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The tubal Arnoldi method for tensor function approximation
Fatima Bouyghf, Mohamed El Ghomari, Alaa El Ichi
TL;DR
Approximating matrix-function actions is challenging when large matrices make spectral or Jordan-form approaches computationally infeasible. The paper introduces a t-product-based tensor t-tubal Arnoldi method, which in experiments achieves same-accuracy approximations with fewer iterations and less CPU time than standard Arnoldi matrix methods.
Problem
For large matrices, spectral and Jordan-form approaches to matrix functions may be computationally infeasible, motivating Krylov methods for approximating f(A)b.
Source: §1 · pp. 1–2Method
The tensor t-tubal Arnoldi method uses the tensor t-product to approximate tensor function actions and parameter-dependent exponential tensor functions.
Source: §1 · p. 2, Conclusion · p. 20Results
The tensor Arnoldi method delivers approximations of the same accuracy as standard Arnoldi matrix methods with fewer iterations and less CPU time.
Source: Conclusion · p. 20Takeaways & Limitations
The experiments demonstrate the method in solving well-known time-integration ODEs through estimates of parameter-dependent exponential tensor functions.
Source: Conclusion · p. 20For the matrix Cauchy–Stieltjes integral to be well-defined, its contour must lie in the resolvent set and avoid the spectrum of A.
Source: §2 · p. 4
Abstract
from arXiv · showhide
This paper describes Krylov subspace methods based on the tensor t-product for approximating quantities associated with third-order tensor functions. Specifically, it introduces the tensor-tubal Arnoldi method, which projects highdimensional tensor problems onto lower-dimensional tensor Krylov subspaces. The derived method exploits novel algebraic properties of the t-product to address large-scale tensor computations efficiently. Applications include the evaluation of parameter-dependent tensor functions and the solution of multidimensional ordinary differential equations. Numerical experiments illustrate the effectiveness and accuracy of the proposed methods compared with other approaches.
1 Introduction
Large matrices can make spectral and Jordan-form approaches computationally infeasible, motivating Krylov methods for matrix-function actions. The paper introduces a t-product-based Arnoldi method for tensor function problems.
1 Introduction
- For large matrices, spectral and Jordan-form approaches may be computationally infeasible, while Krylov methods offer an alternative for approximating f(A)b.The standard Arnoldi process uses subspaces with nonnegative powers of A.Source: §1 · pp. 1–2
- The tensor t-product extends matrix multiplication to third-order tensors, a framework used for tensor functions and multidimensional differential equations.The paper introduces the tensor t-tubal Arnoldi method for a tensor function problem.Source: §1 · p. 2
- The proposed method projects a tensor problem onto an increasing sequence of t-product Krylov subspaces, yielding a reduced problem that can be solved by direct methods.The paper also applies the method to parameter-dependent tensor functions.Source: §1 · p. 2
2 Matrix Stieltjes functions
This section defines matrix functions through interpolation and presents the Cauchy–Stieltjes integral representation, whose contour must avoid the matrix spectrum.
2 Matrix Stieltjes functions
- For a matrix with Jordan blocks, the section defines f(A) by evaluating a Hermite interpolating polynomial that matches f and its derivatives at the eigenvalues.The polynomial degree is bounded using the Jordan-block sizes associated with those eigenvalues.Source: §2 · pp. 3–4
- A Cauchy–Stieltjes representation extends to matrices by an operator-valued integral, with an integration contour in the resolvent set of A.The contour must avoid the spectrum for the integral defining f(A) to be well-defined.Source: §2 · p. 4
- The integral representation offers an approach when diagonalization is inapplicable and Jordan-form methods may be unstable or expensive for large matrices.The passage notes its use in rational Krylov approximations and quadrature-based techniques.Source: §2 · p. 4
3 The tensor t-function
The section reviews the third-order tensor t-product and the associated tensor-function framework. It motivates these tools through multidimensional differential equations and the need for practical approximations of tensor function actions.
3 The tensor t-function
- The t-product is defined using third-order tensor operations, and applying FFT along the tubes produces frontal-slice blocks that support its computation.The inverse FFT recovers the original tensor.Source: §3 · p. 5
- The tensor t-exponential represents solutions to the stated multidimensional differential equation, motivating tensor functions based on the t-product.The equation uses a prescribed initial tensor.Source: §3 · p. 6
- The t-function framework generalizes scalar functions to tensor actions and includes definitions based on tensor eigendecomposition.The section motivates the framework with multidimensional differential equations and the tensor t-exponential.Source: §3 · pp. 6–7
- Approximating f(A) ⋆B for large tensors motivates efficient methods because full t-eigendecomposition may be computationally prohibitive.The work focuses on the case where B has one column and notes the equivalence of the tensor-action and matrix-function-action problems.Source: §3 · p. 7
4 The tensor Arnoldi method for Stieltjes functions of tensors
The tensor t-tubal Arnoldi method builds an orthonormal basis for a tensor tubal Krylov subspace and uses reduced problems to approximate tensor functions, including Stieltjes functions. The section also establishes residual and polynomial exactness properties of the approximation.
