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Inverse problems with fractional operators: an analytical study and its numerical linear algebra treatment via the GLT theory

Asim Ilyas

arXiv:2610.33763v1math.NAcs.NA

TL;DR

Inverse problems for fractional differential equations involve recovering unknowns from indirect data, while their discretizations lead to large structured matrix sequences whose spectral behavior matters. This thesis studies GLT analysis of these sequences and geometric means; it reports that the classical geometric-mean relation cannot hold for noncommuting, simultaneously degenerate symbols.

  • Problem

    Inverse problems for fractional differential equations combine ill-posedness with nonlocal operators, while geometric means of GLT sequences raise questions about symbol commutativity, degeneracy, and maximal distribution results.

    Source: Introduction · pp. 15–16

  • Method

    The thesis studies spectral distributions of geometric means for Hermitian positive definite GLT sequences, including conditions on their symbols.

    Source: Introduction · p. 16, Maximal results for geometric means of… · p. 147

  • Results

    For noncommuting, simultaneously degenerate symbols, the classical relation between the geometric-mean sequence and the geometric mean of symbols cannot hold.

    Source: §9.2.4 · p. 156

  • Takeaways & Limitations

    The geometric-mean analysis identifies noncommuting simultaneous degeneracy as a boundary where the classical symbol relation fails.

    Source: Introduction · p. 16, §9.2.4 · p. 156

    The classical relation fails in the noncommuting, simultaneously degenerate case; studying that case within GLT requires more technical GLT and a.c.s. analysis and deeper functional calculus.

    Source: §9.2.4 · p. 156

Abstract

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Fractional derivatives model memory effects and nonlocal interactions beyond classical integer-order models. Fractional differential equations (FDEs) thus describe anomalous diffusion, transport in heterogeneous media, and hereditary dynamics. Their nonlocality, however, makes inverse problems challenging: unknown sources, coefficients, or initial conditions must be recovered from indirect and noisy data. Inverse problems for FDEs are usually ill-posed in the Hadamard sense, so regularization, such as the quasi-boundary value method, is required for stable reconstruction. Discretization produces large linear systems whose matrices inherit the nonlocal structure of fractional operators. Understanding the asymptotic spectral properties of these matrix sequences is essential for efficient solvers and preconditioners. Generalized Locally Toeplitz (GLT) theory provides a unified framework for the spectral distribution of large structured matrices and for preconditioner design. This thesis uses GLT theory to analyze matrix sequences from inverse problems for FDEs, construct structure-preserving preconditioners, and study nonlinear operations in the GLT algebra. In particular, it proves maximal distribution results for geometric means of GLT sequences: when the symbols commute, the geometric mean sequence remains GLT without symbol invertibility. This resolves a long-standing conjecture and shows maximality in the noncommuting degenerate case. Overall, the thesis integrates regularization, spectral analysis, and preconditioning into a coherent computational framework for inverse problems in nonlocal fractional dynamics, while enriching structured matrix analysis.

Dedication

The dedication expresses gratitude to the author’s family and supervisors for their care, support, guidance, and mentorship.

Dedication

  • The author dedicates the thesis to their parents and son, Muhammad Salim Asim.Source: Dedication · p. 2
  • The author thanks their wife, brother, and sister for patience, love, encouragement, and support.Source: Dedication · p. 2
  • The author acknowledges supervisors Prof. Stefano Serra-Capizzano and Prof. Sean (Yu Sing) Hon for guidance and mentorship.Source: Dedication · p. 2

9.1 Example 1.

This material introduces inverse problems for fractional equations and GLT spectral analysis, alongside captions comparing geometric-mean eigenvalues with symbol-based distributions.

