pymor.parameters.functionals
¶
Module Contents¶
- class pymor.parameters.functionals.BaseMaxThetaParameterFunctional(thetas_prime, thetas, mu_bar, gamma_mu_bar=1.0, name=None)[source]¶
Bases:
ParameterFunctional
Implements a generalization of the max-theta approach from [Haa17] (Exercise 5.12).
Let V denote a Hilbert space and let a: V x V -> K denote a continuous bilinear form or l: V -> K a continuous linear functional, either with affine decomposition
a(u, v, mu) = sum_{q = 1}^Q theta_q(mu) a_q(u, v)
or
l(v, mu) = sum_{q = 1}^Q theta_q(mu) l_q(v)
for Q coefficient
ParameterFunctional
theta_1, …, theta_Q and continuous bilinear forms a_1, …, a_Q: V x V -> K or continuous linear functionals l_q: V -> K. Let mu_bar be a parameter with respect to which the continuity constant of a(., ., mu_bar) or l(., mu_bar) is known, i.e. we known gamma_mu_bar > 0, s.t.a(u, v, mu_bar) <= gamma_mu_bar |u|_V |v|_V
or:
l(v, mu_bar) <= gamma_mu_bar |v|_V.
The max-theta approach (in its generalized form) from [Haa17] (Exercise 5.12) allows to obtain a computable bound for the continuity constant of another bilinear form a_prime(., ., mu) or linear form l_prime(., mu) with the same affine decomposition but different theta_prime_q for arbitrary parameters mu, since
a_prime(u, v, mu=mu) <= |max_{q = 1}^Q theta_prime_q(mu)/theta_q(mu_bar)| |a(u, v, mu=mu_bar)|
or
l_prime(v, mu=mu) <= |max_{q = 1}^Q theta_prime_q(mu)/theta_q(mu_bar)| |l(v, mu=mu_bar)|,
if all theta_q(mu_bar) != 0.
Given a list of the thetas, the
parameter values
mu_bar and the constant gamma_mu_bar, this functional thus evaluates to|gamma_mu_bar * max_{q = 1}^Q theta_prime_q(mu)/theta_q(mu_bar)|
Note that we also get an upper bound if theta_prime_q(mu) == 0 for any q. However, if theta_prime_q(mu) == 0 for at least one q, we need to use the absolute value in the denominator, i.e.
|gamma_mu_bar * max_{q = 1}^Q theta_prime_q(mu)/|theta_q(mu_bar)| |
Parameters
- thetas
List or tuple of
ParameterFunctional
of the base bilinear form a which is used for estimation.- thetas_prime
List or tuple of
ParameterFunctional
of the bilinear form a_prime for the numerator of the MaxThetaParameterFunctional.- mu_bar
Parameter associated with gamma_mu_bar.
- gamma_mu_bar
Known continuity constant of the base bilinear form a.
- name
Name of the functional.
Methods
Return the functionals's derivative with respect to a given parameter.
Evaluate the functional for given
parameter values
mu
.- abstract d_mu(component, index=())[source]¶
Return the functionals’s derivative with respect to a given parameter.
Parameters
Returns
New
ParameterFunctional
representing the partial derivative.
- evaluate(mu=None)[source]¶
Evaluate the functional for given
parameter values
mu
.
- class pymor.parameters.functionals.ConjugateParameterFunctional(functional, name=None)[source]¶
Bases:
ParameterFunctional
Conjugate of a given
ParameterFunctional
Evaluates a given
ParameterFunctional
and returns the complex conjugate of the value.Parameters
- functional
The
ParameterFunctional
of which the complex conjuate is taken.- name
Name of the functional.
Methods
Evaluate the functional for given
parameter values
mu
.- evaluate(mu=None)[source]¶
Evaluate the functional for given
parameter values
mu
.
- class pymor.parameters.functionals.ConstantParameterFunctional(constant_value, name=None)[source]¶
Bases:
ParameterFunctional
ParameterFunctional
returning a constant value for each parameter.Parameters
- constant_value
value of the functional
- name
Name of the functional.
Methods
Return the functionals's derivative with respect to a given parameter.
Evaluate the functional for given
parameter values
mu
.- d_mu(parameter, index=0)[source]¶
Return the functionals’s derivative with respect to a given parameter.
