pymor.analyticalproblems.expressions

This module contains a basic symbolic expression library.

The library is used by ExpressionFunction and ExpressionParameterFunctional by calling parse_expression, which parses str expressions by replacing the names in the string with objects from this module. The result is an Expression object, which can be converted to a NumPy-vectorized function using Expression.to_numpy. In the future, more conversion routines will be added to make the same Expression usable for pymor.discretizers that use external PDE solvers. Further advantages of using this expression library:

• meaningful error messages are generated at parse time of the str expression, instead of hard-to-debug errors in lambda functions at evaluation time,

• expressions are automatically correctly vectorized. In particular, there is no longer a need to add ... to indexing expressions,

• the shape of the resulting expressions is automatically determined.

In the future, we will also provide support for symbolic differentiation of the given Expressions.

Every Expression is built from the following atoms:

• a Constant, which is a fixed value of arbitrary shape,

• a Parameter, which is a variable of a fixed one-dimensional shape.

Note that both Parameters and input variables are treated as a Parameter in the expression. Only when calling, e.g., to_numpy, it is determined which Parameter belongs to the resulting function’s Parameters and which Parameter is treated as an input variable.

More complex expressions can be built using:

• binary operators,

• negation,

• function calls,

• indexing,

• array construction.

For binary operations of Expressions of different shape, the usual broadcasting rules apply.

Module Contents

Classes

Expression	A symbolic math expression
BaseConstant	A constant value.
Constant	A constant value given by a NumPy array.
Parameter	A free parameter in an Expression.
Array	An array of scalar-valued Expressions.
BinaryOp	Compound Expression of a binary operator acting on two sub-expressions.
Neg	Negated Expression.
Indexed	Indexed Expression.
UnaryFunctionCall	Compound Expression of an unary function applied to a sub-expression.
UnaryReductionCall	Compound Expression of an unary function applied to a sub-expression.
Sum	Compound Expression of a binary operator acting on two sub-expressions.
Diff	Compound Expression of a binary operator acting on two sub-expressions.
Prod	Compound Expression of a binary operator acting on two sub-expressions.
Div	Compound Expression of a binary operator acting on two sub-expressions.
Pow	Compound Expression of a binary operator acting on two sub-expressions.
Mod	Compound Expression of a binary operator acting on two sub-expressions.
LE	Compound Expression of a binary operator acting on two sub-expressions.
GE	Compound Expression of a binary operator acting on two sub-expressions.
LT	Compound Expression of a binary operator acting on two sub-expressions.
GT	Compound Expression of a binary operator acting on two sub-expressions.
sin	Compound Expression of an unary function applied to a sub-expression.
cos	Compound Expression of an unary function applied to a sub-expression.
tan	Compound Expression of an unary function applied to a sub-expression.
arcsin	Compound Expression of an unary function applied to a sub-expression.
arccos	Compound Expression of an unary function applied to a sub-expression.
arctan	Compound Expression of an unary function applied to a sub-expression.
sinh	Compound Expression of an unary function applied to a sub-expression.
cosh	Compound Expression of an unary function applied to a sub-expression.
tanh	Compound Expression of an unary function applied to a sub-expression.
arcsinh	Compound Expression of an unary function applied to a sub-expression.
arccosh	Compound Expression of an unary function applied to a sub-expression.
arctanh	Compound Expression of an unary function applied to a sub-expression.
exp	Compound Expression of an unary function applied to a sub-expression.
exp2	Compound Expression of an unary function applied to a sub-expression.
log	Compound Expression of an unary function applied to a sub-expression.
log2	Compound Expression of an unary function applied to a sub-expression.
log10	Compound Expression of an unary function applied to a sub-expression.
sqrt	Compound Expression of an unary function applied to a sub-expression.
abs	Compound Expression of an unary function applied to a sub-expression.
sign	Compound Expression of an unary function applied to a sub-expression.
angle	Compound Expression of an unary function applied to a sub-expression.
norm	Compound Expression of an unary function applied to a sub-expression.
min	Compound Expression of an unary function applied to a sub-expression.
max	Compound Expression of an unary function applied to a sub-expression.
sum	Compound Expression of an unary function applied to a sub-expression.
prod	Compound Expression of an unary function applied to a sub-expression.
Pi	A constant value.
E	A constant value.

