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Inexact versions of several block-splitting preconditioners for indefinite least squares problems

This paper proposes and analyzes inexact block-splitting preconditioners for indefinite least squares problems, establishing their convergence conditions, proving that the preconditioned eigenvalues cluster around unity to ensure rapid GMRES convergence, and validating their effectiveness through numerical experiments.

Original authors: Mohaddese Kaveh Shaldehi, Davod Khojasteh Salkuyeh

Published 2026-03-03

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Mohaddese Kaveh Shaldehi, Davod Khojasteh Salkuyeh

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to find the perfect route for a delivery truck through a massive, chaotic city. You have a map (the data) and a destination (the answer), but the city has some weird, confusing rules: some streets are one-way, some are dead ends, and the traffic laws change depending on which neighborhood you are in. In math terms, this is an Indefinite Least Squares (ILS) problem. It's a way of finding the "best fit" answer when the rules of the game aren't perfectly smooth or predictable.

For a long time, mathematicians have tried to solve these problems using a method called GMRES. Think of GMRES as a very smart, but slightly clumsy, hiker trying to find the bottom of a valley. The hiker takes steps, checks the slope, and tries to get closer to the bottom. The problem is, if the valley is full of confusing bumps and cliffs (which happens with these "indefinite" problems), the hiker might get stuck, walk in circles, or take thousands of steps just to get a few feet closer.

The Problem: The "Inner" Bottleneck

To help the hiker, mathematicians invented preconditioners. You can think of a preconditioner as a pair of magical glasses. When you put them on, the confusing, bumpy valley suddenly looks like a smooth, gentle slope. The hiker can now see the path clearly and run straight to the bottom.

However, there was a catch. To make these glasses work, you had to solve a tiny, difficult math puzzle every single time the hiker took a step.

The Old Way: The puzzle was so hard that solving it exactly took forever. It was like asking the hiker to stop every 10 feet to build a bridge before they could take the next step.
The Result: The "outer" journey (finding the solution) was fast, but the "inner" journey (solving the puzzle) was so slow and error-prone that the whole process crashed or took too long.

The Solution: "Inexact" Glasses

This paper introduces a new idea: Inexact Block-Splitting Preconditioners (IBS).

Instead of demanding that the hiker build a perfect bridge every time, the authors say: "Hey, just build a sturdy plank. It doesn't have to be perfect; it just needs to be good enough to get you across."

Here is the breakdown of their innovation:

The "Approximation" Trick: The difficult puzzle they had to solve involved a matrix (a giant grid of numbers) called $P$ . This matrix was often "sick" or "ill-conditioned" (like a wobbly table). Instead of trying to fix the wobbly table perfectly, they added a little bit of "stability" to it (mathematically, adding a constant $\alpha$ to the diagonal).
- Analogy: Imagine trying to balance a stack of Jenga blocks. The old method tried to balance them perfectly on a shaky floor. The new method puts a thick, stable rubber mat under the blocks. The stack isn't "perfectly" balanced, but it's stable enough to work with, and it's much faster to set up.
The "Inexact" Speed: Because the new puzzle is easier (the "rubber mat" makes it well-behaved), the computer can solve it very quickly and roughly. It doesn't need to be perfect; it just needs to be good enough to guide the hiker.
- The Result: The hiker (the GMRES algorithm) can now take steps much faster because they aren't waiting for perfect bridges. They just get a quick, "good enough" direction and keep moving.

Why It Works So Well

The authors didn't just guess this would work; they did the math to prove it.

The Circle of Safety: They proved that when you use these new "inexact glasses," all the confusing bumps in the valley get squashed into a neat little circle. This means the hiker never gets lost.
The Speed Limit: They calculated a theoretical "speed limit" for the hiker. They proved that no matter how big the city is, the hiker will find the solution in a predictable number of steps, rather than wandering forever.

The Real-World Test

The authors tested this on some very difficult, real-world scenarios:

Aircraft Stability: Checking if a plane will stay in the air.
Oil Reservoirs: Figuring out how oil moves underground.
Hilbert Matrices: These are famous in math for being incredibly difficult and "broken" (ill-conditioned).

The Results:

Old Methods (BS2, BUT): On the hardest problems, they either took thousands of steps or gave up entirely. When they did find an answer, it was often the wrong answer (like finding a valley that doesn't exist).
New IBS Methods: They solved the problems in a fraction of the time (sometimes 10x or 100x faster). Even on the "broken" Hilbert matrices, they worked perfectly.

The Bottom Line

This paper is about smart shortcuts. Instead of trying to solve a math problem perfectly at every single step (which is slow and prone to errors), the authors found a way to solve it "good enough" at every step. By doing this, they made the overall journey to the solution incredibly fast and reliable, even for the most chaotic and difficult problems.

It's like realizing that to get to the store, you don't need a perfectly paved road; you just need a path that doesn't have any giant holes. And sometimes, a slightly rough path gets you there much faster than waiting for the perfect one.

1. Problem Statement

The paper addresses the Indefinite Least Squares (ILS) problem, formulated as:
$\min_{x \in \mathbb{R}^n} (b - Ax)^T H_{p,q} (b - Ax)$
where $A \in \mathbb{R}^{m \times n}$ is a large, sparse matrix ( $m \ge n$ ), $b \in \mathbb{R}^m$ is a vector, and $H_{p,q}$ is a signature matrix with $p$ positive and $q$ negative eigenvalues ( $p+q=m$ ).

