorm(r_h));
fmt::printf("#lInfNorm(e_h) = %.5e\n", lInfNorm(e_h));
fmt::printf("it %3d\n", it);
for (auto [i,ri] : r_h) {
fmt::printf("%d %f\n", i, ri);
}
fmt::printf("\n\n");
}
}


### Code for gnuplot

set terminal svg size 900, 500
set output "restrict_post_smooth.svg"
set xlabel "Grid points"
set ylabel "Residual value"
set title "Smoothing"
set key outside
set pointsize 0.5

plot "restrict_post_smooth.data" index  0 using 1:2 with linespoints lt 1 lw 1, \
"restrict_post_smooth.data" index  1 using 1:2 with linespoints lt 2 lw 3


### How to compile and plot results

theon$g++ -Wall -I /home/numerik/pub/hpc/ws18/session25/ -std=c++17 -o restrict_post_smooth restrict_post_smooth.cpp theon$ ./restrict_post_smooth > restrict_post_smooth.data
theon$ ### Plotting results theon$ gnuplot restrict_post_smooth.gnuplot
theon$gnuplot restrict_post_smooth2.gnuplot theon$ gnuplot restrict_post_smooth3.gnuplot
theon$gnuplot restrict_post_smooth4.gnuplot theon$ gnuplot restrict_post_smooth5.gnuplot
theon$gnuplot restrict_post_smooth6.gnuplot theon$ gnuplot restrict_post_smooth7.gnuplot
theon$gnuplot restrict_post_smooth8.gnuplot theon$ 

## Multigrid with 4 levels

### Code

#include <cmath>
#include <sstream>
#include <memory>
#include <vector>

#include <hpc/matvec/axpy.hpp>
#include <hpc/matvec/copy.hpp>
#include <hpc/matvec/densevector.hpp>
#include <hpc/matvec/iterators.hpp>
#include <hpc/matvec/traits.hpp>
#include <hpc/matvec/print.hpp>

//------------------------------------------------------------------------------

using namespace hpc;
using namespace hpc::matvec;

template <typename T>
const T &
assertEqual(const T &a, const T &b)
{
assert(a==b);
return a;
}

//------------------------------------------------------------------------------

template <
typename F, typename G,
typename T, template<typename> class VectorF,
template<typename> class VectorU,
Require< Dense<VectorF<T>>,
Dense<VectorU<T>> > = true
>
void
discretizeProblem(const F &f, const G &g, VectorF<T> &f_h, VectorU<T> &u_h)
{
auto N  = assertEqual(f_h.length(), u_h.length()) - 2;
auto h  = T(1)/(N+1);

u_h(0)   = g(0);
u_h(N+1) = g(1);

for (auto [i, fi] : f_h) {
fi = f(i*h);
}
}

//------------------------------------------------------------------------------

template <
typename U,
typename T, template<typename> class VectorU,
template<typename> class VectorE,
Require< Dense<VectorU<T>>,
Dense<VectorE<T>> > = true
>
void
evalError(const U &u, const VectorU<T> &u_h, VectorE<T> &e_h)
{
auto N  = assertEqual(u_h.length(), e_h.length()) - 2;
auto h  = T(1)/(N+1);

for (auto [i, ei] : e_h) {
ei = u(i*h) - u_h(i);
}
}

//------------------------------------------------------------------------------

template <
typename T, template<typename> class VectorU,
template<typename> class VectorF,
template<typename> class VectorR,
Require< Dense<VectorF<T>>,
Dense<VectorU<T>>,
Dense<VectorR<T>> > = true
>
void
evalResidual(const VectorU<T> &u_h, const VectorF<T> &f_h, VectorR<T> &r_h)
{
auto N  = assertEqual(u_h.length(), r_h.length()) - 2;
auto h  = T(1)/(N+1);
auto h2 = h*h;

r_h(0) = r_h(N+1) = 0;
for (std::size_t i=1; i<=N; ++i) {
r_h(i) = f_h(i) - 1./h2 * (2*u_h(i) - u_h(i-1) - u_h(i+1));
}
}

//------------------------------------------------------------------------------

template <
typename T, template<typename> class VectorR1,
template<typename> class VectorR2,
Require< Dense<VectorR1<T>>,
Dense<VectorR2<T>> > = true
>
void
restriction(const VectorR1<T> &r_h, VectorR2<T> &r_2h)
{
auto N  = r_h.length() -2;
auto n  = r_2h.length() -2;

assert(2*(n+1) == N+1);

r_2h(0) = r_2h(n+1) = 0;
for (std::size_t i=1; i<=n; ++i) {
r_2h(i) = 0.25*r_h(2*i-1) + 0.5*r_h(2*i) + 0.25*r_h(2*i+1);
}
}

