The main class for interpolation of functors. More...

#include <interpolant.hh>

Inheritance diagram for Interpolant:

Collaboration diagram for Interpolant:

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Classes
class	FuncError_lp

class	FuncError_wp1

Public Member Functions
	Interpolant ()
	Default constructor. More...

template<template< class > class Func, class Type >
	Interpolant (Func< Type > *func, bool interpolate_derivative=false)
	Constructor with functor setting. More...

virtual	~Interpolant (void)
	Destructor. More...

Evaluation.
double	val (double x)
	Returns interpolated value. More...

double	val_test (double x)
	Do NOT use, only for testing purpose. More...

double	val_p1 (double x)
	Do NOT use, unless you are 100% sure. More...

DiffValue	diff (double x)
	Returns interpolated value of the derivation. More...

double	f_val (double x)
	Returns value of the original functor. More...

DiffValue	f_diff (double x)
	Returns 1st derivative of original functor using FADBAD. More...

double	f_diffn (double x, unsigned int n)

Interpolation.
virtual int	interpolate ()
	Creates piecewise interpolation with polynomials of selected degree. More...

template<template< class > class Func, class Type >
void	set_functor (Func< Type > *func, bool interpolate_derivative=false)
	Sets the functor. More...

Public Member Functions inherited from InterpolantBase
	InterpolantBase ()
	Default constructor. More...

virtual	~InterpolantBase ()
	Destructor. More...

double	error ()
	Returns error of the interpolation. More...

double	bound_a () const
	Returns left boundary of the interval. More...

double	bound_b () const
	Returns right boundary of the interval. More...

unsigned int	size () const
	Returns the size of the interpolation table. More...

void	set_interval (double bound_a, double bound_b)
	Sets the interpolation interval boundaries. More...

void	set_size (unsigned int size)
	Sets size of the interpolation table. It is also equal to the number of intervals. More...

void	set_norm (ErrorNorm::Type norm_type=ErrorNorm::max, double p=2)
	Sets the type of norm used for computing estimate of the error of the interpolation. More...

void	set_size_automatic (double user_tol, unsigned int init_size, unsigned int max_size=default_max_size)
	Sets automatic choice of the size of the table. More...

void	set_extrapolation (Extrapolation::Type extrapolation)
	Sets the type of extrapolation. Functor type is default. More...

EvalStatistics	statistics () const
	Returns structure with evaluation statistics. More...

void	reset_stat ()
	Resets all measured statistics. More...

void	check_stats_and_reinterpolate (double percentage=0.3)
	Can be called to check automatically the evaluation statistics and possibly reinterpolate. More...

Protected Member Functions
void	compute_error (double tol, std::vector< double > &f, std::vector< double > &df)
	Creates piecewise linear interpolation. More...

void	compute_error (double tol, double p, ErrorNorm::Type norm_type)
	Computes estimate of interpolation error with given norm. More...

void	swap_middle_values (std::vector< double > &f, std::vector< double > &df)

void	compute_values ()
	Creates table of nodes and function values. More...

DiffValue	diff_p1 (double x)
	Finds interval on which `x` lies. More...

Protected Member Functions inherited from InterpolantBase
long	fact (long x)
	Recursive factorial function (used in Taylor row expansion in n-th derivative computation). More...

void	check_all ()
	Checks that the parameters are set before interpolation. More...

Protected Attributes
FunctorBase< double > *	func
	Pointer to original functor with double type. More...

FunctorBase< B< double > > *	func_diff
	Pointer to original functor with FADBAD B type. More...

FunctorBase< T< double > > *	func_diffn
	Pointer to original functor with FADBAD T type. More...

bool	interpolate_derivative
	Is true if we want to interpolate the derivative too. More...

std::vector< double >	f_vec
	Vector of function values at nodes. More...

std::vector< double >	df_vec
	Vector of function derivatives values at nodes. More...

Protected Attributes inherited from InterpolantBase
double	bound_a_
	Left interval boundary. More...

double	bound_b_
	Right interval boundary. More...

double	step
	Chosen interpolation step. More...

double	a_div_step
	bound_ divided by step - precomputed value for evaluation More...

unsigned int	size_
	Number of dividing intervals. More...

unsigned int	n_nodes
	Number of nodes in the interval . More...

double	user_tol
	User set tolerance which is used during automatic step choice. More...

unsigned int	max_size
	Maximal size of the interpolation table. More...

bool	automatic_size
	Is true if step/size should be chosen automatically. More...

ErrorNorm::Type	norm_type
	Type of norm used to compute error of the interpolation. More...

double	p
	Exponent used in norms and when computing error. More...

double	error_
	Error of the interpolation. More...

Extrapolation::Type	extrapolation

EvalStatistics	stats
	Structure which keeps evaluation statistics. See InterpolantBase::eval_statistics. More...

std::vector< bool >	checks
	Vector of boolean values telling us which parameters are set or not. More...