Challenge: Unlike standard least squares, the ILS objective function is indefinite when $p, q > 0$ . The normal equations $A^T H_{p,q} A x = A^T H_{p,q} b$ require $A^T H_{p,q} A$ to be Symmetric Positive Definite (SPD) for a unique solution, but the matrix $P = A_1^T A_1$ (derived from the block structure of $A$ ) is often ill-conditioned.
Current Limitations: Existing block-splitting preconditioners (e.g., $M_{BS1}$ through $M_{BS4}$ ) require solving inner linear systems involving $P$ . If $P$ is ill-conditioned, exact solves are expensive, and inexact solves accumulate errors, causing the outer Krylov subspace method (GMRES) to fail or converge slowly.

2. Methodology

The authors propose Inexact Block-Splitting (IBS) preconditioners to overcome the conditioning issues of existing methods.

A. Reformulation

The ILS problem is reformulated into a nonsingular block 3x3 linear system:
$\mathcal{A} \mathbf{x} = \begin{pmatrix} I & A_1 & 0 \\ 0 & P & A_2^T \\ 0 & A_2 & I \end{pmatrix} \begin{pmatrix} \delta_1 \\ x \\ \delta_2 \end{pmatrix} = \begin{pmatrix} b_1 \\ A_1^T b_1 \\ b_2 \end{pmatrix}$
where $A$ is partitioned into $A_1$ (full column rank) and $A_2$ .

B. Inexact Preconditioner Construction

The core innovation is replacing the potentially ill-conditioned matrix $P$ in the diagonal blocks of the preconditioner with a well-conditioned SPD approximation $\hat{P}$ .

Approximation: $\hat{P} = \alpha I + P$ , where $\alpha = 1/\|A_1\|_2$ . This shift ensures $\hat{P}$ is well-conditioned and $\hat{P} - P = \alpha I$ is SPD.
Four Variants: The authors define four inexact preconditioners ( $M_1, M_2, M_3, M_4$ ) corresponding to four different block-splitting strategies ( $M_{BS1}$ to $M_{BS4}$ ) found in literature, but with $P$ replaced by $\hat{P}$ .
Implementation: Solving systems with $M_i$ requires solving linear systems with $\hat{P}$ . Since $\hat{P}$ is SPD, these inner systems are solved inexactly using a few iterations of the Conjugate Gradient (CG) method. The outer solver is the Flexible GMRES (FGMRES) to accommodate the varying inner solver accuracy.

3. Key Contributions & Theoretical Analysis

A. Convergence of Stationary Iterative Methods

The authors establish sufficient conditions for the convergence of the stationary iterative methods induced by the IBS preconditioners.

Theorem 1: Proves that if specific combinations of $\hat{P}$ , $P$ , and $A_2^T A_2$ are SPD (e.g., $2\hat{P} - P - A_2^T A_2 \succ 0$ ), the spectral radius of the iteration matrices $G_i$ is less than 1.
Corollary 1: Shows that if $\hat{P} - P$ is SPD (which is guaranteed by the choice $\hat{P} = \alpha I + P$ ), all IBS methods converge. Furthermore, for IBS2 and IBS4, all eigenvalues of the iteration matrix lie in the real interval $(0, 1)$ .

B. Spectral Clustering and GMRES Convergence

A critical theoretical result is the spectral distribution of the preconditioned matrices $M_i^{-1}\mathcal{A}$ .

Spectral Clustering: Under the convergence conditions, all eigenvalues of the preconditioned matrices are contained within a circle centered at (1, 0) with radius 1.
Eigenstructure Analysis: Theorems 2–5 detail the eigenpairs, showing that the eigenvalue 1 has a high geometric multiplicity (related to $p, q$ , and the rank of $A_2$ ), while non-unit eigenvalues are clustered near 1.
Iteration Bound (Theorem 6): The authors derive an upper bound on the degree of the minimal polynomial for the preconditioned matrices as $n + q + 1$ .
- Implication: This implies that the GMRES method will terminate in at most $n + q + 1$ iterations (in exact arithmetic), providing a strong theoretical guarantee for fast convergence.

4. Numerical Results

The authors tested the IBS preconditioners against recent competitors (BS2 and BUT) on three sets of problems:

TOLS Matrices (Flight Stability):
- IBS methods required significantly fewer iterations (e.g., ~31 vs. >300 for competitors) and less CPU time.
- Competitors failed to converge for larger matrices (TOLS2000, TOLS4000).
SHERMAN Matrices (Oil Reservoir Simulation):
- IBS methods converged rapidly with low relative errors.
- Critical Finding: While competitors (BS2, BUT) achieved low residual norms (RES), they failed to approximate the true solution (Relative Error $\approx 1$ ). The IBS methods provided both low residuals and accurate solutions, demonstrating robustness.
Hilbert Matrices (Severely Ill-Conditioned):
- For extremely ill-conditioned problems ( $n=10,000$ ), competitors failed to converge within the iteration limit.
- All IBS variants converged efficiently, with IBS2 and IBS4 performing best in terms of iteration count and CPU time.

5. Significance

Robustness: The proposed inexact approach effectively handles ill-conditioned matrices where exact block-splitting methods fail due to the difficulty of inverting $P$ .
Efficiency: By using a simple shift ( $\alpha I + P$ ) and inexact inner solves, the method drastically reduces computational cost while maintaining high accuracy.
Theoretical Guarantee: The derivation of the eigenvalue clustering and the minimal polynomial bound provides a rigorous theoretical basis for the observed speed of GMRES convergence.
Practical Application: The method is particularly effective for large-scale, sparse, and ill-conditioned ILS problems arising in engineering and optimization (e.g., robust smoothing, total least squares), outperforming state-of-the-art preconditioners in both speed and reliability.