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Mixed precision solvers for the all-at-once Runge--Kutta discretization of the heat equation
Santolo Leveque, Luca Bergamaschi, Ángeles Martínez, Erin Carson
TL;DR
The paper studies how mixed precision can accelerate the solution of large all-at-once Runge–Kutta systems for the heat equation. It uses a preconditioned iterative solver with a mixed-precision preconditioner, retaining comparable iteration counts while reducing CPU time by up to about 50%.
Problem
All-at-once Runge–Kutta discretization of the heat equation produces large sparse systems whose solution motivates faster high-performance numerical algorithms.
Method
The paper applies a parallel-capable preconditioned iterative method to the full space-time system, evaluating the preconditioner in mixed precision.
Results
The mixed-precision strategy is comparable to full double precision in iteration counts and achieves up to about 50% CPU-time speed-up.
Takeaways & Limitations
Mixed precision can accelerate the parallel solution process while preserving the preconditioner’s robustness over the reported tests.
Takeaways & Limitations
Some mixed-precision approaches lose convergence order or preconditioner robustness, so precision must be chosen carefully.
Abstract
from arXiv · showhide
We study the effect of mixed precision on the numerical integration of the heat equation discretized with a Runge--Kutta method in time. A full space-time discretization is applied, which results in a very large and sparse linear system to be solved for the numerical approximations and the Runge--Kutta stages of all time steps. The linear system is solved by applying a suitable preconditioned iterative method that can be run in parallel. In order to speed up the solution process, the preconditioner is applied using a mixed precision framework. Sequential results show the robustness of the preconditioner, even when applied in mixed precision. Finally, we present numerical evidence of the improved performance of the mixed precision strategy when applied in a parallel environment, achieving up to a 50% reduction in CPU time.
1. Introduction.
The paper motivates implicit Runge–Kutta methods and develops a mixed-precision parallel solver for the large sparse systems arising from all-at-once heat-equation discretizations.
- Implicit Runge–Kutta methods provide favorable stability properties for stiff problems and can achieve arbitrary convergence order while remaining A-stable.
- All-at-once Runge–Kutta discretizations require fast, robust solvers for the resulting large and sparse space-time linear systems.
- The adopted parallel-in-time preconditioner exploits a block-diagonal leading block, a block-lower triangular Schur complement, and an inner stage solver.
- The work applies this preconditioning strategy in mixed precision and evaluates its efficiency numerically.
2. The Heat Equation.
The paper considers the heat equation and discretizes it with an s-stage Runge–Kutta method in time and finite elements in space. It then collects the solution updates and stage equations for all time steps into one all-at-once linear system.
- Finite-element discretization produces mass and stiffness matrices M and K, while the Runge–Kutta matrix contains method coefficients and vectors bRK and cRK contain weights and nodes.
- The model is the heat equation on a spatial domain Ω⊂R^d, d=1,2,3, with forcing, Dirichlet boundary data, and an initial condition.
- An s-stage Runge–Kutta scheme partitions (0,T) into nt subintervals of length τ and computes each time-step solution from stage variables V_i,n.
- The resulting matrix system includes a right-hand-side vector f from the discretized forcing and a vector b containing initial- and boundary-condition information.
- Unlike sequential time stepping, the adopted all-at-once formulation solves the updates and stage equations simultaneously for every time step.
3. Preconditioning Strategy.
The paper uses block-upper triangular preconditioners for the all-at-once system. The exact preconditioner is theoretically ideal but impractical or sometimes undefined, motivating invertible approximations whose application uses inner GMRES and flexible GMRES.
- In exact arithmetic, the block-upper triangular preconditioner yields σ(P^-1A)={1} and a minimal polynomial of degree two, implying convergence in at most two iterations when S is invertible.
- The exact preconditioner is impractical because forming S may be too costly, while singular Θ can make the Schur complement undefined and P noninvertible.
