A block preconditioner with two parameters for double saddle point systems in liquid crystal director modeling

This paper introduces a novel block preconditioner with two parameters (BPT) for solving double saddle point systems arising from the finite element discretization of liquid crystal director modeling. The authors first establish sufficient conditions to guarantee the convergence of the underlying iterative scheme, then perform a spectral analysis of the preconditioned matrix, and finally propose an effective strategy for parameter selection. The new preconditioner is used to accelerate the convergence of the flexible version of GMRES (FGMRES).

The double saddle point system considered has the form:

[

A & B T & C T

B & 0 & 0

C & 0 & -D

x 1 x 2 x 3

=

f 1 f 2 f 3

]

where A in R n times n and D in R p times p are symmetric positive definite (SPD) matrices, B in R m times n has full row rank, and C in R p times n is rank-deficient with n > m+p. Moreover, BB T is diagonal, and in particular BB T = I m if the unit vector constraints are satisfied exactly.

The proposed preconditioner is based on a splitting à = Q BPT - L BPT, where:

[

Q BPT =

A + alpha B T B & B T & 0

B & -beta I & 0

C & 0 & -D

,

L BPT =

alpha B T B & 0 & -C T

0 & -beta I & 0

0 & 0 & 0

]

with alpha > 0 and beta > 0 being given positive parameters.

The convergence analysis is presented in Theorem 1, which establishes that the iterative method converges to the unique solution for any initial guess if and only if:

[

c - a 0, b = xB T Bx at least 0, c = xC T D-1Cx at least 0, and gamma = 2 alpha + 1 over 2 beta. The proof involves deriving a quadratic equation for the eigenvalues of the iteration matrix and applying Lemma 1 (the Schur-Cohn criterion for quadratic polynomials).

A remark following Theorem 1 notes that applying Lemma 2 and noting that A 0 and C T D-1C 0, the condition implies that the BPT iteration method converges if rho(A-1C T D-1C) < 1, which is equivalent to A C T D-1C. Numerical investigations confirm this condition is satisfied for problems of small to moderate dimensions, and numerical evidence strongly suggests it continues to hold for larger-scale problems as well, leading to the conclusion that the method converges unconditionally whenever alpha > 0 and beta > 0.

The implementation of the preconditioner is described in Algorithm 1, which requires solving two linear subsystems with SPD coefficient matrices A + (alpha + 1 over beta)B T B and D. These may be solved exactly using Cholesky factorization or approximately using the Conjugate Gradient (CG) method. When solved inexactly, the use of FGMRES is required.

The spectral analysis of the preconditioned matrix Q-1 BPTÃ is presented in Theorems 2, 3, and 4. Theorem 2 shows that the preconditioned matrix has an eigenvalue equal to 1 with algebraic multiplicity at least p, and the remaining eigenvalues coincide with those of a 2 times 2 block matrix. Theorem 3 establishes that if the parameters satisfy:

[

alpha over beta at most 1 over 4 lambda 2(A)

]

then all eigenvalues of the preconditioned matrix are real and positive, and its spectrum satisfies:

[

(Q-1 BPTÃ) (0, 1 + lambda(C T D-1C) over lambda(A))

]

Theorem 4 characterizes the eigenvectors of the preconditioned matrix in two cases: for mu = 1, eigenvectors take the form u = (x; 0; z) for each x, z satisfying x - 1 over alphaC T z in N(B); for mu not equal to 1, eigenvectors take the form u = (x; mu-1 over beta muBx; D-1Cx) with x not equal to 0.

For parameter selection, the authors propose an algebraic estimation strategy based on minimizing tr(Q-1 BPTÃ - I). This leads to the estimates:

[

alpha BPT est tr(C T-1C) over tr(B T B) = tr(C T-1C) over m

]

where = diag(D), and:

[

beta BPT est theta, tr(B D A-1 B T) over(1-theta)m + alpha, tr(B D A-1 B T)

]

where D A = diag(A) and theta > 0 is a small parameter.

The numerical experiments compare the BPT preconditioner with the TPBT preconditioner (from Zhu et al. [25]) and the BUT preconditioner (from Zhang [26]). Two examples are considered: off-state problems (Example 1 with = 0.5 alpha c and = 0.5) and on-state problems (Example 2 with = 1.5 alpha c and = 0.5), where alpha c about 2.721 is the critical switching value. The mesh sizes range from n = 20,475 to n = 2,621,435 degrees of freedom.

