Linear Solver
An algorithm that solves a system of linear equations, the core computation in most simulations.
Definition
A linear solver finds the vector x satisfying Ax = b for a given matrix A and vector b. Solving such systems is the innermost, most time-consuming step of most physics simulations, which reduce continuous equations to large linear systems.
Preconditioning, transforming the system into an equivalent one that iterative methods solve faster, is often the difference between a solver that converges in minutes and one that never does. Designing a good preconditioner is frequently the hardest part of a large simulation.
Because solving Ax equals b sits at the innermost loop of most simulations, its efficiency often decides whether a computation is feasible. Direct methods are robust but scale poorly to large systems, while iterative methods handle large sparse problems if they converge, and their convergence hinges on preconditioning. Designing a good preconditioner, one that makes the system easy to solve without costing too much itself, is frequently the hardest part of a large-scale simulation.
Two families
- Direct methods (LU, Cholesky): exact up to rounding, costly for large systems.
- Iterative methods (conjugate gradient, GMRES): refine a guess, efficient for large sparse systems.
- Preconditioning accelerates iterative convergence.
Why it matters
The efficiency of the linear solver often determines whether a simulation is feasible. Exploiting sparsity, the fact that most matrix entries are zero, is what makes solving systems with millions of unknowns possible. Solver speed depends heavily on the matrix's condition number.
Fusion connection
Every implicit time step and equilibrium calculation in a Kronos plasma simulation invokes a sparse linear solver, making its performance central to overall simulation cost.