# Technical Overview¶

## Three Central Classes¶

From a bird’s eye perspective, pyMOR is a collection of generic algorithms operating on objects of the following types:

`VectorArrays`

Vector arrays are ordered collections of vectors. Each vector of the array must be of the same `dimension`. Vectors can be `copied` to a new array, `appended` to an existing array or `deleted` from the array. Basic linear algebra operations can be performed on the vectors of the array: vectors can be `scaled` in-place, the BLAS `axpy` operation is supported and `inner products` between vectors can be formed. Linear combinations of vectors can be formed using the `lincomb` method. Moreover, various norms can be computed and selected `dofs` of the vectors can be extracted for `empirical interpolation`. To act on subsets of vectors of an array, arrays can be `indexed` with an integer, a list of integers or a slice, in each case returning a new `VectorArray` which acts as a modifiable view onto the respective vectors in the original array. As a convenience, many of Python’s math operators are implemented in terms of the interface methods.

Note that there is not the notion of a single vector in pyMOR. The main reason for this design choice is to take advantage of vectorized implementations like `NumpyVectorArray` which internally store the vectors as two-dimensional `NumPy` arrays. As an example, the application of a linear matrix based operator to an array via the `apply` method boils down to a call to `NumPy`’s optimized `dot` method. If there were only lists of vectors in pyMOR, the above matrix-matrix multiplication would have to be expressed by a loop of matrix-vector multiplications. However, when working with external solvers, vector arrays will often be given as lists of individual vector objects. For this use-case we provide `ListVectorArray`, a `VectorArray` based on a Python list of vectors.

Associated to each vector array is a `VectorSpace` which acts as a factory for new arrays of a given type. New vector arrays can be created using the `zeros` and `empty` methods. To wrap the raw objects of the underlying linear algebra backend into a new `VectorArray`, `make_array` is used.

The data needed to define a new `VectorSpace` largely depends on the implementation of the underlying backend. For `NumpyVectorSpace`, the only required datum is the dimension of the contained vectors. `VectorSpaces` for other backends could, e.g., hold a socket for communication with a specific PDE solver instance. Additionally, each `VectorSpace` has a string `id`, defaulting to `None`, which is used to signify the mathematical identity of the given space.

Two arrays in pyMOR are compatible (e.g. can be added) if they are from the same `VectorSpace`. If a `VectorArray` is contained in a given `VectorSpace` can be tested with the `in` operator.

`Operators`

The main property of operators in pyMOR is that they can be `applied` to `VectorArrays` resulting in a new `VectorArray`. For this operation to be allowed, the operator’s `source` `VectorSpace` must be identical with the `VectorSpace` of the given array. The result will be a vector array from the `range` space. An operator can be `linear` or not. The `apply_inverse` method provides an interface for (linear) solvers.

Operators in pyMOR are also used to represent bilinear forms via the `apply2` method. A functional in pyMOR is simply an operator with `NumpyVectorSpace(1)` as `range`. Dually, a vector-like operator is an operator with `NumpyVectorSpace(1)` as `source`. Such vector-like operators are used in pyMOR to represent `Parameter` dependent vectors such as the initial data of an `InstationaryDiscretization`. For linear functionals and vector-like operators, the `as_vector` method can be called to obtain a vector representation of the operator as a `VectorArray` of length 1.

Linear combinations of operators can be formed using a `LincombOperator`. When such a linear combination is `assembled`, `assemble_lincomb` is called to ensure that, for instance, linear combinations of operators represented by a matrix lead to a new operator holding the linear combination of the matrices.

Default implementations for many methods of the operator interface can be found in `OperatorBase`. Base classes for `NumPy`-based operators can be found in `pymor.operators.numpy`. Several methods for constructing new operators from existing ones are contained in `pymor.operators.constructions`.

`Discretizations`

Discretizations in pyMOR encode the mathematical structure of a given discrete problem by acting as container classes for operators. Each discretization object has `operators`, `products` dictionaries holding the `Operators` which appear in the formulation of the discrete problem. The keys in these dictionaries describe the role of the respective operator in the discrete problem.

Apart from describing the discrete problem, discretizations also implement algorithms for `solving` the given problem, returning `VectorArrays` from the `solution_space`. The solution can be `cached`, s.t. subsequent solving of the problem for the same parameter reduces to looking up the solution in pyMOR’s cache.

While special discretization classes may be implemented which make use of the specific types of operators they contain (e.g. using some external high-dimensional solver for the problem), it is generally favourable to implement the solution algorithms only through the interfaces provided by the operators contained in the discretization, as this allows to use the same discretization class to solve high-dimensional and reduced problems. This has been done for the simple stationary and instationary discretizations found in `pymor.discretizations.basic`.

