ILU

ILU is an acronym that can refer to several concepts in mathematics, computing, and informal communication. The most widely used meaning in technical contexts is incomplete LU factorization, a family of matrix factorizations that approximate LU factorization while preserving a given sparsity pattern. ILU factors, L and U, are intended as a drop-in preconditioner for iterative solvers of sparse linear systems. Variants include ILU(0), which preserves only the original sparsity pattern, ILU(k), which allows k levels of fill-in, and ILUT, which combines fill-in control with a threshold-based drop rule. ILU preconditioners are employed to accelerate methods such as GMRES, CGS, and BiCGStab.

In practice, ILU preconditioning helps reduce iteration counts for large, sparse, well-conditioned problems arising from discretizations of partial differential equations and other engineering simulations. Performance depends on the matrix structure, the chosen variant, and tuning of parameters like drop tolerances and fill-in levels. Potential drawbacks include instability or breakdowns when the factorization fails to exist without pivoting, sensitivity to scaling, and memory growth if fill-in is too generous.

Beyond numerical linear algebra, ILU may also appear in informal online communication as a shorthand for I love you, used in casual messages or posts. However, this usage is less formal and context-dependent.