9.1 Example 1.

  • The listed geometric-mean figures compare eigenvalues of G(An, Bn) with symbol-based distributions across examples and matrix sizes.The captions identify sorted or ordered eigenvalues and rearranged symbol distributions, but do not state a comparative outcome.Source: §9.2 · p. 7, §9.1 · p. 7

Introduction

  • Fractional operators model nonlocal memory and long-range interactions, while inverse problems seek unknown quantities from indirect, often noisy measurements and can be ill-posed.Source: Introduction · pp. 14–15
  • GLT symbols describe asymptotic spectral behavior of discretization matrix sequences and support analysis of conditioning and preconditioner design.Source: Introduction · pp. 15–16

Structure of the thesis

The thesis progresses from analytical foundations and GLT theory to inverse problems, structured solvers, and geometric means of GLT sequences. Its contributions combine well-posedness analysis, regularization, spectral study, and preconditioning.

Structure of the thesis

  • The chapter plan moves from fractional-calculus and matrix-sequence foundations through GLT theory to inverse problems and geometric-mean analysis.The thesis closes its sequence of topics with maximal-setting geometric-mean results, including numerical examples of failure for noncommuting symbols with simultaneous degeneracy.Source: Structure of the thesis · pp. 16–18

Main Contributions of the Thesis

  • Chapter 5 analyzes two inverse source problems for generalized two-dimensional diffusion, proving regular-solution existence and uniqueness and establishing Hadamard ill-posedness.The formulation includes a general memory kernel and nonlocal boundary conditions, with tempered, multi-term, mono-term, and classical models as special cases.Source: Main Contributions of the Thesis · p. 18
  • Chapter 6 uses quasi-boundary regularization and GLT spectral analysis to design a block triangular GMRES preconditioner for a space-time fractional inverse problem.The preconditioned sequence is proved weakly clustered at one in the GLT, singular-value, and eigenvalue senses.Source: Main Contributions of the Thesis · p. 18
  • Chapter 7 develops a preconditioned solver for reconstructing sources in two-dimensional variable-coefficient space-time fractional diffusion problems.Experiments report accurate reconstruction and efficient preconditioning; convergence depends mainly on spatial discretization, with limited sensitivity to temporal refinement.Source: Main Contributions of the Thesis · pp. 18–19
  • Chapter 8 analyzes and preconditions all-at-once systems for variable-coefficient space-fractional equations in one and two spatial dimensions.The structure-preserving preconditioners use circulant approximations, and the GLT analysis covers nonconstant coefficients.Source: Main Contributions of the Thesis · p. 19
  • Chapter 9 proves that commuting GLT symbols yield a GLT geometric-mean sequence without symbol invertibility, resolving a long-standing conjecture.The result covers scalar symbols in any dimension and commuting block symbols; the noncommuting, simultaneously degenerate case shows the result is maximal.Source: Main Contributions of the Thesis · p. 19
  • Together, the contributions combine analytical regularization, spectral analysis, and preconditioning for inverse problems and large structured systems.Source: Main Contributions of the Thesis · p. 19

Fractional operators and Mittag-Leffler type functions

The chapter lays out fractional operators and Mittag–Leffler functions as analytical tools for the thesis’s fractional differential equations. It reviews operator definitions and relationships, then introduces properties of multinomial Mittag–Leffler functions.

Fractional operators and Mittag-Leffler type functions

  • Fractional calculus extends integer-order differentiation and integration to non-integer orders, with nonlocality and memory effects suited to anomalous transport and diffusion.Source: Fractional operators and… · p. 20

1.2 Mittag-Leffler type functions

1.1 Fractional operators

  • The operator review covers Riemann–Liouville, Riesz, tempered, and Caputo derivatives, which need not be equivalent on bounded domains without additional regularity or boundary conditions.Tempering introduces exponential damping; when its parameter is zero, tempered Riemann–Liouville derivatives reduce to the classical Riemann–Liouville derivatives.Source: §1.1 · pp. 21–22

Matrix sequences: spectral distributions and asymptotic structures

Matrix sequences arise as discretizations are refined and retain explicit or hidden structure from the underlying operators. Their asymptotic spectral behavior and approximation theory provide foundations for the GLT framework used in the thesis.

