ConvolutionOperator

This is an introduction to the ConvolutionOperator class. We write a small overarching summary of the class where we define the algorithm/equation/structure reasoning for having this class or where it fits with the rest of the code.

template<int D>
class ConvolutionOperator : public mrcpp::MWOperator<D>

Convolution defined by a Gaussian expansion.

Represents the operator

\[ T = \sum_{m=1}^M \text{sign} (\alpha_m) \bigotimes \limits_{d = 1}^D T_d \left( \beta_m, \sqrt[D]{| \alpha_m |} \right) , \]

where each \( T_d \left( \beta, \alpha \right) \) is the convolution operator with one-dimensional Gaussian kernel \( k(x_d) = \alpha \exp \left( - \beta x_d^2 \right) \). Operator \( T \) is obtained from the Gaussian expansion

\[ \sum_{m=1}^M \alpha_m \exp \left( - \beta_m |x|^2 \right) \]

which is passed as a parameter to the first two constructors.

Todo:

: One may want to change the logic so that \( D \)-root is evaluated on the previous step, namely, when \( \alpha_m, \beta_m \) are calculated.

Note

Every \( T_d \left( \beta_m, \sqrt[D]{| \alpha_m |} \right) \) is the same operator associated with the one-dimensional variable \( x_d \) for \( d = 1, \ldots, D \).

Subclassed by mrcpp::DerivativeConvolution< D >, mrcpp::HeatOperator< D >, mrcpp::IdentityConvolution< D >, mrcpp::TimeEvolutionOperator< D >