Tpetra Matrix/Vector Services  Version of the Day
Trilinos/Tpetra: Next-generation distributed linear algebra

# Introduction

Tpetra implements linear algebra objects, such as sparse matrices and dense vectors. Tpetra is "hybrid parallel," meaning that it uses up to two levels of parallelism:

• MPI (the Message Passing Interface) for distributed-memory parallelism, and
• any of various shared-memory parallel programming models within an MPI process.

We say "distributed linear algebra" because Tpetra objects may be distributed over one or more parallel MPI processes. The shared-memory programming models that Tpetra may use within a process include

• OpenMP
• NVIDIA's CUDA programming model for graphics processing units (GPUs)

Tpetra differs from Epetra, Trilinos' previous distributed linear algebra package, in the following ways:

1. Tpetra has native support for solving very large problems (with over 2 billion unknowns).
2. Tpetra lets you construct matrices and vectors with different kinds of data, such as floating-point types of different precision, or complex-valued types. Our goal is for Tpetra objects to be able to contain any type of data that implements a minimal compile-time interface. Epetra objects only support double-precision floating-point data (of type double).
3. Tpetra can exploit many different kinds of hybrid parallelism, and most of its kernels do so natively. Epetra only supports OpenMP shared-memory parallelism for a few kernels. Tpetra also has optimizations for shared-memory parallel systems with nonuniform memory access (NUMA). All effort in supporting future node-level computer architectures will go into Tpetra.

# Overview of Tpetra

## Templated Types in Tpetra

All of all classes in Tpetra utilize templates, which allows the user to specify any type they want. In some cases, the choice of data type allows increased functionality. For example, 64-bit ordinals allow for problem sizes to break the 2 billion element barrier present in Epetra, whereas complex scalar types allow the native description and solution of complex-valued problems.

Most of the classes in Tpetra are templated according to the data types which constitute the class. These are the following:

- \c Scalar: A \c Scalar is the type of values in the sparse
matrix or dense vector.  This is the type most likely to be
changed by many users. The most common use cases are \c float,
\c double, <tt>std::complex<float></tt> and
<tt>std::complex<double></tt>. However, many other data types
can be used, as long as they have specializations for
Teuchos::ScalarTraits and Teuchos::SerializationTraits, and
support the necessary arithmetic operations, such as addition,
subtraction, division and multiplication.

- \c LocalOrdinal: A \c LocalOrdinal is used to store indices
representing local IDs. The standard use case, as well as the
default for most classes, is \c int. Any signed built-in integer
type may be used.  The reason why local and global ordinals may
have different types is for efficiency.  If the application
allows it, using smaller local ordinals requires less storage
and may improve performance of computational kernels such as
sparse matrix-vector multiply.

- \c GlobalOrdinal: A \c GlobalOrdinal is used to store global
indices and to describe global properties of a distributed
object (e.g., global number of entries in a sparse matrix, or
global number of rows in a vector.) The \c GlobalOrdinal
therefore dictates the maximum size of a distributed object.

- \c Node: Computational classes in %Tpetra will also be templated
on a \c Node type. This node fulfills the \ref kokkos_node_api
"Kokkos Node API" and allows the %Tpetra objects to perform
parallel computation on one of a number of shared-memory nodes,
including multi-core CPUs and GPUs.  You can set the Node type
to control what shared-memory parallel programming model %Tpetra
will use.


The Tpetra::Distributor class is unique in that it is not parametrized by any templated types. However, the class includes some templated member functions. The Tpetra::Distributor::createFromRecvs() method is templated on the ordinal type used to encode IDs, while Tpetra::Distributor::doPosts() and the other post methods are templated on Packet, the data type being communicated by a particular invocation of the Tpetra::Distributor. This allows a single Tpetra::Distributor object (describing a particular communication pattern) to be used to communicate values of different type.

## Tpetra Classes

Tpetra contains a number of classes. The primary parallel classes, employed by most users, are:

## Trilinos and Tpetra

Tpetra can be used mostly as a stand-alone package, with explicit dependencies on Teuchos and Kokkos. There are adapters allowing the use of Tpetra operators and multivectors in both the Belos linear solver package and the Anasazi eigensolver package.