Matrix sequences: spectral distributions and asymptotic structures

  • Refining a discretization produces growing matrix sequences whose structure reflects the differential operator and supports analysis of stability, conditioning, and solver design.Their structure may be explicit, as in Toeplitz matrices, or asymptotic and hidden, as described by GLT theory.Source: Matrix sequences: spectral… · p. 25

2.1 Spectral distributions of matrix sequences

  • Asymptotic eigenvalue and singular-value distributions describe large-matrix spectra; under continuity, eigenvalues behave like uniform symbol samples, with weak clustering at the symbol’s essential range.When that range is a single point, distribution and weak clustering at that point are equivalent for eigenvalues and singular values.Source: §2.1 · pp. 26, 28
  • If every matrix is positive definite, its eigenvalue symbol is nonnegative almost everywhere; a uniform lower bound on singular values implies the symbol’s minimum singular value is positive almost everywhere.Conversely, a symbol with zero minimum singular value on a set of positive measure rules out uniform invertibility.Source: §2.1 · p. 26

2.2 Approximating classes of sequences

  • Approximating classes of sequences formalize asymptotic approximation through negligible-rank perturbations and vanishing-norm remainders, supporting convergence analysis and construction of the GLT *-algebra.As the approximation index increases, these perturbation terms become increasingly insignificant for sufficiently large matrices.Source: §2.2 · pp. 28–29

2.3 Multi-level discretization matrices

  • On a d-dimensional grid with n1,...,nd points per direction, discretization yields matrices of size N × N, where N = n1n2 ··· nd, indexed by refinement in each dimension.Multi-index notation records each grid location by a d-tuple and matches the recursive structure of the matrices.Source: §2.3 · p. 30

Structured matrices

This section surveys structured matrices arising from discretized differential operators, including multilevel block, Toeplitz, circulant and ω-circulant variants, as well as τ and Hankel matrices. It covers their spectral, diagonalization and computational properties, including fast transforms and the conditioning limitation of ω-circulant diagonalization.

Structured matrices

  • Multilevel r-block Toeplitz matrices encode discretized operators through recursively nested Toeplitz blocks, with structure useful for analysis and algorithm design.The number of levels corresponds to spatial dimension, while r > 1 accommodates systems, vector-valued unknowns, and all-at-once discretizations.Source: Structured matrices · p. 31, §3.1 · pp. 32–33

3.1.1 Relationship between the Toeplitz matrix and its generating function

  • For Hermitian generating functions, the fixed number of smallest and largest eigenvalues of associated Toeplitz matrices converge to the symbol’s essential spectral extrema.The symbol also bounds the spectra of the generated matrices.Source: §3.1.1 · p. 34

3.2 Circulant matrices

  • A d-level r-block circulant matrix is recursively built from circulant blocks and is fully determined by its first block row or column.Circulants form a subclass of Toeplitz matrices.Source: §3.2 · pp. 36–37

3.2.1 Spectral structure and fast solvers for circulant matrices

  • The Fourier matrix unitarily block-diagonalizes multilevel block circulants, and FFT-based matrix–vector multiplication costs O(rN(n) log N(n) + r^2N(n)).Their spectrum is the union of the spectra of the diagonal blocks.Source: §3.2.1 · pp. 39–40

3.3 ω-circulant matrices

  • The uni-level ω-circulant class extends standard circulants using a nonzero complex parameter ω that scales selected entries.The section does not consider multilevel ω-circulant matrices.Source: §3.3 · p. 40
  • ω-circulant algorithms cost O(n log n) sequentially and O(log n) in parallel, but diagonalization conditioning tends to max{|ω|, |ω|^-1} and becomes unbounded as ω approaches 0 or |ω| grows without bound.The conditioning follows from the diagonal scaling because the Fourier matrix is unitary.Source: §3.3 · p. 42

3.3.1 Block ω-circulant matrices

  • For fixed block size r, block ω-circulant matrix–vector products cost O(n log n) sequentially or O(log n) parallel steps.If r grows and each inner matrix admits diagonalization costing O(r log r), the total cost is O(nr log(nr)) sequentially or O(log(nr)) parallel steps.Source: §3.3.1 · p. 43