Discretizations can also implement `estimate` and `visualize` methods to estimate the discretization or model reduction error of a computed solution and create graphic representations of `VectorArrays` from the `solution_space`.

## Base Classes¶

While `VectorArrays` are mutable objects, both `Operators` and `Discretizations` are immutable in pyMOR: the application of an `Operator` to the same `VectorArray` will always lead to the same result, solving a `Discretization` for the same parameter will always produce the same solution array. This has two main benefits:

1. If multiple objects/algorithms hold references to the same `Operator` or `Discretization`, none of the objects has to worry that the referenced object changes without their knowledge.
2. It becomes affordable to generate persistent keys for `caching` of computation results by generating state ids which uniquely identify the object’s state. Since the state cannot change, these ids have to be computed only once for the lifetime of the object.

A class can be made immutable in pyMOR by deriving from `ImmutableInterface`, which ensures that write access to the object’s attributes is prohibited after `__init__` has been executed. However, note that changes to private attributes (attributes whose name starts with `_`) are still allowed. It lies in the implementors responsibility to ensure that changes to these attributes do not affect the outcome of calls to relevant interface methods. As an example, a call to `enable_caching` will set the objects private `__cache_region` attribute, which might affect the speed of a subsequent `solve` call, but not its result.

Of course, in many situations one may wish to change properties of an immutable object, e.g. the number of timesteps for a given discretization. This can be easily achieved using the `with_` method every immutable object has: a call of the form `o.with_(a=x, b=y)` will return a copy of `o` in which the attribute `a` now has the value `x` and the attribute `b` the value `y`. It can be generally assumed that calls to `with_` are inexpensive. The set of allowed arguments can be found in the `with_arguments` attribute.

All immutable classes in pyMOR and most other classes derive from `BasicInterface` which, through its meta class, provides several convenience features for pyMOR. Most notably, every subclass of `BasicInterface` obtains its own `logger` instance with a class specific prefix.

## Creating Discretizations¶

pyMOR ships a small (and still quite incomplete) framework for creating finite element or finite volume discretizations based on the NumPy/Scipy software stack. To end up with an appropriate `Discretization`, one starts by instantiating an `analytical problem` which describes the problem we want to discretize. `analytical problems` contain `Functions` which define the analytical data functions associated with the problem and a `DomainDescription` that provides a geometrical definition of the domain the problem is posed on and associates a boundary type to each part of its boundary.

To obtain a `Discretization` from an `analytical problem` we use a `discretizer`. A discretizer will first mesh the computational domain by feeding the `DomainDescription` into a `domaindiscretizer` which will return the `Grid` along with a `BoundaryInfo` associating boundary entities with boundary types. Next, the `Grid`, `BoundaryInfo` and the various data functions of the `analytical problem` are used to instatiate `finite element` or `finite volume` operators. Finally these operators are used to instatiate one of the provided `Discretization` classes.

In pyMOR, `analytical problems`, `Functions`, `DomainDescriptions`, `BoundaryInfos` and `Grids` are all immutable, enabling efficient disk `caching` for the resulting `Discretizations`, persistent over various runs of the applications written with pyMOR.

While pyMOR’s internal discretizations are useful for getting started quickly with model reduction experiments, pyMOR’s main goal is to allow the reduction of discretizations provided by external solvers. In order to do so, all that needs to be done is to provide `VectorArrays`, `Operators` and `Discretizations` which interact appropriately with the solver. pyMOR makes no assumption on how the communication with the solver is managed. For instance, communication could take place via a network protocol or job files. In particular it should be stressed that in general no communication of high-dimensional data between the solver and pyMOR is necessary: `VectorArrays` can merely hold handles to data in the solver’s memory or some on-disk database. Where possible, we favour, however, a deep integration of the solver with pyMOR by linking the solver code as a Python extension module. This allows Python to directly access the solver’s data structures which can be used to quickly add features to the high-dimensional code without any recompilation. A minimal example for such an integration using pybindgen can be found in the `src/pymordemos/minimal_cpp_demo` directory of the pyMOR repository. Bindings for FEnicS and NGSolve packages are available in the `bindings.fenics` and `bindings.ngsolve` modules. The pymor-deal.II repository contains experimental bindings for deal.II.