Parameters
Returns
New
ParameterFunctional
representing the partial derivative.
- evaluate(mu=None)[source]¶
Evaluate the functional for given
parameter values
mu
.
- class pymor.parameters.functionals.ExpressionParameterFunctional(expression, parameters, name=None, derivative_expressions=None, second_derivative_expressions=None)[source]¶
Bases:
GenericParameterFunctional
Turns a Python expression given as a string into a
ParameterFunctional
.Some
NumPy
arithmetic functions likesin
,log
,min
are supported. For a full list see thefunctions
class attribute.Warning
eval
is used to evaluate the given expression. Using this class with expression strings from untrusted sources will cause mayhem and destruction!Parameters
- expression
A Python expression in the parameter components of the given
parameters
.- parameters
The
Parameters
the functional depends on.- name
The name of the functional.
- derivative_expressions
A dict containing a Python expression for the partial derivatives of each parameter component.
- second_derivative_expressions
A dict containing a list of dicts of Python expressions for all second order partial derivatives of each parameter component i and j.
- class pymor.parameters.functionals.GenericParameterFunctional(mapping, parameters, name=None, derivative_mappings=None, second_derivative_mappings=None)[source]¶
Bases:
ParameterFunctional
A wrapper making an arbitrary Python function a
ParameterFunctional
Note that a GenericParameterFunctional can only be
pickled
if the function it is wrapping can be pickled. For this reason, it is usually preferable to useExpressionParameterFunctional
instead ofGenericParameterFunctional
.Parameters
- mapping
The function to wrap. The function has signature
mapping(mu)
.- parameters
The
Parameters
the functional depends on.- name
The name of the functional.
- derivative_mappings
A dict containing all partial derivatives of each
Parameter
and index with the signaturederivative_mappings[parameter][index](mu)
- second_derivative_mappings
A dict containing all second order partial derivatives of each
Parameter
and index with the signaturesecond_derivative_mappings[parameter_i][index_i][parameter_j][index_j](mu)
Methods
Return the functionals's derivative with respect to a given parameter.
Evaluate the functional for given
parameter values
mu
.- d_mu(parameter, index=0)[source]¶
Return the functionals’s derivative with respect to a given parameter.
Parameters
Returns
New
ParameterFunctional
representing the partial derivative.
- evaluate(mu=None)[source]¶
Evaluate the functional for given
parameter values
mu
.
- class pymor.parameters.functionals.LincombParameterFunctional(functionals, coefficients, name=None)[source]¶
Bases:
ParameterFunctional
A
ParameterFunctional
representing a linear combination ofParameterFunctionals
.The coefficients must be provided as scalars.
Parameters
- functionals
List of
ParameterFunctionals
whose linear combination is formed.- coefficients
A list of scalar coefficients.
- name
Name of the functional.
Methods
Return the functionals's derivative with respect to a given parameter.
Evaluate the functional for given
parameter values
mu
.- d_mu(parameter, index=0)[source]¶
Return the functionals’s derivative with respect to a given parameter.
Parameters
Returns
New
ParameterFunctional
representing the partial derivative.
- evaluate(mu=None)[source]¶
Evaluate the functional for given
parameter values
mu
.
- class pymor.parameters.functionals.MaxThetaParameterFunctional(thetas, mu_bar, gamma_mu_bar=1.0, name=None)[source]¶
Bases:
BaseMaxThetaParameterFunctional
ParameterFunctional
implementing the max-theta approach from [Haa17] (Exercise 5.12).This is a specialized version of BaseMaxThetaParameterFunctional which allows to obtain a computable bound for the continuity constant of the actual a(., ., mu) or l(., mu) for arbitrary parameters mu, since
a(u, v, mu=mu) <= |max_{q = 1}^Q theta_q(mu)/theta_q(mu_bar)| |a(u, v, mu=mu_bar)|
or
l(v, mu=mu) <= |max_{q = 1}^Q theta_q(mu)/theta_q(mu_bar)| |l(v, mu=mu_bar)|,
if all theta_q(mu_bar) != 0.