//------------------------------------------------------------------------------

template <
typename T, template<typename> class VectorU1,
template<typename> class VectorU2,
Require< Dense<VectorU1<T>>,
Dense<VectorU2<T>> > = true
>
void
updateProlongation(const VectorU1<T> &u_2h, VectorU2<T> &u_h)
{
auto N  = u_h.length() -2;
auto n  = u_2h.length() -2;

assert(2*(n+1) == N+1);

for (std::size_t i=1; i<=n; ++i) {
u_h(2*i-1) += 0.5 * u_2h(i);
u_h(2*i  ) += 1.0 * u_2h(i);
u_h(2*i+1) += 0.5 * u_2h(i);
}
}

//------------------------------------------------------------------------------

template <
typename T, template<typename> class Vector,
Require< Dense<Vector<T>> > = true
>
void
{
auto N  = x_h.length()-2;
auto h  = T(1)/(N+1);

for (std::size_t i=1; i<=N; ++i) {
x_h(i) += sin(M_PI*k*i*h);
}
}

//------------------------------------------------------------------------------

template <
typename T, template<typename> class Vector,
Require< Dense<Vector<T>> > = true
>
T
lInfNorm(const Vector<T> &x_h)
{
T res = 0;

for (auto [i, xi] : x_h) {
i=i;
if (std::abs(xi) > res) {
res = std::abs(xi);
}
}
return res;
}

template <
typename T, template<typename> class Vector1,
template<typename> class Vector2,
Require< Dense<Vector1<T>>,
Dense<Vector2<T>> > = true
>
T
lInfNorm(const Vector1<T> &x1_h, const Vector2<T> &x2_h)
{
auto N  = assertEqual(x1_h.length(), x2_h.length());

T res = 0;

for (std::size_t i=0; i<N; ++i) {
if (std::abs(x1_h(i) - x2_h(i)) > res) {
res = std::abs(x1_h(i) - x2_h(i));
}
}
return res;
}

//------------------------------------------------------------------------------

template <
typename T, template<typename> class Vector,
Require< Dense<Vector<T>> > = true
>
T
l1Norm(const Vector<T> &x_h)
{
auto N  = x_h.length()-2;
auto h  = T(1)/(N+1);

T res = 0;

for (std::size_t i=1; i<=N; ++i) {
i=i;
res += 0.5*(std::abs(x_h(i-1)) + std::abs(x_h(i)));
}
return h*res;
}

//------------------------------------------------------------------------------

template <
typename T, template<typename> class VectorF,
template<typename> class VectorU,
template<typename> class VectorS,
Require< Dense<VectorF<T>>,
Dense<VectorU<T>>,
Dense<VectorS<T>> > = true
>
void
jacobi_step(double omega, const VectorF<T> &f_h, const VectorU<T> &u_h,
VectorS<T> &s_h)
{
auto N  = assertEqual(f_h.length(), u_h.length()) - 2;
auto h  = T(1)/(N+1);
auto h2 = h*h;

for (std::size_t i=1; i<=N; ++i) {
s_h(i) = (1-omega)*u_h(i) + omega*0.5*(u_h(i-1) +u_h(i+1)+h2*f_h(i));
}
s_h(0)   = u_h(0);
s_h(N+1) = u_h(N+1);
}

//------------------------------------------------------------------------------

template <
typename T, template<typename> class VectorF,
template<typename> class VectorU,
Require< Dense<VectorF<T>>,
Dense<VectorU<T>> > = true
>
void
direct_solver(const VectorF<T> &f_h, VectorU<T> &u_h)
{
auto N  = assertEqual(f_h.length(), u_h.length()) - 2;
auto h  = T(1)/(N+1);
auto h2 = h*h;

// forward substitution
u_h(1) = h2*f_h(1) + u_h(0);
for (std::size_t k=2; k<N; ++k) {
u_h(k) = h2*f_h(k) + u_h(k-1)*(k-1)/k;
}
u_h(N) = h2*f_h(N) + u_h(N+1) + u_h(N-1)*(N-1)/N;

// backward substitution
u_h(N) = u_h(N)*N/(N+1);
for (std::size_t k=N-1; k>=1; --k) {
u_h(k) = (u_h(k)+u_h(k+1))*k/(k+1);
}
}

//------------------------------------------------------------------------------

template <typename T, template<typename> class Vector>
void
init_zero(Vector<T> &u_h)
{
for (auto [i, ui] : u_h) {
i = i;
ui = T(0);
}
}