## Parameters¶

pyMOR classes implement dependence on a parameter by deriving from the `Parametric` mix-in class. This class gives each instance a `parameter_type` attribute describing the form of `Parameters` the relevant methods of the object (`apply`, `solve`, `evaluate`, etc.) expect. A `Parameter` in pyMOR is basically a Python `dict` with strings as keys and `NumPy arrays` as values. Each such value is called a `Parameter` component. The `ParameterType` of a `Parameter` is simply obtained by replacing the arrays in the `Parameter` with their shape. I.e. a `ParameterType` specifies the names of the parameter components and their expected shapes.

The `ParameterType` of a `Parametric` object is determined by the class implementor during `__init__` via a call to `build_parameter_type`, which can be used to infer the `ParameterType` from the `ParameterTypes` of objects the given object depends upon. I.e. an `Operator` implementing the L2-product with some `Function` will inherit the `ParameterType` of the `Function`.

Reading the `reference documentation` on pyMOR’s parameter handling facilities is strongly advised for implementors of `Parametric` classes.

## Defaults¶

pyMOR offers a convenient mechanism for handling default values such as solver tolerances, cache sizes, log levels, etc. Each default in pyMOR is the default value of an optional argument of some function. Such an argument is made a default by decorating the function with the `defaults` decorator:

```@defaults('tolerance')
def some_algorithm(x, y, tolerance=1e-5)
...
```

Default values can be changed by calling `set_defaults`. By calling `print_defaults` a summary of all defaults in pyMOR and their values can be printed. A configuration file with all defaults can be obtained with `write_defaults_to_file`. This file can then be loaded, either programmatically or automatically by setting the `PYMOR_DEFAULTS` environment variable.

As an additional feature, if `None` is passed as value for a function argument which is a default, its default value is used instead of `None`. This allows writing code of the following form:

```def method_called_by_user(U, V, tolerance_for_algorithm=None):
...
algorithm(U, V, tolerance=tolerance_for_algorithm)
...
```

See the `defaults` module for more information.

## RuleTables¶

Many algorithms in pyMOR can be seen as transformations acting on trees of `Operators`. One example is the structure-preserving (Petrov-)Galerkin projection of `Operators` performed by the `project` method. For instance, a `LincombOperator` is projected by replacing all its children (the `Operators` forming the affine decomposition) with projected `Operators`.

During development of pyMOR, it turned out that using inheritance for selecting the action to be taken to project a specific operator (i.e. single dispatch based on the class of the to-be-projected `Operator`) is not sufficiently flexible. With pyMOR 0.5 we have introduced algorithms which are based on `RuleTables` instead of inheritance. A `RuleTable` is simply an ordered list of `rules`, i.e. pairs of conditions to match with corresponding actions. When a `RuleTable` is `applied` to an object (e.g. an `Operator`), the action associated with the first matching rule in the table is executed. As part of the action, the `RuleTable` can be easily `applied recursively` to the children of the given object.

This approach has several advantages over an inheritance-based model:

• Rules can match based on the class of the object, but also on more general conditions, i.e. the name of the `Operator` or being linear and non-`parametric`.
• The entire mathematical algorithm can be specified in a single file even when the definition of the possible classes the algorithm can be applied to is scattered over various files.
• The precedence of rules is directly apparent from the definition of the `RuleTable`.
• Generic rules (e.g. the projection of a linear non-`parametric` `Operator` by simply applying the basis) can be easily scheduled to take precedence over more specific rules.
• Users can implement or modify `RuleTables` without modification of the classes shipped with pyMOR.

## The Reduction Process¶

The reduction process in pyMOR is handled by so called `reductors` which take arbitrary `Discretizations` and additional data (e.g. the reduced basis) to create reduced `Discretizations`. If proper offline/online decomposition is achieved by the reductor, the reduced `Discretization` will not store any high-dimensional data. Note that there is no inherent distinction between low- and high-dimensional `Discretizations` in pyMOR. The only difference lies in the different types of operators, the `Discretization` contains.

This observation is particularly apparent in the case of the classical reduced basis method: the operators and functionals of a given discrete problem are projected onto the reduced basis space whereas the structure of the problem (i.e. the type of `Discretization` containing the operators) stays the same. pyMOR reflects this fact by offering with `GenericRBReductor` a generic algorithm which can be used to RB-project any discretization available to pyMOR. It should be noted however that this reductor is only able to efficiently offline/online-decompose affinely `Parameter`-dependent linear problems. Non-linear problems or such with no affine `Parameter` dependence require additional techniques such as `empirical interpolation`.

If you want to further dive into the inner workings of pyMOR, we highly recommend to study the source code of `GenericRBReductor` and to step through calls of this method with a Python debugger, such as ipdb.