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 Expression.

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

class pymor.analyticalproblems.expressions.Array(array)

Bases: Expression

An array of scalar-valued Expressions.

Methods

fenics_expr	Called by to_fenics.
numpy_expr	Called by to_numpy.

fenics_expr(params)

Called by to_fenics.

numpy_expr()

Called by to_numpy.

class pymor.analyticalproblems.expressions.BaseConstant

Bases: Expression

A constant value.

Methods

fenics_expr	Called by to_fenics.
numpy_expr	Called by to_numpy.

fenics_symbol = None

numpy_symbol = None

shape = ()

fenics_expr(params)

Called by to_fenics.

numpy_expr()

Called by to_numpy.

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

Bases: Expression

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

Methods

fenics_expr	Called by to_fenics.
numpy_expr	Called by to_numpy.

fenics_symbol = None

numpy_symbol = None

fenics_expr(params)

Called by to_fenics.

numpy_expr()

Called by to_numpy.

class pymor.analyticalproblems.expressions.Constant(value)

Bases: BaseConstant

A constant value given by a NumPy array.

Methods

fenics_expr

Called by to_fenics.

shape = ()

fenics_expr(params)

Called by to_fenics.

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

Bases: BinaryOp

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

fenics_conditional = False

fenics_symbol

numpy_symbol = '-'

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

Bases: BinaryOp

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

fenics_conditional = False

fenics_symbol

numpy_symbol = '/'

class pymor.analyticalproblems.expressions.E

Bases: BaseConstant

A constant value.

fenics_symbol = 'e'

numpy_symbol = 'e'

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.

Methods

fenics_expr	Called by to_fenics.
numpy_expr	Called by to_numpy.
to_fenics
to_numpy	Convert expression to a NumPy-vectorized callable.

abstract fenics_expr(params)

Called by to_fenics.

abstract numpy_expr()

Called by to_numpy.

to_fenics(mesh, variable='x')

to_numpy(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.

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

Bases: BinaryOp

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

fenics_conditional = True

fenics_symbol = 'ge'

numpy_symbol = '>='

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

Bases: BinaryOp

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

fenics_conditional = True

fenics_symbol = 'gt'

numpy_symbol = '>'

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

Bases: Expression

Indexed Expression.

Methods

fenics_expr	Called by to_fenics.
numpy_expr	Called by to_numpy.

fenics_expr(params)

Called by to_fenics.

numpy_expr()

Called by to_numpy.

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

Bases: BinaryOp

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

fenics_conditional = True

fenics_symbol = 'le'

numpy_symbol = '<='

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

Bases: BinaryOp

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

fenics_conditional = True

fenics_symbol = 'lt'

numpy_symbol = '<'

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

Bases: BinaryOp

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

fenics_conditional = None

fenics_symbol = None

numpy_symbol = '%'

class pymor.analyticalproblems.expressions.Neg(operand)

Bases: Expression

Negated Expression.

Methods

fenics_expr	Called by to_fenics.
numpy_expr	Called by to_numpy.

fenics_expr(params)

Called by to_fenics.

numpy_expr()

Called by to_numpy.

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.

Methods

fenics_expr	Called by to_fenics.
numpy_expr	Called by to_numpy.

fenics_expr(params)

Called by to_fenics.

numpy_expr()

Called by to_numpy.

class pymor.analyticalproblems.expressions.Pi

Bases: BaseConstant

A constant value.

fenics_symbol = 'pi'

numpy_symbol = 'pi'

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

Bases: BinaryOp

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

fenics_conditional = False

fenics_symbol = 'elem_pow'

numpy_symbol = '**'

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

Bases: BinaryOp

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

fenics_conditional = False

fenics_symbol

numpy_symbol = '*'

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

Bases: BinaryOp

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

fenics_conditional = False

fenics_symbol

numpy_symbol = '+'

class pymor.analyticalproblems.expressions.TransformChainedComparison

Bases: ast.NodeTransformer

Transformer for chained comparisons.

Methods

visit_Compare

visit_Compare(node)

class pymor.analyticalproblems.expressions.TransformLiterals

Bases: ast.NodeTransformer

Transformer for literals.

Methods

visit_Constant
visit_List
visit_Subscript

in_subscript = False

visit_Constant(node)

visit_List(node)

visit_Subscript(node)

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

Bases: Expression

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

The function is applied component-wise.

Methods

fenics_expr	Called by to_fenics.
numpy_expr	Called by to_numpy.

fenics_symbol = None

numpy_symbol = None

fenics_expr(params)

Called by to_fenics.

numpy_expr()

Called by to_numpy.

