Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.30357

A unified variational framework for augmented Lagrangian preconditioning of saddle point systems: applications to mixed-dimensional and fictitious domain methods

Michele Benzi·Marco Feder·Luca Heltai·Federica Mugnaioni
View original source ↗Read source document ↗Export metadata ↓

Abstract

We present a general framework to derive augmented Lagrangian preconditioners for saddle point systems arising from mixed finite element discretizations. Our approach is based on augmenting the variational formulation at the continuous level prior to discretization, providing a unified strategy for designing efficient preconditioners that are robust under mesh refinement for a broad class of problems leading to symmetric saddle point systems. The robustness of the proposed preconditioners is established through a detailed spectral analysis. This framework is applied to different model problems, focusing on mixed-dimensional and fictitious domain formulations involving Lagrange multipliers, with some configurations featuring large coefficient jumps. Extensive numerical experiments on different test cases and geometries are presented, validating the effectiveness and robustness of our preconditioners when used in conjunction with flexible GMRES.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michele Benzi, Marco Feder, Luca Heltai, Federica Mugnaioni. 2026-10-21. A unified variational framework for augmented Lagrangian preconditioning of saddle point systems: applications to mixed-dimensional and fictitious domain methods. https://arxiv.org/abs/2610.30357

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Spectral deferred corrections parallelized across the method for differential-algebraic equations

In this work, the performance of different spectral deferred corrections (SDC) methods applied to initial value problems for differential-algebraic equations (DAEs) of index one is analyzed. The SDC method solves a series of correction equations, and after each iteration, the numerical solution is corrected by adding the obtained approximation of the error. The formulation of the SDC method allows for a parallelization across the method to obtain small-scale parallelism, for which a number of processes equal to the number of collocation nodes can be used. Recently, an analytical approach to finding optimal diagonal coefficients for parallel SDC was proposed. So far, no analysis with the new coefficients for parallel SDC applied to DAEs was done. We demonstrate that parallel SDC methods solve DAE problems with high accuracy faster than the associated sequential SDC schemes, obtaining speedup in a small-scale parallel regime.

math.NA↗
arXiv2026

Deep Truncated FBSDE Method for High-Dimensional Nonlinear PDEs and Fully Coupled FBSDEs

In this paper, we introduce a deep truncated forward-backward stochastic differential equation (FBSDE) method for high-dimensional partial differential equations (PDEs) and fully coupled FBSDEs. The proposed method combines gradient-truncated iterative decoupling with fictitious-play averaging to separate the forward and backward processes in a coupled framework. This preserves the coupled dynamics while reducing the unstable feedback induced by parameter-dependent forward paths during optimization. Furthermore, we incorporate a pathwise consistency term to create explicit local gradient shortcuts, thereby providing a structural mechanism that may mitigate gradient vanishing. We also derive a residual-based error estimate and establish conditional convergence of the fully discrete numerical approximations under suitable conditions, in which the pathwise consistency loss is not required. Our approach is particularly effective for convection-dominated equations, where the coupled formulation provides a stable representation of nonlinear transport without introducing singular terms into the BSDE. Numerical experiments demonstrate improved accuracy and stability in both low- and high-dimensional problems and robust performance for strongly coupled problems.

math.NA↗
arXiv2026

Sum of Squares and Non Sum of Squares Regimes for the Toeplitz Bottcher Wenzel Form

Toeplitz matrices are among the simplest structured models of translation invariance, arising naturally in convolution, stationary covariance models, and discretizations of translation-invariant operators. For a pair of real matrices, the Böttcher--Wenzel form (BW form) is a nonnegative quartic measuring the gap in the corresponding commutator inequality. László conjectured that, when both matrices are Toeplitz, this quartic is a sum of squares for every matrix order. We show that the conjecture is false. More precisely, we identify an explicit family of corner-supported Toeplitz pairs, retaining only the outermost diagonals on the two sides, for which the associated BW form is not a sum of squares at all sufficiently large orders. The proof reduces the complete family of Gram representations to a finite complex test: its Gram-invariant part develops a strictly negative limiting direction, while the remaining Gram-dependent contribution vanishes. We also give an explicit sufficient threshold for the matrix order. The negative result is complemented by two positive regimes. If either Toeplitz factor is symmetric or skew-symmetric, then the corresponding BW form is SoS for every order, with the other factor allowed to be an arbitrary real Toeplitz matrix. For unrestricted Toeplitz pairs, we further prove SoS representability for every $2\le N\le50$, with formal Lean~4 verification through order 20. The large-order non-SoS theorem, including the explicit sufficient threshold, has also been formally verified in Lean 4.

math.NA↗