ATSolver: Agentic Tuning of Sparse Linear Solvers with Numerical Feedback

Sparse linear solver configuration is a sequential decision problem under numerical, resource, and workload constraints. Each attempted configuration reveals execution evidence that can inform subsequent choices, while setup cost, solve cost, accuracy requirements, and search budgets jointly shape the tuning objective. We introduce ATSolver (Agentic Tuning of Solvers), a novel verifier-grounded framework for agentic sparse-solver tuning that iteratively proposes, executes, and revises configurations over a typed portfolio of classical sparse solvers and preconditioners. A pretrained language model with fixed parameters serves as the proposal policy, while trusted numerical software controls legality, execution, solution verification, resource accounting, and numerical-state reuse. Under matched six proposal budgets in an iterative-solver study, ATSolver accepts 75/105 runs versus 64/105 for independent Best-of-6 and selects configurations with 30.1% lower independently replayed solve time on matrices solved by both policies in all repetitions. Across repeated RHS workloads, ATSolver shifts toward greater setup investment as batch size grows, adapting configuration choices to changing setup–solve trade-offs. Feedback memory studies show that trial history avoids observed repeated proposals and that adding retrieved experience to knowledge improves configuration quality on held-out parameters within two PDE families. These results demonstrate effective agentic tuning under explicit accuracy, runtime, and memory constraints without updating the underlying language model.