3.4 τ matrices

  • A uni-level r-block τ matrix satisfies a cross-sum condition; symmetric generating blocks make it symmetric and persymmetric, and its blocks are determined by its first block row or column.For r = 1, the definition reduces to the classical scalar case.Source: §3.4 · p. 44

3.4.1 Spectral properties and the discrete sine transform

  • Multi-level block τ matrices are real, and with symmetric generating blocks their spectrum is the union of r×r diagonal-block spectra; core operations cost O(N(n) log N(n)) sequentially.The diagonal blocks are obtained by applying the DST to the first block column, and the corresponding invariant subspaces are spanned by the columns of Qn,r; parallel operations take O(log N(n)) steps.Source: §3.4.1 · p. 46

3.5 Hankel matrices

  • A multi-level r-block Hankel matrix has blocks constant along block antidiagonals and a recursive block structure, and is specified by its first block row and last block column.It can be interpreted as a mirrored Toeplitz structure.Source: §3.5 · pp. 46–47

GLT matrix sequences and their algebraic structure

The section presents GLT theory as a spectral framework for structured matrix sequences arising from differential and nonlocal operators, and as a guide to numerical solver design. It develops the sequence classes, symbols, and operative properties used to analyze spectral distributions and preconditioners.

GLT matrix sequences and their algebraic structure

  • GLT theory provides a unified framework for the asymptotic spectral analysis of structured matrices from differential and nonlocal operators and informs efficient solver design.The section focuses on d-level sequences with r × r matrix-valued symbols and the operative GLT properties used in later analysis.Source: GLT matrix sequences and their… · p. 48

4.1 The GLT class

  • LT sequences extend Toeplitz structure to variable-coefficient discretizations by modulating Toeplitz matrices with a spatially varying weight.Their symbol combines the weight a(x) with the generating function f(θ) through pointwise matrix multiplication.Source: §4.1 · pp. 48–49

4.2 Matrix sequences with explicit or hidden (asymptotic) structure

  • The GLT ∗-algebra combines Toeplitz, sampling-diagonal, and zero-distributed sequences, with reduced versions also modeling fractional integro-differential operators on general domains.These classes provide explicit and hidden asymptotic structure within maximal algebras of matrix sequences.Source: §4.2 · p. 50

4.3 Zero-distributed sequences

  • A sequence is zero-distributed exactly when it decomposes into a vanishing-relative-rank part and a part whose spectral norm tends to zero.Equivalently, its normalized Schatten p-norm tends to zero under the stated scaling.Source: §4.3 · p. 50

4.4 Multi-level block Toeplitz matrices

  • Multi-level block Toeplitz matrices are generated by matrix-valued functions whose Fourier coefficients form the matrix blocks.The associated family is indexed by the multi-level matrix size.Source: §4.4 · p. 51

4.5 Block diagonal sampling matrices

  • Block diagonal sampling matrices place evaluations of a spatial function at uniform-mesh points along the diagonal in lexicographical order.Source: §4.5 · p. 51

4.6 The ∗-algebra of d-level r-block GLT matrix sequences

  • The GLT symbol is unique almost everywhere and determines singular-value distributions, as well as eigenvalue distributions for Hermitian sequences.The class is generated by zero-distributed, Toeplitz, and sampling-diagonal sequences, and its operative properties include algebraic and continuous-function rules.Source: §4.6 · pp. 51–52

4.6.1 GLT theory and preconditioning: a practical perspective

  • A preconditioned sequence with constant GLT symbol 1 has singular values asymptotically distributed at 1; Hermitian matrices also have eigenvalues clustering there.For non-Hermitian matrices, clustering may follow under additional conditions such as normality or a suitable low-rank perturbation structure; clustering typically supports fast Krylov convergence.Source: §4.6.1 · p. 53

Inverse source problems for a 2D generalized diffusion equation

This section analyzes two inverse source problems for a two-dimensional generalized diffusion equation, establishing their ill-posedness and developing solution representations and well-posedness results. It also connects the analysis to regularization for stable reconstruction.

