The MW operators discussed in this chapter is available to the application program by including:
#include "MRCPP/MWOperators"Note
The convolution operators have separate precision parameters for their construction and application. The build_prec argument to the operator constructors will affect e.g. the number of terms in the separated representations of the Poisson/Helmholtz approximations, as well as the operator bandwidth. The apply_prec argument to the apply function relates only to the adaptive construction of the output function, based on a wavelet norm error estimate.
mrcpp::IdentityConvolution
mrcpp::DerivativeConvolution
mrcpp::PoissonOperator
mrcpp::HelmholtzOperator
mrcpp::apply(double prec, FunctionTree<D> &out, ConvolutionOperator<D> &oper, FunctionTree<D> &inp, int maxIter, bool absPrec)
mrcpp::apply(double prec, FunctionTree<D> &out, ConvolutionOperator<D> &oper, FunctionTree<D> &inp, FunctionTreeVector<D> &precTrees, int maxIter, bool absPrec)
Note
The derivative operators have clearly defined requirements on the output grid structure, based on the grid of the input function. This means that there is no real grid adaptivity, and thus no precision parameter is needed for the application of such an operator.
mrcpp::ABGVOperator
mrcpp::PHOperator
mrcpp::BSOperator
mrcpp::apply(FunctionTree<D> &out, DerivativeOperator<D> &oper, FunctionTree<D> &inp, int dir)
mrcpp::divergence(FunctionTree<D> &out, DerivativeOperator<D> &oper, FunctionTreeVector<D> &inp)
mrcpp::gradient(DerivativeOperator<D> &oper, FunctionTree<D> &inp)
The electrostatic potential g arising from a charge distribution f are related through the Poisson equation
− ∇2g(r) = f(r)
This equation can be solved with respect to the potential by inverting the differential operator into the Green's function integral convolution operator
This operator is available in the MW representation, and can be solved with arbitrary (finite) precision in linear complexity with respect to system size. Given an arbitrary charge dirtribution f_tree in the MW representation, the potential is computed in the following way:
double apply_prec; // Precision for operator application
double build_prec; // Precision for operator construction
mrcpp::PoissonOperator P(MRA, build_prec); // MW representation of Poisson operator
mrcpp::FunctionTree<3> f_tree(MRA); // Input function
mrcpp::FunctionTree<3> g_tree(MRA); // Output function
mrcpp::apply(apply_prec, g_tree, P, f_tree); // Apply operator adaptivelyThe Coulomb self-interaction energy can now be computed as the dot product:
double E = mrcpp::dot(g_tree, f_tree);The Helmholtz operator is a generalization of the Poisson operator and is given as the integral convolution
The operator is the inverse of the shifted Laplacian
[ − ∇2 + μ2]g(r) = f(r)
and appears e.g. when solving the SCF equations. The construction and application is similar to the Poisson operator, with an extra argument for the μ parameter
double apply_prec; // Precision for operator application
double build_prec; // Precision for operator construction
double mu; // Must be a positive real number
mrcpp::HelmholtzOperator H(MRA, mu, build_prec);// MW representation of Helmholtz operator
mrcpp::FunctionTree<3> f_tree(MRA); // Input function
mrcpp::FunctionTree<3> g_tree(MRA); // Output function
mrcpp::apply(apply_prec, g_tree, H, f_tree); // Apply operator adaptivelyThe ABGV (Alpert, Beylkin, Gines, Vozovoi) derivative operator is initialized with two parameters a and b accounting for the boundary conditions between adjacent nodes, see Alpert et al.
double a = 0.0, b = 0.0; // Boundary conditions for operator
mrcpp::ABGVOperator<3> D(MRA, a, b); // MW derivative operator
mrcpp::FunctionTree<3> f(MRA); // Input function
mrcpp::FunctionTree<3> f_x(MRA); // Output function
mrcpp::FunctionTree<3> f_y(MRA); // Output function
mrcpp::FunctionTree<3> f_z(MRA); // Output function
mrcpp::apply(f_x, D, f, 0); // Operator application in x direction
mrcpp::apply(f_y, D, f, 1); // Operator application in y direction
mrcpp::apply(f_z, D, f, 2); // Operator application in z directionThe tree structure of the output function will depend on the choice of parameters a and b: if both are zero, the output grid will be identical to the input grid; otherwise the grid will be widened by one node (on each side) in the direction of application.
The PH derivative operator is based on the noise reducing derivative of Pavel Holoborodko. This operator is also available as a direct second derivative.
mrcpp::PHOperator<3> D1(MRA, 1); // MW 1st derivative operator
mrcpp::PHOperator<3> D2(MRA, 2); // MW 2nd derivative operator
mrcpp::FunctionTree<3> f(MRA); // Input function
mrcpp::FunctionTree<3> f_x(MRA); // Output function
mrcpp::FunctionTree<3> f_xx(MRA); // Output function
mrcpp::apply(f_x, D1, f, 0); // Operator application in x direction
mrcpp::apply(f_xx, D2, f, 0); // Operator application in x directionSpecial thanks to Prof. Robert J. Harrison (Stony Brook University) for sharing the operator coefficients.
The BS derivative operator is based on a pre-projection onto B-splines in order to remove the discontinuities in the MW basis, see Anderson et al. This operator is also available as a direct second and third derivative.
mrcpp::BSOperator<3> D1(MRA, 1); // MW 1st derivative operator
mrcpp::BSOperator<3> D2(MRA, 2); // MW 2nd derivative operator
mrcpp::BSOperator<3> D3(MRA, 3); // MW 3nd derivative operator
mrcpp::FunctionTree<3> f(MRA); // Input function
mrcpp::FunctionTree<3> f_x(MRA); // Output function
mrcpp::FunctionTree<3> f_yy(MRA); // Output function
mrcpp::FunctionTree<3> f_zzz(MRA); // Output function
mrcpp::apply(f_x, D1, f, 0); // Operator application in x direction
mrcpp::apply(f_yy, D2, f, 1); // Operator application in x direction
mrcpp::apply(f_zzz, D3, f, 2); // Operator application in x direction