Additional Inherited Members
Static Protected Attributes inherited from InterpolantBase
static const unsigned int	n_derivatives = 10
	Defines how many derivatives we allow to be returned from Taylor's coeficients. More...

static const unsigned int	default_max_size = 10*1000
	Default maximal size of the interpolation table. More...

static const double	simpson_tolerance = 1e-10
	Tolerance in Adaptive Simpson intergration. More...

Detailed Description

The main class for interpolation of functors.

This class is a wrapper of a functor (defined by class FunctorBase) which provides computation of derivates and interpolation of the functor. We will describe the functionality in the following paragrahphs.

Evaluation of a functor and its derivatives

Beside the interpolation, the class provides calling of the functor itself. All functions evaluating directly the functor starts with 'f_', e.g. function f_val computes the functor value.

We use the FADBAD++ library (website) to compute derivatives of the functor. First derivatives can be obtained by function f_diff which uses backward automatic differentiation. Higher order derivatives can be obtained via f_diffn which uses Taylor expansion method.

Interpolation

Sofar we provide only a piecewise linear interpolation. Both the function and its derivative can be interpolated. User has to provide the functor object to the constructor or later to the function set_functor. Then one must set at least the interval $ [a,b]$ by set_interval and the size of the interpolation table by set_size before calling interpolate. Otherwise assert will crash the program in debug build.

When user wants the size to be chosen automatically then set_size_automatic must be called at first. An initial guess of size of the interpolation table and a tolerance which is assumed to be the maximum relative difference between the function and the interpolant (sum of value and derivative difference) must be provided. Futher user can set the maximum size of the interpolation (default 1000) table which will not be exceeded and the type of norm in which the estimate of the error of interpolation will be computed (enumerated in ErrorNorm, default maximal norm). See one of the following paragraphs to learn how we deal with the error of the interpolation.

Outside the given interval the interpolation is extrapolated. User can choose among several types of extrapolation which are enumerated in Extrapolation. Constant and linear extrapolation use the first and last interpolating polynomials to compute the value. Functor type means that the functor itself will be evaluated outside the interval. Type of extrapolation can be changed by set_extrapolation function. Calling functor is the default type of extrapolation.

Error of the interpolation

We are interested in the maximum difference between the function value and its interpolation. When interpolating also the derivative, we need to compute the difference between derivative and its interpolation too. Therefore we would like to compute something like the L-infinity norm, relative to function value and its derivative. Let $ f(x), f'(x) $ be the function and its derivative and $ g(x), g'(x)$ interpolation of the function and interpolation of the derivative on the interval $ I $ . We suggest an approximation of the infinity norm:

$\|f-g\|_\infty = \sup_{x\in I} F(x) = \sup_{x\in I} \left( \frac{|f(x)-g(x)|}{|f(x)| + tol} + \frac{|f'(x)-g'(x)|}{|f'(x)| + tol} \right) \approx \max_{x\in J} \left( \frac{|f(x)-g(x)|}{|f(x)| + tol} + \frac{|f'(x)-g'(x)|}{|f'(x)| + tol} \right)$

where $ J $ is set of points that lies in the middle of two neighboring nodes. $ tol $ is a given tolerance for zero (avoids zero division). When computing the size of the interpolation table automatically, we compare the user tolerance $ TOL $ given by set_size_automatic funtion directly with the estimated value of the supremum norm.

This approach is fast and when computing the size automatically, we use the computed values (middle points, function and derivative values) in next interpolation. On the other hand this does not work good around zero points of the function where the value is small and the fraction becomes large or does not converge when dividing the intervals ( $ x^2 $ is a good example).

Therefore we provide also error estimation with $ L_p $ and $ W_p^1 $ norm. Then the user given tolerance is compared as it is stated in the following inequation:

$\left( \frac{\int \limits_I F^p(x) \,\rm{d} x}{|I|} \right)^{\frac{1}{p}} \leq TOL.$

This approach is much more expensive but can give better results.

TODO: Improve error computation. Suggest more robust method.

Evaluation statistics

We collect some statistics during the evaluation of the interpolant in the struct InterpolantBase::EvalStatistics – total number of calls, number of evaluations inside the interpolation interval and number of evaluation outside the interval. One can use these statistics to decide whether it is necessary to widen the interval and recompute the interpolation. Function check_stats_and_reinterpolate is provided to do this automatically according to the given percentage of evaluation outside the interval.

Definition at line 291 of file interpolant.hh.

Constructor & Destructor Documentation

Interpolant::Interpolant ( )

Default constructor.

Definition at line 155 of file interpolant.cc.

template<template< class > class Func, class Type >

Interpolant::Interpolant	(	Func< Type > *	func,
		bool	interpolate_derivative = `false`
	)

Constructor with functor setting.

Parameters

func	is the pointer to functor
interpolate_derivative	is true when derivate is also interpolated

Template Parameters

Func	is the functor type
Type	is the template type of the functor (e.g. double)

Definition at line 56 of file interpolant_impl.hh.

Interpolant::~Interpolant ( void )

virtual

Destructor.

