Compiler Support for Sparse Tensor Computations in MLIR

Sparse tensors arise in problems in science, engineering, machine learning, and data analytics. Programs that operate on such tensors can exploit sparsity to reduce storage requirements and computational time. Developing and maintaining sparse software by hand, however, is a complex and error-prone task. Therefore, we propose treating sparsity as a property of tensors, not a tedious implementation task, and letting a sparse compiler generate sparse code automatically from a sparsity-agnostic definition of the computation. This paper discusses integrating this idea into MLIR

Combinatorial problems in solving linear systems

42 pages, available as LIP research report RR-2009-15Numerical linear algebra and combinatorial optimization are vast subjects; as is their interaction. In virtually all cases there should be a notion of sparsity for a combinatorial problem to arise. Sparse matrices therefore form the basis of the interaction of these two seemingly disparate subjects. As the core of many of today's numerical linear algebra computations consists of the solution of sparse linear system by direct or iterative methods, we survey some combinatorial problems, ideas, and algorithms relating to these computations. On the direct methods side, we discuss issues such as matrix ordering; bipartite matching and matrix scaling for better pivoting; task assignment and scheduling for parallel multifrontal solvers. On the iterative method side, we discuss preconditioning techniques including incomplete factorization preconditioners, support graph preconditioners, and algebraic multigrid. In a separate part, we discuss the block triangular form of sparse matrices

THREE-DIMENSIONAL COUPLED FEM MODELLING AND PROGRAMMING OF PARTIALLY SATURATED POROUS MEDIA

The purpose of the work presented in this thesis is to investigate the fully and partially saturated behaviour of soils, behaviour that can be extended also to geomaterials like concrete. The physical - mathematical approach proposed within this manuscript is a coupled thermo-hydro-mechanical model, suitable for consolidation / subsidence analyses of unsaturated soils. This coupled formulation, can therefore be qualified as u – pw – pg – (T), by the introduction of basic state variables involved in the processes, that here are: the displacements field u, the liquid (water) pressure field pw, the gas (dry air and water vapour) pressure field pg, and eventually the temperature T that is involved on the modelling of non – isothermal process. Due to the coexistence of two different fluid phases, liquid and gaseous one, this model can be regarded as a multiphase approach to a deforming porous medium as proposed by Lewis and Schrefler in the framework of the hybrid mixture theory for porous media firstly presented by Hassanizadeh and Gray and Zienkiewicz et al. The evolution at macroscopic scale of the state variables above mentioned, in particular of pressures of both liquid and gas, is basically influenced by the microstructure of the material that characterizes the behaviour of a soil with relation on capillary effects and deformability. The physical approach proposed here is based on averaging techniques applied to the physical quantities that can be estimated in a representative elementary volume (REV) . With the addition of water retention functions that provide a description of the relation that exists among capillary pressure and the degree of water saturation, a complete set of fluid balance equations and mechanical and thermodynamic equilibrium equations can be obtained for the medium in a macroscopic scale. A coupled (thermo)-hydro-mechanical formulation u – p – (T) that deals with a fully saturated porous medium has been implemented with success in the past in the F.E. two-dimensional program PLASCON and its further extensions to three- dimensionality with PLASCON3D. The present work focused on the extension and upgrading of the relative simple single phase theory along with its numerical implementations, towards a more realistic multiphase description of the porous material, where voids may be filled up with both liquid and gas that interacts each other by mean of the concept of capillary pressure. An improved code PLASCON3D_PS based on the fully coupled u – pw – pg – (T) formulation and developed from previous versions has been realized. Due to the lack in literature of three-dimensional coupled numerical and experimental tests, some numerical results of benchmark tests and real case problems, that derive from two-dimensional domains, will be presented