Inverse source problems for a 2D generalized diffusion equation

  • Both inverse source problems have established existence and uniqueness of regular solutions, yet are ill-posed because small data perturbations can cause unbounded reconstruction errors.The chapter derives explicit series representations using a bi-orthogonal system of functions.Source: Inverse source problems for a 2D… · p. 54

5.1 Mathematical formulation and preliminaries

5.1.3 Special cases

  • Suitable memory-kernel choices recover tempered, multi-term, mono-term, and classical diffusion equations within the generalized framework.The chapter describes n-tempered, multi-term, mono-term, and standard diffusion special cases.Source: Inverse source problems for a 2D… · p. 54, §5.1.3 · p. 56

5.1.1 Generalized Caputo-type fractional derivative

  • The generalized Caputo-type derivative is formulated and its main analytical properties are recalled as preliminaries to defining the diffusion model and inverse problems.The derivative formulation is attributed to Kochubei.Source: §5.1 · p. 54, §5.1.1 · p. 55

5.1.4 Mathematical formulation of the ISPs

  • The first ISP recovers a spatial source f(x, y) and solution u from final-time data, while the second recovers time-dependent w(t) from an integral condition.The integral observation supplies information for recovering the source; its physical interpretation includes total particle concentration or heat energy.Source: §5.1.4 · pp. 56–57

5.2 Spectral problem

  • The spectral problem yields bi-orthogonal eigenfunction systems, and its eigenfunctions form a Riesz basis in L2((0, 1) × (0, 1)).The bi-orthogonality is stated for systems X_j,k and Y_j,k; the Riesz-basis result applies to the X_j,k system.Source: §5.2 · p. 58

5.3.1 Series solution construction for first-ISP

  • For the first ISP, the solution is constructed as a series, and its impulse-response function V_j,k(t) is completely monotone.The first ISP seeks the spatial source and state from the governing system and final-time observation.Source: §5.3.1 · pp. 59–60

5.3.2 Ill-posedness of first-ISP

  • The chapter’s first-ISP analysis specifically treats its ill-posedness, consistent with the broader finding that small measurement perturbations can destabilize reconstruction.The supplied passage identifies the ill-posedness analysis but does not provide a quantitative instability result.Source: Inverse source problems for a 2D… · p. 54, §5.3.2 · p. 62
  • The first-ISP is ill-posed, as concluded from Eqs. (5.38) and (5.39).Source: §5.3.2 · p. 62

5.3.3 The main results of first-ISP

  • The first-ISP is ill-posed, while a regular solution exists under the stated smoothness and compatibility conditions.The results also establish continuity of u_yy and the generalized derivative term; the uniqueness argument concludes the source terms agree when the solutions agree at an interior point.Source: §5.3.2 · p. 62, §5.3.3 · pp. 63–66

5.4 Second-ISP

  • The second-ISP seeks to recover w(t) together with u(x, y, t) for a source term F(x, y, t) = w(t)f(x, y, t), using an additional condition.Source: §5.4 · p. 66

5.4.1 Series solution construction for second-ISP

  • The second-ISP solution is constructed as a series with w(t) to be determined, and its modal expressions specialize to multi-term, mono-, and standard diffusion equations.The p = 0 case gives the multi-term FDE expression; further parameter choices yield mono-FDE and SDE cases.Source: §5.4.1 · p. 67

5.4.2 The main results of second-ISP

  • Under Theorem 5.3’s assumptions, the second-ISP has a regular solution, and Theorem 5.4 establishes its uniqueness.A contraction mapping and Banach’s fixed-point theorem establish the unique w(t); the solution and its second spatial derivatives are continuous.Source: §5.4.2 · pp. 67, 69, 70
  • For the second-ISP main result, the modal quantities U_j,k(t) vanish, yielding u(x, y, t) = v(x, y, t).The passage attributes this deduction to the initial conditions, uniqueness of w(t), and completeness of Yj,k(x, y).Source: §5.4.2 · p. 70

