WP3: Solvers for Linear Algebra and Multiphysics

WP3 designs novel scalable solvers for sparse linear systems and coupled multiphysics problems, optimized for exascale architectures through communication reduction, mixed precision, and data compression techniques.

Recent Highlights (2024-2025)

50× Speedup Achievement

GenEO preconditioner: up to 50× speedup on large problems (45M DoFs, 2916 cores)

Multilevel Schwarz

Robust multilevel Schwarz preconditioners with hierarchies of coarse spaces

HPDDM Framework

HPDDM framework with high-performance domain decomposition methods

Mixed Precision

Mixed precision pathways integrated into solver kernels

1. Objectives

WP3 aims to design numerical kernels for solving linear systems that are mostly agnostic of the underlying models and approximation techniques:

  • Generic solution techniques that don’t depend on function representation

  • Reduction of computational complexity, memory footprint, and data movement

  • Communication avoiding/hiding strategies with mixed arithmetic and data compression

  • Exploitation of extreme parallelism for coupled physics solvers

2. Approach

Diagram

3. Key Tasks

T3.1: Domain Decomposition Methods with Subspace-Correction

  • Enhance scalability of multilevel domain decomposition methods

  • Investigate robustness with inexact setups and preconditioner applications

  • Study mixed arithmetic, inexact local solves, low-rank approximations

  • Validation in HPDDM library and Maphys++

T3.2: Data-Sparsity, Multiple Precision and Compression

T3.2.1: Modular Mixed Precision Krylov Solvers

  • Develop subspace Krylov methods using mixed precision arithmetic

  • Create modular analyses for composable parallel implementations

  • Ensure final numerical quality of solutions

T3.2.2: Variable Accuracy Paradigm

  • Decouple data representation from arithmetic

  • Control memory footprint and communication volume

  • Ensure user-prescribed accuracy guarantees

T3.2.3: Precision Auto-Tuning Tools

  • Provide mixed precision versions of numerical codes

  • Improve precision auto-tuning performance

  • Develop methodologies for autotuning in coupled physics simulations

T3.2.4: Silent Errors in Solution Techniques

  • Study and design silent error detection and correction at scale

  • Use intrinsic properties of numerical schemes or statistical techniques

T3.3: Adaptive Solution Strategies for Multiphysics and Multiscale

T3.3.1: Saddle Point Problems

  • Investigate parallel efficiency of domain decomposition for saddle point problems

  • Address coupled multiphysics problems

T3.3.2: Exascale Resolution for Partitioned Coupling

  • Automatic tuning of performance parameters

  • Handle coupling strengths, physics paths, adaptive convergence criteria

  • Joint strategy with auto-tuning work from T3.2.3

4. Leads & Partners

Lead Institution

Inria

Co-Leaders

UNISTRA, Sorbonne Université, CEA, École Polytechnique

Duration

Months 1-60

5. Addressed Exascale Bottlenecks

WP3 targets bottlenecks B7 (Exascale Algorithms) and B9 (Resilience, robustness and accuracy):

  • Communication reduction through domain decomposition methods

  • Avoiding/hiding synchronization in parallel solvers

  • Improved computational efficiency with mixed precision and data compression

  • Ensuring correctness and verifiability with silent error detection

6. Deliverables

ID Title Due Dates

D3.1-MR

Activity reports (included in annual report D0.2-TR)

M12, M24, M36, M48, M60

D3.2-S

Software packages HPDDM, Maphys++, PROMISE

M24, M36, M48, M60

D3.3-B

Benchmarking analysis report (bottlenecks and breakthroughs)

M12, M24, M36, M48, M60

7. Collaborations

  • ExaSoft: Low-level software infrastructure and performance portability

  • WP1: Integration with nonconforming methods (T1.5) and multiphysics coupling (T1.7)

  • WP2: Collaboration on mixed precision and variable accuracy paradigm

  • WP7: Benchmarking analysis and integration