Tpetra parallel linear algebra
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Different ways to add entries to Tpetra sparse matrix
This lesson explains how to add entries to a Tpetra sparse matrix (Tpetra::CrsMatrix). It explains how you can choose any of various ways to add entries to the matrix. Some are more flexible but less efficient, and others are more efficient but less flexible. In general, knowing the sparse matrix structure in advance will reduce both peak memory usage and total run time.
This lesson does not currently include code examples. The section Exercise at the end suggests modifying the example Tpetra Lesson 03: Power method in order to learn about the different ways to add or change the entries of a sparse matrix.
We call adding entries to a sparse matrix filling it, and call the general procedure "sparse matrix fill" or just fill. Often people have finite element assembly in mind when they use this expression, but fill here means adding entries for any reason. For example, you might want to
Also, by extension, "fill" might refer to modifying existing entries of a sparse matrix.
CrsMatrix inherits from the RowMatrix interface; it "is a" RowMatrix. Some people have the misconception that "RowMatrix" (and therefore CrsMatrix) refers to a rowwise distribution of the sparse matrix over parallel processes. It does not. CrsMatrix supports an arbitrary "twodimensional" distribution of rows and columns over processes. The term instead refers to rowwise access to the data. That is, the methods in this interface and in CrsMatrix let you add or access entries on each process by row.
This distinction matters because two or more processes might share entries in a row. Asking for "all the entries in a row" on a particular process only accesses the entries owned by that process, which is not necessarily all the entries in a row.
Whether adding entries or modifying existing ones, you may always do so for any number of entries in the row, such that their columns are owned by the calling process. You should always modify as many entries with one method call as possible, in order to amortize function call and data access overhead.
A sparse matrix may be in any of three different states:
fillComplete()
fillComplete()
has been called after construction or resumeFill()
resumeFill()
has been called In States 1 and 3, you are allowed to set or modify values in the matrix. If the matrix was not created with a constant graph (see below), and if there is room in the row (see "static profile" discussion below), you are allowed to insert entries into the matrix.
We call a matrix in State 2 "fill complete." In State 2, you are not allowed to modify the matrix at all. (This is different than Epetra_CrsMatrix, which allows modifications to the values (but not the structure) of a fill complete matrix.)
Calling fillComplete()
does a lot of work. In general, it requires distributedmemory communication as well as local rearrangement of data. The intent is that you should exploit this work for as long as possible before calling resumeFill()
. You can do so by performing multiple sparse matrixvector multiplies or triangular solves.
In all the examples below, we assume the following "preamble" of code:
CrsMatrix lets you make tradeoffs between flexibility and performance. All of these tradeoffs revolve around how much you know about the structure of the sparse matrix – that is, its graph – in advance. In the most general case, all you know is which process owns which row – the row Map.
This code snippet says that you want to create a sparse matrix with the given row distribution. It also says that you intend each process to own all the entries in a given row. If you want to allow a more general "2D" distribution, you must also tell the constructor which process owns which columns of the matrix. You do so by supplying a "column Map":
All the examples that follow allow you to supply a column Map if you which, so we omit this for brevity.
Both of these ways to create a sparse matrix allocate memory for the entries in each row dynamically. This means that it reallocates space as necessary. We call this dynamic profile, where "profile" refers to the structure (the graph) of the sparse matrix. (Static profile, which we describe later, does not reallocate; it fixes an upper bound on the number of entries allowed per row.) CrsMatrix implements this currently as an array of arrays, one array per row, though this may change. You might hear this array of arrays structure referred to as "2D storage," versus the "1D" storage of the conventional compressed sparse row format.
In the above examples, the matrix has no information about how many entries you want to add to each row. Thus, it has to make some default assumption. The "size hint" defaults to zero, to ensure that empty or nearly empty sparse matrices take as little memory as possible. An initial reservation of zero entries per row may cause frequent reallocation if you add entries to a row many times. Reallocation takes time, because it requires talking to the C++ runtime library or the system library. It also tends to fragment memory and thus disturb locality, further reducing performance.
If you have a pretty good idea how many entries will be in each row, you may give a "size hint" to the constructor. You may do so either with a single number, which is an expected upper bound on the number of entries in any one row, or with an upper bound for each row of the matrix. The typical use case is a single upper bound, which we illustrate below.
If you have an upper bound for each row, you probably could use static profile, but it is still legal to use dynamic profile in this case.
Giving an upper bound on the number of entries per row allows the matrix to preallocate space for each row. Dynamic profile still allows you to put more entries in a row than that, but if you don't, then the matrix will only allocate space once per row during the fill phase.
Static allocation, what we call static profile, means that the "size hint" becomes a hard constraint. You may either specify a single upper bound on the number of entries in any row, or an array with a separate upper bound for each row. Here is an example of the former:
What happens when you specify static profile is that the matrix allocates "1D" storage. That is, rather than the aforementioned "2D" array of arrays, the matrix allocates the usual three compressed sparse row arrays (ptr, ind, val), with fixed space in each row (given either by the upper bound, or by the aforementioned array). When you're done filling and call fillComplete()
, this means the matrix need only "pack" the 1D storage into its final representation by copying the entries that you filled in (and not any of the "slack
space" you might have left). Static profile also may use less memory than dynamic profile, because you don't have to store a complicated dynamic data structure for each row of the matrix.
The most efficient but least flexible fill method is to create the CrsMatrix with a constant graph. That is, you create a CrsGraph separately, fill it, call its fillComplete()
method, then pass the graph to the CrsMatrix constructor. This completely constrains the structure of the CrsMatrix. You may only set or modify values in the matrix, not the structure. (This means you may not call insertGlobalValues()
or insertLocalValues()
, only the replace*Values()
, sumInto*Values
, scale()
, and setAllToScalar()
methods.
Filling the CrsGraph works much like filling a CrsMatrix, except that you only specify structure, not values. Furthermore, the same size hint and dynamic profile vs. static profile options apply to CrsGraph.
Please see Tpetra Lesson 02: Map and Vector for a detailed explanation of global and local indices.
In the sparse matrix's final state, it represents column indices as local indices. This is faster for operations like sparse matrixvector multiply. It saves storage as well, especially if sizeof(LocalOrdinal) < sizeof(GlobalOrdinal)
. However, accessing or inserting entries by local index is only allowed under certain conditions. In particular, the matrix must have a column Map, which tells it how to convert between global and local indices for the column indices of the matrix. If you didn't create the matrix with a precomputed column Map (either in its constructor, or in the constant graph), it has to compute the column Map on its own. It does so in
fillComplete()
.
If the matrix doesn't have a column Map yet, you must use global indices, since local indices don't exist yet. However, if you can use local indices, you should. For global indices, the matrix must go to the Map to look up the corresponding local index for every global index. Local indices only require an array access.
Remember that if column indices are local, then they are counted with respect to the column Map. Local row and column indices may not necessarily be the same. For example, the matrix entry at position
(i_local, i_local)
, where the first i_local
is the local row index and the second i_local
the local column index, may not necessarily be a diagonal entry of the matrix. However, the matrix entry at global position (i_global, i_global)
is always a diagonal entry of the matrix.
Start with the Tpetra Lesson 03: Power method example, which uses a dynamic profile and global indices. Try all of the different fill techniques described in this lesson. If you like, you may also try cyclic boundary conditions instead of Dirichlet boundary conditions in the 1D Poisson discretization. (This makes the number of entries per row the same for all rows.)