Sine Transform Preconditioner

• Sine Transform-Based Preconditioner is an operator constructed using discrete sine transforms to diagonalize structured matrices, improving spectral properties.
• The preconditioner optimizes convergence by ensuring eigenvalue clustering through spectral equivalence in Toeplitz and τ-matrix systems.
• Its efficient O(n log n) implementation and adaptability to multi-dimensional fractional PDEs make it a valuable tool in numerical analysis.

A Sine Transform-Based Preconditioner is an operator or matrix, constructed explicitly via the (discrete) sine transform or analytically via sine-kernel integral operators, whose purpose is to improve the spectral properties and hence the convergence behavior of iterative solvers for structured linear systems or inverse problems. The methodology fundamentally rests on the ability of the sine transform to diagonalize key classes of structured operators—particularly Toeplitz, Toeplitz-plus-diagonal, and related algebraic systems with boundary conditions or nonlocal operators—and extends into areas such as harmonic analysis on manifolds, convex geometry, and numerical solutions of (fractional) partial differential equations.

1. Mathematical Foundations and Algebraic Structure

A sine transform-based preconditioner exploits the fact that the discrete sine transform (DST) diagonalizes tridiagonal matrices (such as the discrete Laplacian) and, with appropriate modifications, can diagonalize a broader class of structured matrices—most notably τ-matrices, which are Toeplitz matrices with a Hankel correction. For a Toeplitz matrix $T_n$ of size $n$, the τ-matrix is constructed as: $\tau(T_n) = T_n - H_n$ where $H_n$ is a symmetric Hankel matrix matching the antidiagonals of $T_n$ (Noutsos et al., 2014, Huang et al., 2021, Lin et al., 17 Apr 2024).

The DST matrix $S_n$ has entries

$$[S_n]_{j,k} = \sqrt{\frac{2}{n+1}} \sin\left(\frac{\pi j k}{n+1}\right), \quad 1 \leq j,k \leq n$$

and provides the eigendecomposition

$\tau(T_n) = S_n \Lambda_n S_n$

where $\Lambda_n$ is diagonal with explicit entries derived from the Fourier expansion of $T_n$’s generating function (Noutsos et al., 2014, Huang et al., 2021).

In the context of integral geometry, the sine transform is an integral operator on manifolds such as the Stiefel or Grassmannian, defined for a function $f$ by

$$(S_{m,k}^\alpha f)(u) = \int_{V_{n,m}} f(v) \left| I_m - v^T u u^T v \right|^{(\alpha + k - n)/2} dv$$

for $u \in V_{n,k}$, parameter $\alpha$, and suitable $f$ (1103.1161). Analytic continuation with respect to $\alpha$ leads to inversion and identity relations that are crucial to preconditioning strategies in more abstract settings.

2. Spectral Properties, Diagonalization, and Implementation

The sine transform-based preconditioner’s effectiveness arises from its ability to induce spectral equivalence or essential spectral equivalence between the original system matrix and the preconditioned operator:

• Full Spectral Equivalence: For generating functions with zeros of order $0 < \theta \leq 2$, the spectrum of $P^{-1}A$ is uniformly bounded away from zero and infinity, i.e., $c \leq \lambda_i(P^{-1}A) \leq C$ for all $i, n$ (Noutsos et al., 2014).
• Essential Spectral Equivalence: For higher-order zeros ($\theta > 2$), all but a fixed number of eigenvalues are bounded within $[\alpha_\theta, \beta_\theta]$, with outlier eigenvalues not affecting iterative solver performance (Noutsos et al., 2014).

This diagonalization property translates into efficient numerical algorithms: multiplication by the sine transform matrix and its inverse can be carried out in $O(n \log n)$ operations via fast sine transform algorithms, which remains true in higher dimensions due to tensor product structures (Huang et al., 2021, She et al., 16 Apr 2024, Huang et al., 31 Jul 2025). In fractional PDEs, Kronecker product structure allows the construction of preconditioners as

$$\mathcal{P} = \sum_{i=1}^d I_{n_1} \otimes \cdots \otimes \tau(A_{n_i}) \otimes \cdots \otimes I_{n_d}$$

where each $\tau(A_{n_i})$ is diagonalized individually (Huang et al., 2021, She et al., 16 Apr 2024, Huang et al., 31 Jul 2025).

3. Applications in Fractional and Nonlocal Differential Equations

Sine transform-based preconditioners are foundational in the solution of discretized fractional diffusion equations, Riesz fractional diffusion, and tempered fractional diffusion equations, where the coefficient matrices are often multi-level Toeplitz or Toeplitz-plus-diagonal, and condition numbers grow rapidly with discretization. By replacing the dense (multi-level) Toeplitz blocks with their $\tau$-matrix analogues diagonalized by DST, the preconditioner enables mesh-independent convergence of Krylov subspace methods such as CG or GMRES (Huang et al., 2021, She et al., 16 Apr 2024, Lin et al., 17 Apr 2024, Zhang et al., 31 Jul 2024, Huang et al., 31 Jul 2025).

For example, in the framework of space-fractional diffusion equations with variable coefficients,

$A = I + \sum_i D_i S_i$

with $S_i$ multi-level Toeplitz, the approximating $\tau$-matrix preconditioner leads to spectral bounds on $P^{-1}A$ in the interval $(1/2,\,3/2)$ (or a similar robust interval), ensuring optimal (size-independent) contraction factors for GMRES or CG (Lin et al., 17 Apr 2024, Huang et al., 31 Jul 2025).