4 The tensor Arnoldi method for Stieltjes functions of tensors
- The tensor t-tubal Arnoldi process generates an orthonormal basis by orthogonalizing each new tensor against the existing basis tensors.The algorithm applies A ⋆ V_j, subtracts its projections onto earlier basis tensors, and normalizes the remainder.Source: §4 · pp. 7–8
- The shifted tensor A − tI is invertible exactly when t is outside the spectrum of every Fourier-transformed frontal slice of A.This condition is used to establish invertibility for the Stieltjes-function formulation.Source: §4 · p. 9
- For the shifted tensor system, the residual after m steps has a formula in terms of the next basis tensor and reduced-problem quantities.The expression includes V_{m+1}, h_{m+1,m}, E_m, and Z_m(t).Source: §4 · p. 10
- After m Arnoldi steps, the method provides an approximation for f(A) ⋆B for Stieltjes functions and an expression for its mth error.The error expression is related to the shifted-system residual.Source: §4 · p. 11
- The approximation is exact for polynomials with tube-fiber coefficients of degree at most m−1.Tube fibers serve as coefficients in the polynomial formulation linked to the tensor Krylov subspace.Source: §4 · pp. 11–12
5 Application to the approximation of the tensor t-exponential
The tensor t-tubal Arnoldi method approximates solutions of multidimensional tensor ODEs through a reduced problem. The section gives a residual formula that avoids explicit products with the large tensor and describes convergence estimation and reuse across multiple time values.
5 Application to the approximation of the tensor t-exponential
- The residual formula is arranged to avoid explicit matrix-tensor products involving the large tensor A.The residual is associated with the approximation obtained from the reduced t-ODE.Source: §5 · p. 14
- The reduced t-ODE solution yields an approximation to the full tensor ODE solution after m steps of the tensor t-tubal Arnoldi method.The reduced solution satisfies its initial condition, and the full approximation is formed using the generated basis.Source: §5 · pp. 13–14
- The convergence estimate can guide stopping and is reused across multiple time values, though computing it requires one additional Arnoldi step.The iterative algorithm tracks which requested time values have met the tolerance and terminates when all have converged.Source: §5 · pp. 14–15
6 Numerical experiments
Across function-approximation and exponential-tensor experiments, TTA delivers comparable accuracy with lower runtime; in exponential tests it also uses fewer iterations or smaller projections. The error estimate is compared with relative error across several time-dependent examples.
6 Numerical experiments
- Across tested functions and exponential tensor problems, TTA matches comparator accuracy while reducing CPU time and, in exponential tests, iteration counts or projection dimensions.Table 1 uses m = 10 and n = 10000; exponential tests report comparable estimate errors of order 10^-5 and 10^-7.Source: §6.1 · p. 16, Table 1 · p. 17, §6.2 · pp. 18–19
6.2 Application to the solution of time-dependent partial differential equations
- At n = 900, the estimate error γm(t) coincides with relative error ǫm(t) across iterations for t = 0.25, 0.5, and 0.75.The comparison is shown for the time-dependent PDE example.Source: §6.2 · p. 17, Figure 1 · p. 18
- For t = 0.5, γm(t) provides higher precision for the behavior of ǫm(t) across six graph examples.The plots compare the estimate and relative errors against iteration count.Source: §6.2 · p. 19, Figure 2 · p. 20
Conclusion
The conclusion presents tensor t-tubal Arnoldi as an efficient approach to tensor-function approximation and multidimensional ODE applications, with experiments reporting fewer iterations and less CPU time than standard Arnoldi matrix methods at the same accuracy.
Conclusion
- The experiments report that tensor t-tubal Arnoldi requires fewer iterations and less CPU time than standard Arnoldi matrix methods to deliver approximations of the same accuracy.The paper applies the method to parameter-dependent exponential tensor functions arising in multidimensional ODEs.Source: Conclusion · p. 20
Funding
The paper reports receiving no specific grant from any funding agency.
Funding
- This research received no specific grant from any funding agency.Source: Funding · p. 21