Functions

parse_expression
_convert_to_expression	Used internally to convert literals/list constructors to an Expression.
_broadcastable_shapes

Attributes

builtin_max
pi
e
_numpy_functions

pymor.analyticalproblems.expressions.builtin_max

pymor.analyticalproblems.expressions.parse_expression(expression, parameters={}, values={})

class pymor.analyticalproblems.expressions.Expression

Bases: pymor.parameters.base.ParametricObject

A symbolic math expression

shape

The shape of the object this expression evaluates to in the sense of NumPy.

to_numpy(self, variables)

Convert expression to a NumPy-vectorized callable.

Parameters

variables

List of names of ~Parameters in the expression which shall be treated as input variables.

abstract numpy_expr(self)

Called by to_numpy.

__getitem__(self, index)

__add__(self, other)

__radd__(self, other)

__sub__(self, other)

__rsub__(self, other)

__mul__(self, other)

__rmul__(self, other)

__truediv__(self, other)

__rtruediv__(self, other)

__pow__(self, other)

__rpow__(self, other)

__mod__(self, other)

__rmod__(self, other)

__neg__(self)

__le__(self, other)

Return self<=value.

__ge__(self, other)

Return self>=value.

__lt__(self, other)

Return self<value.

__gt__(self, other)

Return self>value.

class pymor.analyticalproblems.expressions.BaseConstant

Bases: Expression

A constant value.

numpy_symbol

shape = []

numpy_expr(self)

Called by to_numpy.

__str__(self)

Return str(self).

class pymor.analyticalproblems.expressions.Constant(value)

Bases: BaseConstant

A constant value given by a NumPy array.

__str__(self)

Return str(self).

class pymor.analyticalproblems.expressions.Parameter(name, dim)

Bases: Expression

A free parameter in an Expression.

Parameters represent both pyMOR Parameters as well as input variables. Each parameter is a vector of shape (dim,).

Parameters

name

The name of the parameter.

dim

The dimension of the parameter.

numpy_expr(self)

Called by to_numpy.

__str__(self)

Return str(self).

class pymor.analyticalproblems.expressions.Array(array)

Bases: Expression

An array of scalar-valued Expressions.

numpy_expr(self)

Called by to_numpy.

__str__(self)

Return str(self).

class pymor.analyticalproblems.expressions.BinaryOp(first, second)

Bases: Expression

Compound Expression of a binary operator acting on two sub-expressions.

numpy_symbol

numpy_expr(self)

Called by to_numpy.

__str__(self)

Return str(self).

class pymor.analyticalproblems.expressions.Neg(operand)

Bases: Expression

Negated Expression.

numpy_expr(self)

Called by to_numpy.

__str__(self)

Return str(self).

class pymor.analyticalproblems.expressions.Indexed(base, index)

Bases: Expression

Indexed Expression.

numpy_expr(self)

Called by to_numpy.

__str__(self)

Return str(self).

class pymor.analyticalproblems.expressions.UnaryFunctionCall(*arg)

Bases: Expression

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol

numpy_expr(self)

Called by to_numpy.

__str__(self)

Return str(self).

class pymor.analyticalproblems.expressions.UnaryReductionCall(*arg)

Bases: Expression

Compound Expression of an unary function applied to a sub-expression.

The function is applied to the entire vector/matrix/tensor the sub-expression evaluates to, returning a single number.

numpy_expr(self)

Called by to_numpy.

__str__(self)

Return str(self).

pymor.analyticalproblems.expressions._convert_to_expression(obj)

Used internally to convert literals/list constructors to an Expression.

pymor.analyticalproblems.expressions._broadcastable_shapes(first, second)

class pymor.analyticalproblems.expressions.Sum(first, second)

Bases: BinaryOp

Compound Expression of a binary operator acting on two sub-expressions.

numpy_symbol = +

class pymor.analyticalproblems.expressions.Diff(first, second)

Bases: BinaryOp

Compound Expression of a binary operator acting on two sub-expressions.

numpy_symbol = -

class pymor.analyticalproblems.expressions.Prod(first, second)

Bases: BinaryOp

Compound Expression of a binary operator acting on two sub-expressions.

numpy_symbol = *

class pymor.analyticalproblems.expressions.Div(first, second)

Bases: BinaryOp

Compound Expression of a binary operator acting on two sub-expressions.

numpy_symbol = /

class pymor.analyticalproblems.expressions.Pow(first, second)

Bases: BinaryOp

Compound Expression of a binary operator acting on two sub-expressions.

numpy_symbol = **

class pymor.analyticalproblems.expressions.Mod(first, second)

Bases: BinaryOp

Compound Expression of a binary operator acting on two sub-expressions.

numpy_symbol = %

class pymor.analyticalproblems.expressions.LE(first, second)