4. Theoretical Analysis and Uniform Convergence Guarantees

Rigorous spectral analysis demonstrates that preconditioned matrices exhibit eigenvalue clustering independent of problem size, grid refinement, or differential order. This property is established via careful estimation of the action of the preconditioner, using properties of the τ algebra and the DST diagonalization (Noutsos et al., 2014, Huang et al., 2021, She et al., 16 Apr 2024, Huang et al., 31 Jul 2025). For multi-dimensional operators, Kronecker product structure and tensor product τ-matrix analogues preserve these spectral properties.

In all-at-once time-parallel methodologies, spectral analysis shows eigenvalue clustering around ±1 (MINRES setting) or $1$ (CG on normal equations) for the preconditioned systems, which robustly explains the mesh- and parameter-independent convergence observed in large-scale evolutionary PDE discretizations (Hon et al., 2022).

Key theoretical estimates include bounds of the form: $$\min\left\{\frac{r_{\min}}{r_{\max}},\, \frac{3}{8}\right\} \leq \lambda(P^{-1}A) \leq \max\left\{\frac{r_{\max}}{r_{\min}},\,2\right\}$$ with $r_{\min}$, $r_{\max}$ denoting bounds on the variable coefficients, valid across mesh refinements (Huang et al., 31 Jul 2025).

5. Numerical Performance and Practical Considerations

Empirical results across a wide range of fractional diffusion and related PDEs consistently demonstrate that the use of a sine transform-based preconditioner yields

• Nearly constant Krylov iteration counts as spatial or temporal discretization increases (Huang et al., 2021, She et al., 16 Apr 2024, Lin et al., 17 Apr 2024, Huang et al., 31 Jul 2025).
• Substantial reductions in CPU time compared to both standard and classical circulant/Toeplitz-based preconditioners, due to the $O(n \log n)$ cost per solve and reduced number of iterations (She et al., 16 Apr 2024, Zhang et al., 31 Jul 2024).
• Robustness to coefficient variability and non-separability, with optimal convergence persisting for high-dimensional, non-separable variable coefficient problems and nonlinear extensions (e.g., fractional NLS equations) (Chen et al., 4 Oct 2024).

Typical iteration numbers in numerical tables range from 5–11, showing negligible increase as mesh resolution is refined. Eigenvalue histograms and spectra plots visually confirm sharp clustering around the predicted intervals.

6. Extensions, Challenges, and Broader Impact

While the sine transform-based preconditioning approach is optimally effective for problems where the system is Toeplitz-like or closely approximated by τ-matrices, certain systems with nonsymmetric, more general, or irregular structure may require schema extensions. In such scenarios, block diagonalization, two-sided preconditioning, and adaptation to nonuniform grids have been considered (Huang et al., 2021, Hon et al., 2022, She et al., 16 Apr 2024).

In harmonic analysis and integral geometry, sine transform-based preconditioning underlies stable inversion mechanisms for generalized Radon transforms on manifolds, as analytic continuation of normalized sine transforms directly yields the identity operator in certain cases, enabling geometrically optimal regularization (1103.1161).

Furthermore, in convex geometry, the sine transform of isotropic measures, through sharp isoperimetric and reverse inequalities, serves as a regularizing mechanism ensuring well-controlled operator norms and robust performance in inverse convex body recovery (Maresch et al., 2012).

The algebraic signal-processing framework allows the systematic generalization of sine transform-based preconditioners to quantum lattice simulations via tensor networks, where diagrammatic decompositions of DST-I achieve near-optimal computational complexity and inherit the favorable preconditioning properties in many-body settings (Epple et al., 2017).

7. Summary Table: Sine Transform-Based Preconditioner—Contexts and Properties

Application Context	Preconditioner Structure	Spectral/Convergence Guarantee
Riesz/tempered fractional diffusion (multi-d)	τ-matrix via DST	Eigenvalues in (1/2, 3/2) or near 1
Time-parallel (all-at-once) evolutionary PDEs	DST-diagonalizable block symbol	Eigenvalue cluster at ±1 or 1
Convex body recovery/inverse geometry	Sine transform on measures	Sharp geometric operator norm bounds
Fermionic tensor networks (signal processing theory)	DST-I diagrammatic structure	Linear-logarithmic scaling, spectral flattening

• Sine transform on Stiefel and Grassmann manifolds, analytic continuation and identity property (1103.1161)
• Local fractional sine transform, spectral diagonalization, and preconditioning in fractal space (1110.4756)
• Isoperimetric inequalities and preconditioning in convex geometry (Maresch et al., 2012)
• Sine-transform diagonalized τ-algebra preconditioning for ill-conditioned Toeplitz systems (Noutsos et al., 2014)
• Sine-transform-based preconditioners for multi-dimensional Riesz fractional diffusion (Huang et al., 2021)
• Two-sided τ-preconditioners for quasi-compact and non-separable coefficient schemes (She et al., 16 Apr 2024, Lin et al., 17 Apr 2024)
• Preconditioning for fourth-order high-order schemes via the sine transform (Huang et al., 31 Jul 2025)
• Sine transform-based fast solvers in nonlinear Schrödinger equations (Chen et al., 4 Oct 2024)
• DST-based tensor network and block structure in quantum simulation (Epple et al., 2017)
• Sine-transform-based preconditioning for all-at-once evolutionary PDE solvers (Hon et al., 2022)
• Rigorous spectral analysis and optimal convergence in all fields above

A sine transform-based preconditioner thus represents a unifying framework for diagonalization-based numerical regularization, spectral normalization, and efficient iterative solution across a spectrum of applied mathematics, numerical analysis, and geometric analysis applications.