Given a list of the thetas, the
parameter values
mu_bar and the constant gamma_mu_bar, this functional thus evaluates to|gamma_mu_bar * max{q = 1}^Q theta_q(mu)/theta_q(mu_bar)|
Parameters
- thetas
List or tuple of
ParameterFunctional
- mu_bar
Parameter associated with gamma_mu_bar.
- gamma_mu_bar
Known continuity constant.
- name
Name of the functional.
- class pymor.parameters.functionals.MinThetaParameterFunctional(thetas, mu_bar, alpha_mu_bar=1.0, name=None)[source]¶
Bases:
ParameterFunctional
ParameterFunctional
implementing the min-theta approach from [Haa17] (Prop. 2.35).Let V denote a Hilbert space and let a: V x V -> K denote a parametric coercive bilinear form with affine decomposition
a(u, v, mu) = sum_{q = 1}^Q theta_q(mu) a_q(u, v),
for Q positive coefficient
ParameterFunctional
theta_1, …, theta_Q and positive semi-definite component bilinear forms a_1, …, a_Q: V x V -> K. Let mu_bar be a parameter with respect to which the coercivity constant of a(., ., mu_bar) is known, i.e. we known alpha_mu_bar > 0, s.t.alpha_mu_bar |u|_V^2 <= a(u, u, mu=mu_bar).
The min-theta approach from [Haa17] (Proposition 2.35) allows to obtain a computable bound for the coercivity constant of a(., ., mu) for arbitrary parameters mu, since
a(u, u, mu=mu) >= min_{q = 1}^Q theta_q(mu)/theta_q(mu_bar) a(u, u, mu=mu_bar).
Given a list of the thetas, the
parameter values
mu_bar and the constant alpha_mu_bar, this functional thus evaluates toalpha_mu_bar * min_{q = 1}^Q theta_q(mu)/theta_q(mu_bar)
Parameters
- thetas
List or tuple of
ParameterFunctional
- mu_bar
Parameter associated with alpha_mu_bar.
- alpha_mu_bar
Known coercivity constant.
- name
Name of the functional.
Methods
Evaluate the functional for given
parameter values
mu
.- evaluate(mu=None)[source]¶
Evaluate the functional for given
parameter values
mu
.
- class pymor.parameters.functionals.ParameterFunctional[source]¶
Bases:
pymor.parameters.base.ParametricObject
Interface for
Parameter
functionals.A parameter functional is simply a function mapping
Parameters
to a number.Methods
Return the functionals's derivative with respect to a given parameter.
Evaluate the functional for given
parameter values
mu
.- d_mu(parameter, index=0)[source]¶
Return the functionals’s derivative with respect to a given parameter.
Parameters
Returns
New
ParameterFunctional
representing the partial derivative.
- abstract evaluate(mu=None)[source]¶
Evaluate the functional for given
parameter values
mu
.
- class pymor.parameters.functionals.ProductParameterFunctional(factors, name=None)[source]¶
Bases:
ParameterFunctional
Forms the product of a list of
ParameterFunctionals
or numbers.Parameters
- factors
A list of
ParameterFunctionals
or numbers.- name
Name of the functional.
Methods
Return the functionals's derivative with respect to a given parameter.
Evaluate the functional for given
parameter values
mu
.- d_mu(parameter, index=0)[source]¶
Return the functionals’s derivative with respect to a given parameter.
Parameters
Returns
New
ParameterFunctional
representing the partial derivative.
- evaluate(mu=None)[source]¶
Evaluate the functional for given
parameter values
mu
.
- class pymor.parameters.functionals.ProjectionParameterFunctional(parameter, size=1, index=None, name=None)[source]¶
Bases:
ParameterFunctional
ParameterFunctional
returning a component value of the given parameter.For given parameter map
mu
, this functional evaluates tomu[parameter][index]
Parameters
- parameter
The name of the parameter to return.
- size
The size of the parameter.
- index
See above.
- name
Name of the functional.
Methods
Return the functionals's derivative with respect to a given parameter.
Evaluate the functional for given
parameter values
mu
.- d_mu(parameter, index=0)[source]¶
Return the functionals’s derivative with respect to a given parameter.
Parameters
Returns
New
ParameterFunctional
representing the partial derivative.
- evaluate(mu=None)[source]¶
Evaluate the functional for given
parameter values
mu
.