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ScaledGaussianSmoothingKernel Class Reference

#include <ScaledGaussianSmoothingKernel.hpp>

Inheritance diagram for ScaledGaussianSmoothingKernel:
Inheritance graph
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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 ItemgetItemProperty (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)
 
Itemoperator= (const Item &)=delete
 
Itemparent () 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

 ScaledGaussianSmoothingKernel ()
 
void setupSelfBefore () override
 
- Protected Member Functions inherited from SmoothingKernel
 SmoothingKernel ()
 
Randomrandom () 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 = ScaledGaussianSmoothingKernel
 

Private Attributes

Array _Xv
 

Friends

class ItemRegistry
 

Detailed Description

An instance of the ScaledGaussianSmoothingKernel describes a scaled Gaussian smoothing kernel with finite support, as presented by Altay and Theuns 2013 (MNRAS 434,748). For the normalized radius \(u=r/h\), the kernel profile is given by:

\[ W(u) = \begin{cases}\; \mathcal{N}\,\exp(-\frac{u^2}{2\sigma^2}) & \quad\text{if }0\leq u \leq 1, \\ \; 0 & \quad \text{else}. \end{cases} \]

where

\[ \begin{aligned} \sigma &= \frac{1}{2\sqrt{2}\,\pi^{1/6}} \\ \mathcal{N} &= \frac{8}{\pi}\,\left[\mathrm{erf}(t)-\frac{2t\exp(-t^2)}{\sqrt{\pi}} \right]^{-1} \quad \text{with} \quad t=2\pi^{1/6}. \end{aligned} \]

It can be verified that this function satisfies the required normalization

\[ 4\pi \int_0^\infty W(u)\, u^2\, {\text{d}}u = 1. \]

With this scaling and cutoff, the Gaussian profile approximates the standard cubic spline kernel profile to within about three percent for all radii. The reason for using a Gaussion kernel instead of the standard cubic spline kernel in some applications is that a spherical Gaussian profile (assuming infinite support, i.e. not cut off at the smoothing length) can be separated into Gaussian component profiles along each of the coordinate axes, It thus becomes easy to calculate the mass inside a cuboidal box (such as a grid cell) or to determine the surface density for the projection on a rectangle (such as a pixel).

Constructor & Destructor Documentation

◆ ScaledGaussianSmoothingKernel()

ScaledGaussianSmoothingKernel::ScaledGaussianSmoothingKernel ( )
inlineprotected

Default constructor for concrete Item subclass ScaledGaussianSmoothingKernel : "a scaled Gaussian smoothing kernel" .

Member Function Documentation

◆ columnDensity()

double ScaledGaussianSmoothingKernel::columnDensity ( double  q) const
overridevirtual

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

\[\Sigma(q) = \begin{cases} \; {\cal{N}}\,\sqrt{2\pi}\,\sigma \exp\left(-\frac{q^2}{2\sigma^2}\right) \mathrm{erf}\left(\frac{\sqrt{1-q^2}} {\sqrt{2}\sigma}\right) & \quad{\text{if }} 0\leq q\leq 1, \\ \; 0 & \quad{\text{else}}. \end{cases} \]

Implements SmoothingKernel.

◆ density()

double ScaledGaussianSmoothingKernel::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 ScaledGaussianSmoothingKernel::generateRadius ( ) const
overridevirtual

This function generates a random normalized radius \(u\) from the smoothing kernel, by drawing a number from the one-dimensional probability density \(p(u)\,{\text{d}}u = 4\pi\,W(u)\,u^2\, {\text{d}}u\). This is accomplished by generating a uniform deviate \({\cal{X}}\), and solving the equation

\[ {\cal{X}} = \int_0^u 4\pi\,W(u')\,u'^2\, {\text{d}}u' \]

for \(u\). For the scaled gaussian smoothing kernel, we use a precomputed grid with values on which we interpolate to solve this equation.

Implements SmoothingKernel.

◆ setupSelfBefore()

void ScaledGaussianSmoothingKernel::setupSelfBefore ( )
overrideprotectedvirtual

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

Reimplemented from SimulationItem.


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