# Applications¶

HLR contains different matrix kernels. The main purpose of those kernels is to set up the actual test problems for the arithmetic. They are not intended for solving real world problems.

## LogKernel¶

The model problem from HLIB is the 1d integral equation

$\int_0^1 \operatorname{log} \|x-y\| u(x) dy = f(x), \quad x \in [0,1]$

With piecewise constant ansatz functions on the equally spaced partitioned interval $$[0,1]$$, good low-rank approximation is possible for almost all clustering methods.

## Matern Covariance¶

The Matern kernel

$C(h, \theta ) := \frac{\sigma^2}{2^{\nu-1} \Gamma(\nu)} \left(\frac{h}{\ell}\right)^{\nu} K_{\nu}\left(\frac{h}{\ell}\right)$

with either random coordinates in $$\mathbb{R}^2$$ or $$\mathbb{R}^3$$ or coordinates defined by a given grid. The approximation depends stronger on the distance $$h$$ and such, on the coordinates.

The parameters $$\theta = (\sigma,\ell,\nu)$$ are by default chosen as $$\theta = (1, 1, 0.5)$$.

## Laplace Single Layer Potential¶

This application implements the integral equation

$\int_{\Gamma} \frac{1}{\|x-y\|} u(x) dy = f(x), \quad x \in \Gamma$

with $$\Gamma$$ defined by grids implemented in HLIBpro or user defined grids. For the discretization, piece-wise constant ansatz functions are used.

## Helmholtz Single Layer Potential¶

This application implements the integral equation

$\int_{\Gamma} \frac{e^{i \kappa \|x-y\|}}{\|x-y\|} u(x) dy = f(x), \quad x \in \Gamma$

with $$\kappa \in \mathbb{C}$$ being the wave number and $$\Gamma$$ as above. Again, for the discretization, piece-wise constant ansatz functions are used.

## Exponential Kernel¶

The same integral equation as in the last two applications is used here but the kernel is the exponental kernel:

$\int_{\Gamma} e^{\|x-y\|} u(x) dy = f(x), \quad x \in \Gamma$

The same grids as for the previous applications can be used for this integral equation.