5.4.3 Ill-posedness of second-ISP

  • Based on Eq. (5.67), the passage concludes that the second-ISP is ill-posed.Source: §5.4.3 · p. 72

5.5 Examples

  • The examples provide solutions for both first-ISP and second-ISP, including a second-ISP example with a specified source term and zero initial condition.The supplied passages do not expose the source formula or resulting solution values.Source: §5.5 · pp. 72–74
  • The chapter establishes severe Hadamard ill-posedness for inverse source problems and derives explicit solution series using bi-orthogonal eigenfunction systems.It identifies regularization as necessary for stability and notes the challenge of recovering source coefficients from noisy observations.Source: §5.5 · p. 75

Krylov preconditioning for inverse problems in space-time fractional diffusion equations

This section studies GLT-based Krylov preconditioning for inverse source problems in space-time fractional diffusion, following regularization and discretization with analysis of the resulting matrix sequences. It examines the preconditioner’s effect on GMRES, reporting spectral distribution as 1 and reduced iteration counts.

Krylov preconditioning for inverse problems in space-time fractional diffusion equations

  • The chapter addresses an ill-posed inverse problem for a space-dependent source in a model with tempered time and Caputo space fractional derivatives using quasi-boundary value regularization.The resulting large block linear system is analyzed with GLT tools, and numerical experiments confirm that the proposed GMRES preconditioner reduces iteration counts and computational cost.Source: Krylov preconditioning for inverse… · p. 76

6.1 Discretization and matrix structures

  • The finite-difference scheme converges to the regularized problem as the space and time mesh sizes tend to zero, and the coefficient matrix is invertible when its recursively defined diagonal criterion holds.Under the stated regularity assumptions, the L1 time and space discretizations have orders O(∆t^(2−ξ)) and O(∆x^(2−η)); weak initial-time singularity can reduce temporal convergence to first order. The matrix is block lower triangular up to a rank-m correction, and the invertibility check uses the diagonal entries of β^(m).Source: §6.1 · pp. 81–82

6.2 GLT analysis

  • GLT analysis yields the overall coefficient-matrix sequence’s symbol and singular-value distribution, while eigenvalue distribution requires additional justification for these non-Hermitian matrices.The chapter establishes weak clustering at 1 for the preconditioned sequence using the low-rank perturbation between the preconditioner and coefficient matrix.Source: §6.2 · p. 87

6.3 GLT preconditioning

  • The proposed preconditioner is designed to cluster the eigenvalues of P_N^-1 A_N at 1 and has the same GLT symbol and singular-value distribution as A_N.Its block-triangular structure and invertible diagonal blocks ensure that P_N is invertible.Source: §6.3 · p. 87
  • The preconditioned matrix sequence has GLT symbol 1, singular-value distribution 1, and eigenvalues weakly clustered at 1.The distribution result is derived without assuming particular discretization parameters.Source: §6.3 · p. 88
  • For constant a(x), replacing the Toeplitz matrices in the preconditioner with Strang preconditioners enables FFT-based inversion at O(nm log(nm)) cost.The approach uses circulant matrices that are uniformly diagonalizable by the Fourier matrix.Source: §6.3 · p. 89

6.4 Numerical results

  • GMRES preconditioning reduces iteration counts for every tested fractional-order pair, remains stable across problem sizes and orders, and performs best at ξ=0.2, η=0.8.The comparison is reported for the preconditioned and nonpreconditioned systems in Table 6.1.Source: §6.4 · p. 90, Table 6.1 · p. 91
  • When n and m grow proportionally, preconditioned iteration growth in n is less than linear; for fixed n, counts vary little as m increases.The near-constant counts across m are attributed to the number of outliers in the cluster at 1 depending only on n.Source: §6.4 · pp. 90–91
  • The reported CPU timings may be misleading because MATLAB implementation-specific overhead prevents a meaningful comparison of intrinsic algorithm efficiency.The timings are reported alongside iteration counts in Table 6.1.Source: §6.4 · p. 91

Efficient Krylov solvers for inverse source problem in 2D space-time fractional diffusion equation

This section develops a quasi-boundary regularization and finite-difference framework for reconstructing a source in a two-dimensional space-time fractional diffusion equation, using GLT analysis to characterize the resulting matrices and guide GMRES preconditioning. It examines a block triangular preconditioner’s spectral properties and computational cost, alongside a cheaper Strang-like alternative.

