Section: Scientific Foundations

Scheduling for Parallel Sparse Direct Solvers

Participants : Guillaume Joslin, Maurice Brémond, Indranil Chowdhury, Jean-Yves L'Excellent, Bora Uçar.

The solution of sparse systems of linear equations (symmetric or unsymmetric, most often with an irregular structure) is at the heart of many scientific applications arising in various domains such as geophysics, chemistry, electromagnetism, structural optimization, and computational fluid dynamics. The importance and diversity of the fields of application are our main motivation to pursue research on sparse linear solvers. Furthermore, in order to solve hard problems that result from ever-increasing demand for accuracy in simulations, special attention must be paid to both memory usage and execution time on the most powerful parallel platforms (whose usage is necessary because of the volume of data and amount of computation induced). This is done by specific algorithmic choices and scheduling techniques. From a complementary point of view, it is also necessary to be aware of the functionality requirements from the applications and from the users, so that robust solutions can be proposed for a large range of problems.

Because of their efficiency and robustness, direct methods (based on Gaussian elimination) are methods of choice to solve these types of problems. In this context, we are particularly interested in the multifrontal method  [104] , [105] for symmetric positive definite, general symmetric or unsymmetric problems, with numerical pivoting in order to ensure numerical accuracy. The existence of numerical pivoting induces dynamic updates in the data structures where the updates are not predictable with a static or symbolic analysis approach.

The multifrontal method is based on an elimination tree  [110] which results (i) from the graph structure corresponding to the nonzero pattern of the problem to be solved, and (ii) from the order in which variables are eliminated. This tree provides the dependency graph of the computations and is exploited to define tasks that may be executed in parallel. In this method, each node of the tree corresponds to a task (itself can be potentially parallel) that consists in the partial factorization of a dense matrix. This approach allows for a good locality and hence efficient use of cache memories.

We are especially interested in approaches that are intrinsically dynamic and asynchronous [1] , [100] , as these approaches can encapsulate numerical pivoting and can be adopted to various computer architectures. In addition to their numerical robustness, the algorithms are based on a dynamic and distributed management of the computational tasks, not so far from today's peer-to-peer approaches: each process is responsible for providing work to some other processes and at the same time it acts as a worker for others. These algorithms are very interesting from the point of view of parallelism and in particular for the study of mapping and scheduling strategies for the following reasons:

• the associated task graphs are very irregular and can vary dynamically,

• they are currently used inside industrial applications, and

• the evolution of high performance platforms, to the more heterogeneous and less predictable ones, requires that applications adapt themselves, using a mixture of dynamic and static approaches, as our approach allows.

Our research in this field is strongly linked to the software package Mumps (see Section  5.2 ) which is our main platform to experiment and validate new ideas and pursue new research directions. We are facing new challenges for very large problems (tens to hundreds of millions of equations) that occur nowadays in various application fields: in that case, either parallel out-of-core approaches are required, or direct solvers should be combined with iterative schemes, leading to hybrid direct-iterative methods.