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

Bases: UnaryFunctionCall

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.

Methods

fenics_expr	Called by to_fenics.
numpy_expr	Called by to_numpy.

fenics_op = None

numpy_symbol = None

fenics_expr(params)

Called by to_fenics.

numpy_expr()

Called by to_numpy.

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

Bases: UnaryFunctionCall

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

The function is applied component-wise.

Methods

fenics_expr

Called by to_fenics.

numpy_symbol = 'abs'

fenics_expr(params)

Called by to_fenics.

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.

Methods

fenics_expr

Called by to_fenics.

fenics_op = None

numpy_symbol = 'angle'

fenics_expr(params)

Called by to_fenics.

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

Bases: UnaryFunctionCall

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

The function is applied component-wise.

fenics_symbol = 'acos'

numpy_symbol = 'arccos'

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

Bases: UnaryFunctionCall

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

The function is applied component-wise.

Methods

fenics_expr

Called by to_fenics.

numpy_symbol = 'arccosh'

fenics_expr(params)

Called by to_fenics.

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

Bases: UnaryFunctionCall

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

The function is applied component-wise.

fenics_symbol = 'asin'

numpy_symbol = 'arcsin'

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

Bases: UnaryFunctionCall

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

The function is applied component-wise.

Methods

fenics_expr

Called by to_fenics.

numpy_symbol = 'arcsinh'

fenics_expr(params)

Called by to_fenics.

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

Bases: UnaryFunctionCall

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

The function is applied component-wise.

fenics_symbol = 'atan'

numpy_symbol = 'arctan'

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

Bases: UnaryFunctionCall

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

The function is applied component-wise.

Methods

fenics_expr

Called by to_fenics.

numpy_symbol = 'arctanh'

fenics_expr(params)

Called by to_fenics.

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

Bases: UnaryFunctionCall

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

The function is applied component-wise.

fenics_symbol = 'cos'

numpy_symbol = 'cos'

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

Bases: UnaryFunctionCall

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

The function is applied component-wise.

fenics_symbol = 'cosh'

numpy_symbol = 'cosh'

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

Bases: UnaryFunctionCall

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

The function is applied component-wise.

fenics_symbol = 'exp'

numpy_symbol = 'exp'

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

Bases: UnaryFunctionCall

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

The function is applied component-wise.

Methods

fenics_expr

Called by to_fenics.

numpy_symbol = 'exp2'

fenics_expr(params)

Called by to_fenics.

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

Bases: UnaryFunctionCall

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

The function is applied component-wise.

fenics_symbol = 'ln'

numpy_symbol = 'log'

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

Bases: UnaryFunctionCall

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

The function is applied component-wise.

Methods

fenics_expr

Called by to_fenics.

numpy_symbol = 'log10'

fenics_expr(params)

Called by to_fenics.

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

Bases: UnaryFunctionCall

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

The function is applied component-wise.

Methods

fenics_expr

Called by to_fenics.

numpy_symbol = 'log2'

fenics_expr(params)

Called by to_fenics.

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

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.

fenics_op = 'max_value'

numpy_symbol = 'max'

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

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.

fenics_op = 'min_value'

numpy_symbol = 'min'

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

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.

Methods

fenics_expr

Called by to_fenics.

numpy_symbol = 'norm'

fenics_expr(params)

Called by to_fenics.

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

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.

fenics_op

numpy_symbol = 'prod'

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

Bases: UnaryFunctionCall

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

The function is applied component-wise.

fenics_symbol = 'sign'

numpy_symbol = 'sign'

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

Bases: UnaryFunctionCall

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

The function is applied component-wise.

fenics_symbol = 'sin'

numpy_symbol = 'sin'

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

Bases: UnaryFunctionCall

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

The function is applied component-wise.

fenics_symbol = 'sinh'

numpy_symbol = 'sinh'

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

Bases: UnaryFunctionCall

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

The function is applied component-wise.

fenics_symbol = 'sqrt'

numpy_symbol = 'sqrt'

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

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.

fenics_op

numpy_symbol = 'sum'

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

Bases: UnaryFunctionCall

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

The function is applied component-wise.

fenics_symbol = 'tan'

numpy_symbol = 'tan'

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

Bases: UnaryFunctionCall

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

The function is applied component-wise.

fenics_symbol = 'tanh'

numpy_symbol = 'tanh'

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

pymor.analyticalproblems.expressions.NUMPY_COPY_IF_NEEDED = None

pymor.analyticalproblems.expressions.builtin_max

pymor.analyticalproblems.expressions.e

pymor.analyticalproblems.expressions.pi