Parallel Inexact Levenberg-Marquardt Method for Nearly-Separable Nonlinear Least Squares

TL;DR

This work addresses large-scale nonlinear least squares with nearly-separable structure, common in localization tasks, by introducing Parallel Inexact Levenberg–Marquardt (PILM). PILM exploits block separability to solve the LM linear system via inner fixed-point iterations, distributing block solves across a master–worker parallel framework and communicating sparsely across blocks. The authors prove global convergence to stationary points under standard LM assumptions with a nonmonotone line search and establish local convergence with rates (linear, superlinear, or quadratic) under stronger near-separability and regularity conditions. Numerical experiments on million-scale cadastral map refinement problems demonstrate substantial speedups over centralized LM and robustness to partition choices, with an open-source Python/MPI implementation available. Overall, PILM provides a scalable, provably convergent approach for distributed NLS in nearly-separable settings, enabling efficient localization computations at massive scales.

Abstract

Motivated by localization problems such as cadastral maps refinements, we consider a generic Nonlinear Least Squares (NLS) problem of minimizing an aggregate squared fit across all nonlinear equations (measurements) with respect to the set of unknowns, e.g., coordinates of the unknown points' locations. In a number of scenarios, NLS problems exhibit a nearly-separable structure: the set of measurements can be partitioned into disjoint groups (blocks), such that the unknowns that correspond to different blocks are only loosely coupled. We propose an efficient parallel method, termed Parallel Inexact Levenberg Marquardt (PILM), to solve such generic large scale NLS problems. PILM builds upon the classical Levenberg-Marquard (LM) method, with a main novelty in that the nearly-block separable structure is leveraged in order to obtain a scalable parallel method. Therein, the problem-wide system of linear equations that needs to be solved at every LM iteration is tackled iteratively. At each (inner) iteration, the block-wise systems of linear equations are solved in parallel, while the problem-wide system is then handled via sparse, inexpensive inter-block communication. We establish strong convergence guarantees of PILM that are analogous to those of the classical LM; provide PILM implementation in a master-worker parallel compute environment; and demonstrate its efficiency on huge scale cadastral map refinement problems.

Figures (9)

• Figure 1: Considered measurements.
• Figure 2: Sparsity plot of the coefficient matrix of the system for one of the test problems.
• Figure 3: Percentage of residuals within 1, 2 and 3 standard deviations. Values of the percentages at each iteration
• Figure 4: Execution time for different number of processors
• Figure 5: Distribution plot of the coordinate error at the initial guess (left) and at termination (right).

Theorems & Definitions (22)

• Remark 2.1
• Remark 2.2
• Lemma 3.1
• proof
• Theorem 3.1
• proof
• Lemma 3.2
• Lemma 3.3
• proof
• Lemma 3.4