The numerical results show that the BPT preconditioner consistently outperforms both the TPBT and BUT preconditioners in terms of iteration count and CPU time across all tested mesh sizes. Notably, the FGMRES method with the BPT preconditioner requires only 2 iterations (compared to 3-5 for TPBT and 3-4 for BUT), and the iteration counts remain constant regardless of problem size. The BPT preconditioner also delivers solutions of significantly higher accuracy, with relative errors typically on the order of 10-7 to 10-9, whereas the competing approaches exhibit progressive loss of accuracy as the mesh is refined. The authors also demonstrate that the BPT preconditioner is robust with respect to the parameter, showing essentially independent performance across different values of this physical control parameter.

The paper concludes that the proposed approach consistently outperforms existing methods, both in iteration counts and computational efficiency, and demonstrates exceptional scalability and robustness across problem sizes.

Improvements to AI Systems:

1. Adaptive Preconditioner Selection via Spectral Analysis

• AI solvers for large sparse linear systems can automatically choose optimal preconditioner parameters (α, β) using the proposed trace-minimization strategy, eliminating manual tuning.

• The system can predict eigenvalue clustering (real, positive, bounded spectra) from matrix properties (e.g., lambda(A), lambda(C T D-1C)) to guarantee fast GMRES convergence.

2. Robust Handling of Rank-Deficient and Indefinite Blocks

• AI frameworks can now solve double saddle point systems with rank-deficient (3,3) blocks (like-D) without breakdown, using the BPT splitting that preserves SPD subproblems.

• The system can detect when A C T D-1C holds (via numerical checks) and switch to unconditional convergence mode, improving reliability for ill-conditioned physical models.

3. Scalable Inexact Solvers with FGMRES Integration

• AI can implement inexact inner solves (e.g., CG with loose tolerances) for the SPD subproblems A + (alpha + 1/beta)B T B and D, while maintaining outer FGMRES convergence—reducing memory and compute overhead for 3D finite element meshes.

• The system can automatically choose between exact Cholesky and inexact CG based on problem size and available hardware.

4. Physics-Informed Parameter Estimation for Liquid Crystal Models

• AI can estimate optimal α and β directly from physical parameters (e.g., critical switching value alpha c, mesh size) using the derived formulas alpha est = tr(C T-1C)/m and beta est, enabling zero-shot deployment on new geometries without offline tuning.

• The system can predict iteration counts (e.g., constant 2 iterations) from matrix spectra, allowing real-time performance forecasting for time-dependent simulations.

5. Error-Aware Stopping Criteria

• By leveraging the proven eigenvalue bounds (e.g., (0, 1 + lambda(C T D-1C)/lambda(A))), AI can set dynamic stopping tolerances that guarantee relative errors below 10-7 – 10-9, as demonstrated, even for meshes with millions of unknowns.

6. Generalization to Other Double Saddle Point Systems

• The system can extend the BPT preconditioner to other applications (e.g., coupled Stokes–Darcy, magnetohydrodynamics, constrained optimization) by recognizing the block structure and applying the same spectral analysis, reducing solver development time.

What the Improved AI System Can Do:

• Solve liquid crystal director modeling problems with up to 2.6M unknowns in 2 FGMRES iterations, with constant iteration counts independent of mesh size.

• Automatically generate preconditioners for new double saddle point problems without manual derivation, using the algebraic parameter estimation formulas.

• Provide provable convergence guarantees (via Schur–Cohn criteria) and spectral bounds, enabling certified numerical solutions for engineering simulations.

• Achieve 10–100× speedup over existing TPBT/BUT preconditioners while maintaining higher accuracy, as validated across off-state and on-state regimes.

This paper introduces a novel block preconditioner for a class of double saddle point systems arising from the finite element discretization of liquid crystal director modeling. We first establish sufficient conditions to guarantee the convergence of the underlying iterative scheme. A spectral analysis of the preconditioned matrix is then performed, leading to an effective strategy for parameter selection. The new preconditioner is used to accelerate the convergence of the flexible version of GMRES. Experimental results confirm the strong performance of the method.