//------------------------------------------------------------------------------

template <typename T, template<typename> class Vector>
void
print_record(const Vector<T> &r_h, const char *descr)
{
auto N  = r_h.length() -2;

fmt::printf("\"%s\"\n", descr);
for (auto [i, ri] : r_h) {
fmt::printf("%f %f\n", double(i)/(N+1), ri);
}
fmt::printf("\n\n");
}

int
main()
{
/*
auto f = [](double x) { return 0; };
auto g = [](double x) { return 0; };
auto u = [](double x) { return 0; };
*/

/*
auto f = [](double x) { return -2; };
auto g = [](double x) { return 1 + 3*x; };
auto u = [](double x) { return (x+1)*(x+1); };
*/

auto f = [](double x) {
return -8*M_PI*cos(2*M_PI*exp(2*x))
+16*M_PI*M_PI*exp(2*x)*exp(2*x)*sin(2*M_PI*exp(2*x));
};

auto g = [](double x) {
return x*sin(2*M_PI*exp(2.));
};

auto u = [](double x) {
return sin(2*M_PI*exp(2*x));
};

std::size_t N0 = 512;
std::size_t N1 = 256;
std::size_t N2 = 128;
std::size_t N3 = 64;

DenseVector<double> f_h(N0+1), u_h(N0+1), tmp_h(N0+1), r_h(N0+1), e_h(N0+1);
DenseVector<double> f_2h(N1+1), u_2h(N1+1), tmp_2h(N1+1), r_2h(N1+1);
DenseVector<double> f_4h(N2+1), u_4h(N2+1), tmp_4h(N2+1), r_4h(N2+1);
DenseVector<double> f_8h(N3+1), u_8h(N3+1), tmp_8h(N3+1), r_8h(N3+1);

discretizeProblem(f, g, f_h, u_h);

// First initialize u_h with exact solution
direct_solver(f_h, u_h);

double omega = 0.5;     // smoothing parameter

evalResidual(u_h, f_h, r_h);
print_record(r_h, "Initial residual");

for (int it=0; it<5; ++it) {
// On Omega_{h}
// ------------
jacobi_step(omega, f_h, u_h, tmp_h);
jacobi_step(omega, f_h, tmp_h, u_h);
evalResidual(u_h, f_h, r_h);

// On Omega_{2h}
// -------------
restriction(r_h, f_2h);
init_zero(u_2h);
evalResidual(u_2h, f_2h, r_2h);

jacobi_step(omega, f_2h, u_2h, tmp_2h);
jacobi_step(omega, f_2h, tmp_2h, u_2h);

evalResidual(u_2h, f_2h, r_2h);

// On Omega_{4h}
// -------------
restriction(r_2h, f_4h);
init_zero(u_4h);
evalResidual(u_4h, f_4h, r_4h);

jacobi_step(omega, f_4h, u_4h, tmp_4h);
jacobi_step(omega, f_4h, tmp_4h, u_4h);

evalResidual(u_4h, f_4h, r_4h);

// On Omega_{8h}
// -------------
restriction(r_4h, f_8h);
direct_solver(f_8h, u_8h);

// On Omega_{4h}
// -------------
updateProlongation(u_8h, u_4h);
jacobi_step(omega, f_4h, u_4h, tmp_4h);
jacobi_step(omega, f_4h, tmp_4h, u_4h);

// On Omega_{2h}
// -------------
updateProlongation(u_4h, u_2h);
jacobi_step(omega, f_2h, u_2h, tmp_2h);
jacobi_step(omega, f_2h, tmp_2h, u_2h);

// On Omega_{h}
// ------------
updateProlongation(u_2h, u_h);
jacobi_step(omega, f_h, u_h, tmp_h);
jacobi_step(omega, f_h, tmp_h, u_h);

evalResidual(u_h, f_h, r_h);
std::stringstream ss;
ss << "r_{h} after " << (it+1) << " V-cycle(s)";
print_record(r_h, ss.str().c_str());
}

evalError(u, u_h, e_h);
print_record(e_h, "Error e_h = u - u_h after last V-cycle");
print_record(u_h, "u_h after last V-cycle");
}

theon$g++ -Wall -I /home/numerik/pub/hpc/ws18/session25/ -std=c++17 -o mg_with_4_levels mg_with_4_levels.cpp theon$ ./mg_with_4_levels > mg_with_4_levels.data
theon$ ### Plotting results theon$ gnuplot mg_with_4_levels.gnuplot
theon$gnuplot mg_with_4_levels2.gnuplot theon$ gnuplot mg_with_4_levels3.gnuplot
theon$gnuplot mg_with_4_levels4.gnuplot theon$ gnuplot mg_with_4_levels5.gnuplot
theon$gnuplot mg_with_4_levels6.gnuplot theon$ gnuplot mg_with_4_levels7.gnuplot
theon\$