CubicSplineSmoothingKernel Class Reference

#include <CubicSplineSmoothingKernel.hpp>

Inheritance diagram for CubicSplineSmoothingKernel:

Public Member Functions
double	columnDensity (double q) const override
double	density (double u) const override
double	generateRadius () const override
virtual double	columnDensity (double q) const =0
virtual double	density (double u) const =0
virtual double	generateRadius () const =0
Public Member Functions inherited from SimulationItem
template<class T >
T *	find (bool setup=true) const
template<class T >
T *	interface (int levels=-999999, bool setup=true) const
virtual string	itemName () const
void	setup ()
string	typeAndName () const
Public Member Functions inherited from Item
	Item (const Item &)=delete
virtual	~Item ()
void	addChild (Item *child)
const vector< Item * > &	children () const
virtual void	clearItemListProperty (const PropertyDef *property)
void	destroyChild (Item *child)
virtual bool	getBoolProperty (const PropertyDef *property) const
virtual vector< double >	getDoubleListProperty (const PropertyDef *property) const
virtual double	getDoubleProperty (const PropertyDef *property) const
virtual string	getEnumProperty (const PropertyDef *property) const
virtual int	getIntProperty (const PropertyDef *property) const
virtual vector< Item * >	getItemListProperty (const PropertyDef *property) const
virtual Item *	getItemProperty (const PropertyDef *property) const
virtual string	getStringProperty (const PropertyDef *property) const
int	getUtilityProperty (string name) const
virtual void	insertIntoItemListProperty (const PropertyDef *property, int index, Item *item)
Item &	operator= (const Item &)=delete
Item *	parent () const
virtual void	removeFromItemListProperty (const PropertyDef *property, int index)
virtual void	setBoolProperty (const PropertyDef *property, bool value)
virtual void	setDoubleListProperty (const PropertyDef *property, vector< double > value)
virtual void	setDoubleProperty (const PropertyDef *property, double value)
virtual void	setEnumProperty (const PropertyDef *property, string value)
virtual void	setIntProperty (const PropertyDef *property, int value)
virtual void	setItemProperty (const PropertyDef *property, Item *item)
virtual void	setStringProperty (const PropertyDef *property, string value)
void	setUtilityProperty (string name, int value)
virtual string	type () const

Protected Member Functions
	CubicSplineSmoothingKernel ()
void	setupSelfBefore () override
Protected Member Functions inherited from SmoothingKernel
	SmoothingKernel ()
Random *	random () const
void	setupSelfBefore () override
Protected Member Functions inherited from SimulationItem
	SimulationItem ()
virtual bool	offersInterface (const std::type_info &interfaceTypeInfo) const
virtual void	setupSelfAfter ()
virtual void	setupSelfBefore ()
Protected Member Functions inherited from Item
	Item ()

Private Types
using	BaseType = SmoothingKernel
using	ItemType = CubicSplineSmoothingKernel

Private Attributes
Array	_Xv

Friends
class	ItemRegistry

Detailed Description

The CubicSpineSmoothingKernel class is a subclass of the abstract SmoothingKernel class, and describes smoothing kernels defined by the standard cubic spline density,

$$W(u)=\{\begin{array}{ll}{\displaystyle \frac{8(1-6u^2+6u^3)}{\pi }}& \text{if }0\le u\le \frac{1}{2},\\ {\displaystyle \frac{16(1-u{)}^3}{\pi }}& \text{if }\frac{1}{2}\le u\le 1,\\ 0& \text{else}.\end{array}$$

It can be checked that this function is continuous at $u=\frac{1}{2}$ and that it satisfies the required normalization

$$4\pi {\int }_0^{\mathrm{\infty }}W(u)u^2\text{d}u=1.$$

Constructor & Destructor Documentation

CubicSplineSmoothingKernel()

CubicSplineSmoothingKernel::CubicSplineSmoothingKernel ( )

inlineprotected

Default constructor for concrete Item subclass CubicSplineSmoothingKernel : "a cubic spline smoothing kernel" .

Member Function Documentation

columnDensity()

double CubicSplineSmoothingKernel::columnDensity ( double  q ) const

overridevirtual

This function returns the column density $\mathrm{\Sigma }(q)=2{\int }_q^1\frac{W(u)u\text{d}u}{\sqrt{u^2-q^2}}$ of the smoothing kernel as a function of the normalized impact radius $q=r_{\text{i}}/h$. For the cubic spline smoothing kernel, we obtain

$$\mathrm{\Sigma }(q)=\{\begin{array}{l}{\displaystyle \frac{2}{\pi }}[(4+26q^2)\sqrt{1-q^2}-(1+26q^2)\sqrt{1-4q^2}\\ -18q^4\ln \left({\displaystyle \frac{2q}{1+\sqrt{1-4q^2}}}\right)-6q^2(4+q^2)\ln \left({\displaystyle \frac{2+2\sqrt{1-q^2}}{1+\sqrt{1-4q^2}}}\right)]& \text{if }0\le q\le \frac{1}{2},\\ {\displaystyle \frac{4}{\pi }}[(2+13q^2)\sqrt{1-q^2}+3q^2(4+q^2)\ln \left({\displaystyle \frac{q}{1+\sqrt{1-q^2}}}\right)]& \text{if }\frac{1}{2}\le q\le 1,\\ 0& \text{else}.\end{array}$$

Implements SmoothingKernel.

density()

double CubicSplineSmoothingKernel::density ( double  u ) const

overridevirtual

This function returns the density $W(u)$ of the smoothing kernel as a function of the normalized radius $u$. It just implements the analytical formula given in the class header.

Implements SmoothingKernel.

generateRadius()

double CubicSplineSmoothingKernel::generateRadius ( ) const

overridevirtual

This function generates a random normalized radius $u$ from the smoothing kernel. This is accomplished by generating a uniform deviate $\mathcal{X}$, and solving the equation

$$\mathcal{X}={\int }_0^{u}4\pi W(u^{\prime })u^{\prime 2}\text{d}{u}^{\prime }$$

for $u$. For the cubic spline smoothing kernel, we obtain

$${\int }_0^{u}4\pi W(u^{\prime })u^{\prime 2}\text{d}{u}^{\prime }=\{\begin{array}{ll}32u^6-{\displaystyle \frac{192}{5}}{u}^5+{\displaystyle \frac{32}{3}}{u}^3& \text{if }0\le u\le \frac{1}{2},\\ -{\displaystyle \frac{32}{3}}{u}^6+{\displaystyle \frac{192}{5}}{u}^5-48u^4+{\displaystyle \frac{64}{3}}{u}^3-{\displaystyle \frac{1}{15}}& \text{if }\frac{1}{2}\le u\le 1.\end{array}$$

We precompute a tabulation of this cumulative distribution function on which we interpolate to solve this equation.

Implements SmoothingKernel.

setupSelfBefore()

void CubicSplineSmoothingKernel::setupSelfBefore ( )

overrideprotectedvirtual

This function sets up a grid that will be used to sample random radii from the smoothing kernel.

Reimplemented from SimulationItem.

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The documentation for this class was generated from the following file:

• CubicSplineSmoothingKernel.hpp