- Practical preconditioners replace the diagonal blocks with suitable invertible approximations eΘ and eS≈Φ−Ψ1 eΘ^-1Ψ2.
- The implemented approximation eP uses a fixed number of GMRES iterations with the strategy from, rather than an SVD-based preconditioner for ARK.
- Because eP is nonlinear through its inner GMRES solver, the outer system is solved with flexible GMRES.
- Mixed precision arithmetic is framed as using different floating-point formats across computation stages to exploit hardware while retaining high accuracy.
4. A Mixed Precision Approach.
The paper develops and analyzes a mixed-precision framework for preconditioned iterative solvers applied to Runge–Kutta heat-equation systems. It combines precision-specific error guidance with parallel-in-time preconditioning and multigrid components.
- The study investigates mixed precision in iterative solvers for linear systems arising from Runge–Kutta discretizations of PDEs, identified as a first study of this setting.
- Four-precision split-preconditioned FGMRES assigns separate precisions to working arithmetic, matrix operations, and left and right preconditioners.Algorithm 1 implements this structure using left and right preconditioning within flexible GMRES.
- Backward-error bounds depend on the working precision and the precisions used for matrix and left-preconditioner operations, under suitable floating-point assumptions.The analysis also derives forward-error estimates through condition numbers of the preconditioned coefficient matrix.
- Precision selection is guided by condition numbers: ill-conditioned matrices and left preconditioners may require substantially higher precision for matrix operations and backward accuracy.The precision used for right-preconditioner operations does not influence backward error under the stated bound, subject to its specified condition.
- The heat-equation solver applies an inexact inverse of M + cK through multigrid and compares double-precision preconditioning with mixed-precision variants for Radau IIA discretizations.The preconditioner is studied within both right- and left-preconditioned FGMRES, while parallel-in-time structure motivates the overall approach.
5. Numerical Results.
Sequential tests assess double-, single-, and mixed-precision preconditioning for all-at-once Runge–Kutta heat-equation solves, while parallel tests compare full double precision with single-precision preconditioning. The strategy remains robust in sequential experiments and reduces CPU time in parallel runs, with reported speedups reaching about 50%.
- 5. Numerical Results.: The experiments solve the full space-time system with right-preconditioned FGMRES sequentially and left-preconditioned FGMRES in parallel, using fixed inner iterations and multigrid-based approximate inversions.The outer solver is restarted every 20 iterations, targets a relative residual reduction of 10^-8, and permits at most 100 iterations.
- 5.1. Sequential Results.: In sequential tests, full-double FGMRES converges in at most 10 iterations independently of discretization parameters and Runge–Kutta stage count.The same convergence behavior is not affected by applying the preconditioner in single precision.
- 5.1. Sequential Results.: Single-precision preconditioning preserves solver convergence, mildly polluting Q2 discretization errors, whereas mixed preconditioning causes more pollution but keeps errors comparable to full double precision.The mixed approach shows some refinement dependence, especially with higher-order finite elements.
- 5.2. Parallel Results.: Single-precision preconditioning increases outer FGMRES iterations by 30%–50% depending on the test case, but CPU times remain significantly lower across processor counts.The convergence profiles differ only moderately, and the speed-up is not influenced by the inner-solver iteration limit or tolerance.
- 5.2. Parallel Results.: About 50% speed-up is achieved with mixed-precision preconditioning, while the full-double approach is 1.5 times slower on average.The comparison uses ratios of CPU times for double-precision versus single-precision preconditioners.
6. Conclusion.
The mixed precision strategy retains iteration robustness comparable to full double precision and delivers parallel speedups. Future work will extend the approach to more complicated differential-algebraic equations.
- Iteration counts with the mixed precision strategy are comparable to those of the full double precision solver.
- Parallel results demonstrate speedups when the iterative solver uses mixed precision.
- Future work will apply the mixed precision preconditioner to more complicated DAEs, including incompressible fluid-flow problems.