blockOperator.md

Block Operator representation

We distinguish two types of operator :

• irreducible : one elementary block operator, that cannot be decomposed into a product, addition, etc ... of other block operators. For instance, F and G in Parareal.
• composite : built from the multiplication, addition, etc ... of several block operators. For instance, F-G in the Parareal iteration.

Both types are represented by the following class :

class BlockOperator(object):
    # Attributes used for representation and performance analysis
    symbol      # sympy symbol
    cost        # real number
    components  # dictionnary

    # Attributes used for evaluation and error analysis
    matrix      # numpy 2D array
    invert      # numpy 2D array

• symbol : is a symbolic representation of the operator, used for representation and performance analysis. For composite block operators, it becomes a sympy expression that can be broken down into symbolic operations and symbols.
• cost : is a theoretical cost for this operator. It is relevant only for irreducible block operators, and is None for any composite block operator,
• components : a dictionnary with key=name and value=BlockOperator containing all irreducible block operators composing the block operator. For an irreducible block operator, it only contains a reference to itself.

Numerical representation and evaluation

For any vector $u$, its evaluation by a block operator is represented by :

$$u_{eval} = A B^{-1} u$$

where $A$ and $B$ are square matrices, with $B$ supposed to ne invertible. This allows to inverse some operator, and avoid computing the inverse of $B$ for evaluation. Instead, evaluation is performed in two stages :

• Solve $By=u$ using numpy.linalg.solve (better numerical accuracy),
• Compute evaluated solution $u_{eval} = Ay$ using numpy.dot matrix-vector product.

The $A$ matrix is stored in the matrix attribute, while the invert attribute stores the $B$ matrix. When a block operator does not have an implicit part (only $A$), then the invert attribute is set to None, and vice-versa. By default, when both invert and matrix are set to None, it means that the BlockOperator represent the identity.

Arithmetic operation between block operator are allowed, and their matrix and invert attributes are modified accordingly.

🔔 For now, addition and substraction between BlockOperators having an invert attribute not set to None is not allowed.