Bases: BinaryOp

Compound Expression of a binary operator acting on two sub-expressions.

numpy_symbol = <=

class pymor.analyticalproblems.expressions.GE(first, second)

Bases: BinaryOp

Compound Expression of a binary operator acting on two sub-expressions.

numpy_symbol = >=

class pymor.analyticalproblems.expressions.LT(first, second)

Bases: BinaryOp

Compound Expression of a binary operator acting on two sub-expressions.

numpy_symbol = <

class pymor.analyticalproblems.expressions.GT(first, second)

Bases: BinaryOp

Compound Expression of a binary operator acting on two sub-expressions.

numpy_symbol = >

class pymor.analyticalproblems.expressions.sin(*arg)

Bases: UnaryFunctionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol = sin

class pymor.analyticalproblems.expressions.cos(*arg)

Bases: UnaryFunctionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol = cos

class pymor.analyticalproblems.expressions.tan(*arg)

Bases: UnaryFunctionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol = tan

class pymor.analyticalproblems.expressions.arcsin(*arg)

Bases: UnaryFunctionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol = arcsin

class pymor.analyticalproblems.expressions.arccos(*arg)

Bases: UnaryFunctionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol = arccos

class pymor.analyticalproblems.expressions.arctan(*arg)

Bases: UnaryFunctionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol = arctan

class pymor.analyticalproblems.expressions.sinh(*arg)

Bases: UnaryFunctionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol = sinh

class pymor.analyticalproblems.expressions.cosh(*arg)

Bases: UnaryFunctionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol = cosh

class pymor.analyticalproblems.expressions.tanh(*arg)

Bases: UnaryFunctionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol = tanh

class pymor.analyticalproblems.expressions.arcsinh(*arg)

Bases: UnaryFunctionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol = arcsinh

class pymor.analyticalproblems.expressions.arccosh(*arg)

Bases: UnaryFunctionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol = arccosh

class pymor.analyticalproblems.expressions.arctanh(*arg)

Bases: UnaryFunctionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol = arctanh

class pymor.analyticalproblems.expressions.exp(*arg)

Bases: UnaryFunctionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol = exp

class pymor.analyticalproblems.expressions.exp2(*arg)

Bases: UnaryFunctionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol = exp2

class pymor.analyticalproblems.expressions.log(*arg)

Bases: UnaryFunctionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol = log

class pymor.analyticalproblems.expressions.log2(*arg)

Bases: UnaryFunctionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol = log2

class pymor.analyticalproblems.expressions.log10(*arg)

Bases: UnaryFunctionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol = log10

class pymor.analyticalproblems.expressions.sqrt(*arg)

Bases: UnaryFunctionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol = sqrt

class pymor.analyticalproblems.expressions.abs(*arg)

Bases: UnaryFunctionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol = abs

class pymor.analyticalproblems.expressions.sign(*arg)

Bases: UnaryFunctionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol = sign

class pymor.analyticalproblems.expressions.angle(arg)

Bases: UnaryFunctionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied component-wise.

numpy_symbol = angle

class pymor.analyticalproblems.expressions.norm(*arg)

Bases: UnaryReductionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied to the entire vector/matrix/tensor the sub-expression evaluates to, returning a single number.

numpy_symbol = norm

class pymor.analyticalproblems.expressions.min(*arg)

Bases: UnaryReductionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied to the entire vector/matrix/tensor the sub-expression evaluates to, returning a single number.

numpy_symbol = min

class pymor.analyticalproblems.expressions.max(*arg)

Bases: UnaryReductionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied to the entire vector/matrix/tensor the sub-expression evaluates to, returning a single number.

numpy_symbol = max

class pymor.analyticalproblems.expressions.sum(*arg)

Bases: UnaryReductionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied to the entire vector/matrix/tensor the sub-expression evaluates to, returning a single number.

numpy_symbol = sum

class pymor.analyticalproblems.expressions.prod(*arg)

Bases: UnaryReductionCall

Compound Expression of an unary function applied to a sub-expression.

The function is applied to the entire vector/matrix/tensor the sub-expression evaluates to, returning a single number.

numpy_symbol = prod

class pymor.analyticalproblems.expressions.Pi

Bases: BaseConstant

A constant value.

numpy_symbol = pi

class pymor.analyticalproblems.expressions.E

Bases: BaseConstant

A constant value.

numpy_symbol = e

pymor.analyticalproblems.expressions.pi

pymor.analyticalproblems.expressions.e

pymor.analyticalproblems.expressions._numpy_functions