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International audience; Numerical simulations continue to enable fast and enormous progress in science and engineering. Writing efficient numerical codes is a difficult challenge that encompasses a variety of tasks from designing the right algorithms to exploiting the full potential of a platform's architecture. Domain-specific languages (DSLs) can ease these tasks by offering the right abstractions for expressing numerical problems. With the aid of domain knowledge, efficient code can then be generated automatically from abstract expressions. In this work, we present the CFDlang DSL for expressing tensor operations that constitute the performance-critical code sections in a class of real numerical applications from fluid dynamics. We demonstrate that CFDlang can be used to generate code automatically that performs as well, if not better, than carefully hand-optimized code.

High-order methods gain increased attention in computational fluid dynamics. However, due to the time step restrictions arising from the semi-implicit time stepping for the incompressible case, the potential advantage of these methods depends critically on efficient elliptic solvers. Due to the operation counts of operators scaling with the polynomial degree p times the number of degrees of freedom n DOF , the runtime of the best available multigrid solvers scales with O ( p ⋅ n DOF ) . This scaling with p significantly lowers the applicability of high-order methods to high orders. While the operators for residual evaluation can be linearized when using static condensation, Schwarz -type smoothers require their inverses on fixed subdomains. No explicit inverse is known in the condensed case and matrix-matrix multiplications scale with p ⋅ n DOF . This paper derives a matrix-free explicit inverse for the static condensed operator in a cuboidal, Cartesian subdomain. It scales with p 3 per element, i.e. n DOF globally, and allows for a linearly scaling additive Schwarz smoother, yielding a p-multigrid cycle with an operation count of O ( n DOF ) . The resulting solver uses fewer than four iterations for all polynomial degrees to reduce the residual by ten orders and has a runtime scaling linearly with n DOF for polynomial degrees at least up to 48. Furthermore the runtime is less than one microsecond per unknown over wide parameter ranges when using one core of a CPU, leading to time-stepping for the incompressible Navier-Stokes equations using as much time for explicitly treated convection terms as for the elliptic solvers.

International audience; Many numerical algorithms are naturally expressed as operations on tensors (i.e. multi-dimensional arrays). Hence, tensor expressions occur in a wide range of application domains , e.g. quantum chemistry and physics; big data analysis and machine learning; and computational fluid dynamics. Each domain, typically, has developed its own strategies for efficiently generating optimized code, supported by tools such as domain-specific languages, compilers, and libraries. However, strategies and tools are rarely portable between domains, and generic solutions typically act as " black boxes " that offer little control over code generation and optimization. As a consequence, there are application domains without adequate support for easily generating optimized code, e.g. computational fluid dynamics. In this paper we propose a generic and easily extensible intermediate language for expressing tensor computations and code transformations in a modular and generative fashion. Beyond being an intermediate language, our solution also offers meta-programming capabilities for experts in code optimization. While applications from the domain of computational fluid dynamics serve to illustrate our proposed solution, we believe that our general approach can help unify research in tensor optimizations and make solutions more portable between domains.

High-order methods gain more and more attention in computational fluid dynamics. However, the potential advantage of these methods depends critically on the availability of efficient elliptic solvers. With spectral-element methods, static condensation is a common approach to reduce the number of degree of freedoms and to improve the condition of the algebraic equations. The resulting system is block-structured and the face-based operator well suited for matrix-matrix multiplications. However, a straight-forward implementation scales super-linearly with the number of unknowns and, therefore, prohibits the application to high polynomial degrees. This paper proposes a novel factorization technique, which yields a linear operation count of just 13N multiplications, where N is the total number of unknowns. In comparison to previous work it saves a factor larger than 3 and clearly outpaces unfactored variants for all polynomial degrees. Using the new technique as a building block for a preconditioned conjugate gradient method resulted in a runtime scaling linearly with N for polynomial degrees $2 \leq p \leq 32$ . Moreover the solver proved remarkably robust for aspect ratios up to 128.