Definition at line 164 of file interpolant.cc.

Member Function Documentation

void Interpolant::compute_error	(	double	tol,
		std::vector< double > &	f,
		std::vector< double > &	df
	)

protected

Creates piecewise linear interpolation.

Computes estimate of interpolation error in maximum norm.

Definition at line 410 of file interpolant.cc.

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void Interpolant::compute_error	(	double	tol,
		double	p,
		ErrorNorm::Type	norm_type
	)

protected

Computes estimate of interpolation error with given norm.

Definition at line 490 of file interpolant.cc.

void Interpolant::compute_values ( )

protected

Creates table of nodes and function values.

Creates vector of nodes according to the table size and computes function and derivative values at the nodes.

Definition at line 335 of file interpolant.cc.

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DiffValue Interpolant::diff ( double x )

inline

Returns interpolated value of the derivation.

Parameters

x	is the point at which we evaluate the interpolation

Definition at line 157 of file interpolant_impl.hh.

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DiffValue Interpolant::diff_p1 ( double x )

inlineprotected

Finds interval on which x lies.

Function that evaluates the derivative of P1 interpolant at x.

Definition at line 281 of file interpolant_impl.hh.

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DiffValue Interpolant::f_diff ( double x )

inline

Returns 1st derivative of original functor using FADBAD.

Parameters

x	is the point at which we evaluate the original functor

Definition at line 213 of file interpolant_impl.hh.

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double Interpolant::f_diffn	(	double	x,
		unsigned int	n
	)

Returns n-th derivative of original functor using FADBAD. Uses coeficients in Taylor's row.

Parameters

x	is the point at which we evaluate the original functor
n	is the order of the derivative we want

Definition at line 171 of file interpolant.cc.

double Interpolant::f_val ( double x )

inline

Returns value of the original functor.

Parameters

x	is the point at which we evaluate the original functor

Definition at line 208 of file interpolant_impl.hh.

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int Interpolant::interpolate ( )

virtual

Creates piecewise interpolation with polynomials of selected degree.

Implements InterpolantBase.

Definition at line 189 of file interpolant.cc.

template<template< class > class Func, class Type >

void Interpolant::set_functor	(	Func< Type > *	func,
		bool	interpolate_derivative = `false`
	)

Sets the functor.

Can be used when the functor is not set in the constructor.

Parameters

func	is the pointer to functor
interpolate_derivative	is true when derivate is also interpolated

Template Parameters

Func	is the functor type
Type	is the template type of the functor (e.g. double)

Definition at line 71 of file interpolant_impl.hh.

void Interpolant::swap_middle_values	(	std::vector< double > &	f,
		std::vector< double > &	df
	)

protected

Definition at line 375 of file interpolant.cc.

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double Interpolant::val ( double x )

inline

Returns interpolated value.

Parameters

x	is the point at which we evaluate the interpolation

Definition at line 84 of file interpolant_impl.hh.

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double Interpolant::val_p1 ( double x )

inline

Do NOT use, unless you are 100% sure.

This function evaluates the P1 interpolant at x. It does not check the interval, does not provide extrapolation and does not collect statistics.

Can be used to POSSIBLY speed the evaluation just a little bit, if you are absolutely sure that you evaluate the interpolant only on the given interval and do not want to collect statistics.

Same can be done with diff_p1 if it is made public.

Used in unit_test benchmark to compare with val function.

Definition at line 250 of file interpolant_impl.hh.

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double Interpolant::val_test ( double x )

inline

Do NOT use, only for testing purpose.

TODO: After testing it can be removed and val_p1 can be made private again.

Definition at line 137 of file interpolant_impl.hh.

Member Data Documentation

std::vector<double> Interpolant::df_vec

protected

Vector of function derivatives values at nodes.

Definition at line 390 of file interpolant.hh.

std::vector<double> Interpolant::f_vec

protected

Vector of function values at nodes.

Definition at line 389 of file interpolant.hh.

FunctorBase<double>* Interpolant::func

protected

Pointer to original functor with double type.

Definition at line 380 of file interpolant.hh.

FunctorBase<B<double> >* Interpolant::func_diff

protected

Pointer to original functor with FADBAD B type.

Definition at line 383 of file interpolant.hh.

FunctorBase<T<double> >* Interpolant::func_diffn

protected

Pointer to original functor with FADBAD T type.

Definition at line 384 of file interpolant.hh.

bool Interpolant::interpolate_derivative

protected

Is true if we want to interpolate the derivative too.

Definition at line 386 of file interpolant.hh.

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

/opt/flow123d/flow123d/src/tools/interpolant.hh
/opt/flow123d/flow123d/src/tools/interpolant.cc
/opt/flow123d/flow123d/src/tools/interpolant_impl.hh

Classes

Public Member Functions

Protected Member Functions

Protected Attributes

Additional Inherited Members

Detailed Description

Evaluation of a functor and its derivatives

Interpolation

Error of the interpolation

Evaluation statistics

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation