void
JLBasis::reconstruct_1(const Index& lambda,
const int j,
InfiniteVector<double, Index>& c) const {
c.clear();
if (lambda.j() >= j)
c.set_coefficient(lambda, 1.0);
else {
#if _JL_PRECOND==0
const double phi0factor = sqrt(35./26.);
const double phi1factor = sqrt(105./2.);
#else
const double phi0factor = sqrt(5./12.);
const double phi1factor = sqrt(15./4.);
#endif
if (lambda.e() == 0) {
// generator
if (lambda.c() == 0) {
// type phi_0
// phi_0(x) = 1/2*phi_0(2*x+1)+phi_0(2*x)+1/2*phi_0(2*x-1)+3/4*phi_1(2*x+1)-3/4*phi_1(2*x-1)
int m = 2*lambda.k()-1; // m-2k=-1
{ // phi_0(2x+1)
InfiniteVector<double, Index> dhelp;
reconstruct_1(Index(lambda.j()+1, 0, 0, m), j, dhelp);
c.add(0.5*M_SQRT1_2, dhelp);
}
if (m >= 0) { // phi_1(2x+1)
InfiniteVector<double, Index> dhelp;
reconstruct_1(Index(lambda.j()+1, 0, 1, m), j, dhelp);
c.add(0.75*M_SQRT1_2 * phi0factor/phi1factor, dhelp);
}
// m=2k <-> m-2k=0
m++;
{ // phi_0(2x)
InfiniteVector<double, Index> dhelp;
reconstruct_1(Index(lambda.j()+1, 0, 0, m), j, dhelp);
c.add(M_SQRT1_2, dhelp);
}
// m=2k+1 <-> m-2k=1
m++;
{ // phi_0(2x-1)
InfiniteVector<double, Index> dhelp;
reconstruct_1(Index(lambda.j()+1, 0, 0, m), j, dhelp);
c.add(0.5*M_SQRT1_2, dhelp);
}
if (m <= (1<<(lambda.j()+1))-1) { // phi_1(2x-1)
InfiniteVector<double, Index> dhelp;
reconstruct_1(Index(lambda.j()+1, 0, 1, m), j, dhelp);
c.add(-0.75*M_SQRT1_2 * phi0factor/phi1factor, dhelp);
}
} else { // lambda.c() == 1
// type phi_1
// phi_1(x) = -1/8*phi_0(2*x+1)+1/8*phi_0(2*x-1)-1/8*phi_1(2*x+1)+1/2*phi_1(2*x)-1/8*phi_1(2*x-1)
int m = 2*lambda.k()-1; // m-2k=-1
{ // phi_0(2x+1)
InfiniteVector<double, Index> dhelp;
reconstruct_1(Index(lambda.j()+1, 0, 0, m), j, dhelp);
c.add(-0.125*M_SQRT1_2 * phi1factor/phi0factor, dhelp);
}
if (m >= 0) { // phi_1(2x+1)
InfiniteVector<double, Index> dhelp;
reconstruct_1(Index(lambda.j()+1, 0, 1, m), j, dhelp);
c.add(-0.125*M_SQRT1_2, dhelp);
}
// m=2k <-> m-2k=0
m++;
{ // phi_1(2x)
InfiniteVector<double, Index> dhelp;
reconstruct_1(Index(lambda.j()+1, 0, 1, m), j, dhelp);
c.add(0.5*M_SQRT1_2, dhelp);
}
// m=2k+1 <-> m-2k=1
m++;
{ // phi_0(2x-1)
InfiniteVector<double, Index> dhelp;
reconstruct_1(Index(lambda.j()+1, 0, 0, m), j, dhelp);
c.add(0.125*M_SQRT1_2 * phi1factor/phi0factor, dhelp);
}
if (m <= (1<<(lambda.j()+1))) { // phi_1(2x-1)
InfiniteVector<double, Index> dhelp;
reconstruct_1(Index(lambda.j()+1, 0, 1, m), j, dhelp);
c.add(-0.125*M_SQRT1_2, dhelp);
}
} // end if (lambda.c() == 0)
} else { // lambda.e() == 1
// wavelet
if (lambda.c() == 0) {
// type psi_0
// psi_0(x) = -2*phi_0(2*x+1)+4*phi_0(2*x)-2*phi_0(2*x-1)-21*phi_1(2*x+1)+21*phi_1(2*x-1)
#if _JL_PRECOND==0
const double factor = sqrt(35./352.);
#else
const double factor = sqrt(5./3648.);
#endif
int m = 2*lambda.k()-1; // m-2k=-1
{ // phi_0(2x+1)
InfiniteVector<double, Index> dhelp;
reconstruct_1(Index(lambda.j()+1, 0, 0, m), j, dhelp);
c.add(-2.0*M_SQRT1_2 * factor/phi0factor, dhelp);
}
if (m >= 0) { // phi_1(2x+1)
InfiniteVector<double, Index> dhelp;
reconstruct_1(Index(lambda.j()+1, 0, 1, m), j, dhelp);
c.add(-21.0*M_SQRT1_2 * factor/phi1factor, dhelp);
}
// m=2k <-> m-2k=0
m++;
{ // phi_0(2x)
InfiniteVector<double, Index> dhelp;
reconstruct_1(Index(lambda.j()+1, 0, 0, m), j, dhelp);
c.add(4.0*M_SQRT1_2 * factor/phi0factor, dhelp);
}
// m=2k+1 <-> m-2k=1
m++;
{ // phi_0(2x-1)
InfiniteVector<double, Index> dhelp;
reconstruct_1(Index(lambda.j()+1, 0, 0, m), j, dhelp);
c.add(-2.0*M_SQRT1_2 * factor/phi0factor, dhelp);
}
if (m <= (1<<(lambda.j()+1))) { // phi_1(2x-1)
InfiniteVector<double, Index> dhelp;
reconstruct_1(Index(lambda.j()+1, 0, 1, m), j, dhelp);
c.add(21.0*M_SQRT1_2 * factor/phi1factor, dhelp);
}
} else { // lambda.c() == 1
// type psi_1
// psi_1(x) = phi_0(2*x+1)-phi_0(2*x-1)+ 9*phi_1(2*x+1)+12*phi_1(2*x)+ 9*phi_1(2*x-1)
#if _JL_PRECOND==0
const double factor = sqrt(35./48.);
#else
const double factor = sqrt(5./768.);
#endif
int m = 2*lambda.k()-1; // m-2k=-1
{ // phi_0(2x+1)
InfiniteVector<double, Index> dhelp;
reconstruct_1(Index(lambda.j()+1, 0, 0, m), j, dhelp);
c.add(M_SQRT1_2 * factor/phi0factor, dhelp);
}
if (m >= 0) { // phi_1(2x+1)
InfiniteVector<double, Index> dhelp;
reconstruct_1(Index(lambda.j()+1, 0, 1, m), j, dhelp);
c.add(9.0*M_SQRT1_2 * factor/phi1factor, dhelp);
}
// m=2k <-> m-2k=0
m++;
{ // phi_1(2x)
InfiniteVector<double, Index> dhelp;
reconstruct_1(Index(lambda.j()+1, 0, 1, m), j, dhelp);
c.add(12.0*M_SQRT1_2 * factor/phi1factor, dhelp);
}
// m=2k+1 <-> m-2k=1
m++;
{ // phi_0(2x-1)
InfiniteVector<double, Index> dhelp;
reconstruct_1(Index(lambda.j()+1, 0, 0, m), j, dhelp);
c.add(-M_SQRT1_2 * factor/phi0factor, dhelp);
}
if (m <= (1<<(lambda.j()+1))) { // phi_1(2x-1)
InfiniteVector<double, Index> dhelp;
reconstruct_1(Index(lambda.j()+1, 0, 1, m), j, dhelp);
c.add(9.0*M_SQRT1_2 * factor/phi1factor, dhelp);
}
}
}
}
}
Array1D<SampledMapping<2> >
evaluate(const LDomainJLBasis& basis,
const Index& lambda,
const int N)
{
Array1D<SampledMapping<2> > r(3);
// supp(psi_{j,e,c,k}) = 2^{-j}[k1-1,k1+1]x[k2-1,k2+1] cap Omega
FixedArray1D<Array1D<double>,2> values;
values[0].resize(N+1); // values in x-direction
values[1].resize(N+1); // values in y-direction
Array1D<double> knots;
knots.resize(N+1);
const double h = 1./N;
// patch Omega_0 = [-1,0]x[0,1]
for (int i = 0; i <= N; i++) {
values[0][i] = 0.;
values[1][i] = 0.;
}
if (lambda.k()[0] <= 0 && lambda.k()[1] >= 0) {
for (int i = 0; i <= N; i++)
knots[i] = -1.0+i*h;
evaluate(0, lambda.j(), lambda.e()[0], lambda.c()[0], lambda.k()[0], knots, values[0]);
for (int i = 0; i <= N; i++)
knots[i] = i*h;
evaluate(0, lambda.j(), lambda.e()[1], lambda.c()[1], lambda.k()[1], knots, values[1]);
}
r[0] = SampledMapping<2>(Point<2>(-1, 0), Point<2>(0,1), values);
// patch Omega_1 = [-1,0]x[-1,0]
for (int i = 0; i <= N; i++) {
values[0][i] = 0.;
values[1][i] = 0.;
}
if (lambda.k()[0] <= 0 && lambda.k()[1] <= 0) {
for (int i = 0; i <= N; i++)
knots[i] = -1.0+i*h;
evaluate(0, lambda.j(), lambda.e()[0], lambda.c()[0], lambda.k()[0], knots, values[0]);
evaluate(0, lambda.j(), lambda.e()[1], lambda.c()[1], lambda.k()[1], knots, values[1]);
}
r[1] = SampledMapping<2>(Point<2>(-1,-1), Point<2>(0,0), values);
// patch Omega_2 = [0,1]x[-1,0]
for (int i = 0; i <= N; i++) {
values[0][i] = 0.;
values[1][i] = 0.;
}
if (lambda.k()[0] >= 0 && lambda.k()[1] <= 0) {
for (int i = 0; i <= N; i++)
knots[i] = i*h;
evaluate(0, lambda.j(), lambda.e()[0], lambda.c()[0], lambda.k()[0], knots, values[0]);
for (int i = 0; i <= N; i++)
knots[i] = -1.0+i*h;
evaluate(0, lambda.j(), lambda.e()[1], lambda.c()[1], lambda.k()[1], knots, values[1]);
}
r[2] = SampledMapping<2>(Point<2>( 0,-1), Point<2>(1,0), values);
return r;
}
double
LDomainJLGramian::a(const Index& lambda,
const Index& mu) const
{
double r = 0;
// First we compute the support intersection of \psi_\lambda and \psi_\mu:
typedef WaveletBasis::Support Support;
Support supp;
if (intersect_supports(basis_, lambda, mu, supp))
{
// use that both \psi_\lambda and \psi_\mu are a tensor product of 1D bases;
// the entry of the Gramian on each square subpatch is a product of 2 1D integrals
// iterate through the subsquares of supp and compute the integral shares
const double h = ldexp(1.0, -supp.j); // sidelength of the subsquare
const int N_Gauss = 6;
FixedArray1D<Array1D<double>,2> gauss_points, gauss_weights;
for (int i = 0; i <= 1; i++) {
gauss_points[i].resize(N_Gauss);
gauss_weights[i].resize(N_Gauss);
for (int ii = 0; ii < N_Gauss; ii++)
gauss_weights[i][ii] = h*GaussWeights[N_Gauss-1][ii];
}
FixedArray1D<int,2> k;
FixedArray1D<Array1D<double>,2>
psi_lambda_values, // values of the components of psi_lambda at gauss_points[i]
psi_mu_values; // -"-, for psi_mu
Array1D<double> dummy;
for (k[0] = supp.xmin; k[0] < supp.xmax; k[0]++) {
if (k[0] >= -(1<<supp.j) && k[0] < (1<<supp.j)) {
for (int ii = 0; ii < N_Gauss; ii++)
gauss_points[0][ii] = h*(2*k[0]+1+GaussPoints[N_Gauss-1][ii])/2.;
evaluate(lambda.j(), lambda.e()[0], lambda.c()[0], lambda.k()[0],
gauss_points[0], psi_lambda_values[0], dummy);
evaluate(mu.j(), mu.e()[0], mu.c()[0], mu.k()[0],
gauss_points[0], psi_mu_values[0], dummy);
double factor0 = 0;
for (int i0 = 0; i0 < N_Gauss; i0++)
factor0 += gauss_weights[0][i0] * psi_lambda_values[0][i0] * psi_mu_values[0][i0];
for (k[1] = supp.ymin; k[1] < supp.ymax; k[1]++) {
// check whether 2^{-supp.j}[k0,k0+1]x[k1,k1+1] is contained in Omega
if ((k[1] >= -(1<<supp.j) && k[1] < 0)
|| (k[0] < 0 && k[1] >= 0 && k[1] < (1<<supp.j))) {
for (int ii = 0; ii < N_Gauss; ii++)
gauss_points[1][ii] = h*(2*k[1]+1+GaussPoints[N_Gauss-1][ii])/2.;
evaluate(lambda.j(), lambda.e()[1], lambda.c()[1], lambda.k()[1],
gauss_points[1], psi_lambda_values[1], dummy);
evaluate(mu.j(), mu.e()[1], mu.c()[1], mu.k()[1],
gauss_points[1], psi_mu_values[1], dummy);
double factor1 = 0;
for (int i1 = 0; i1 < N_Gauss; i1++)
factor1 += gauss_weights[1][i1] * psi_lambda_values[1][i1] * psi_mu_values[1][i1];
r += factor0 * factor1;
}
}
}
}
}
return r;
}