7.3.1 Block Triangular preconditioning

  • Numerical experiments confirm that symbol-based preconditioning reduces GMRES iteration counts and computational cost; block triangular preconditioners target clustering of preconditioned eigenvalues at one.The block triangular preconditioner is invertible and can be solved by forward elimination; its diagonal blocks can use Toeplitz or circulant approximations.Source: Efficient Krylov solvers for inverse… · p. 92, §7.3.1 · pp. 101–102

Problem Formulation

  • The inverse problem reconstructs the source from a noisy final-time datum; quasi-boundary regularization yields parameter-dependent well-posed problems and stabilizes the reconstruction.Source: Problem Formulation · p. 93

7.1 Discretization and matrix structures

  • L1 finite differences for the tempered time derivative and both spatial Caputo derivatives produce an all-at-once linear system with nonlocal matrix structure.Source: §7.1 · pp. 94–96

7.2 GLT analysis

  • Under a mild relation among discretization rates, GLT theory gives an asymptotic symbol for the full coefficient-matrix sequence.Source: §7.2 · pp. 99–100
  • Because the coefficient matrices are non-Hermitian, their GLT symbol yields singular-value distribution, while low-rank structure supports weak eigenvalue clustering at one after preconditioning and fast GMRES convergence.Source: §7.2 · pp. 100–101

7.3.1 Block Triangular preconditioning

  • The proposed block triangular preconditioner complements evolutionary-PDE preconditioning and builds on block-structure techniques for fractional diffusion inverse problems.The cited approaches exploit time-stepping structure or matching Schur complements.Source: §7.3.1 · p. 102

7.3.2 GLT preconditioning analysis

  • The preconditioner has the same GLT symbol and singular-value distribution as the original matrix sequence, while the preconditioned eigenvalues are spectrally distributed as 1.The eigenvalue characterization is given without specific conditions on discretization parameters.Source: §7.3.2 · pp. 102–103
  • The analysis extends to bounded simply Peano-Jordan measurable domains, higher-order finite elements or isogeometric discretizations, and higher spatial dimensions.These extensions use reduced GLT theory for general domains and can be combined.Source: §7.3.2 · pp. 103–104

7.4 Numerical results and study of the computational cost

  • At (n1, n2, m) = (25, 25, 25), preconditioning reduced GMRES iterations from 186 to 39 (almost 80%); counts stayed 39–40 as m increased for the highlighted fractional orders.Across tested fractional orders, preconditioned iteration counts remained nearly constant as m increased and grew moderately with spatial refinement; CPU-time gains were less pronounced in some cases.Source: §7.4 · p. 105, Table 7.1 · p. 106

Study of the computational costs and further proposals

  • Both matrix-vector products with A and solves with P have arithmetic cost m^2ν(n)+o(m^2ν(n)) under log[ν(n)] = o(m), with constant preconditioned iteration counts.The solve-cost bound assumes the number of preconditioned GMRES iterations remains bounded independently of matrix size.Source: Study of the computational costs and… · pp. 105–106
  • The cheaper Strang-like preconditioner replaces block Toeplitz matrices by circulant approximations, but lacks GLT-guaranteed clustering at 1 and is less effective in iteration count.Its eigenvalue asymptotics are also described as unfavorable.Source: Study of the computational costs and… · pp. 108–109

Structure-preserving preconditioning for variable-coefficient space-fractional diffusion equations and θ-Method

This section uses GLT analysis to study spatial and space-time discretizations of variable-coefficient space-fractional diffusion systems in one and two dimensions, and develops structure-preserving circulant preconditioners. It examines their spectral properties and numerical performance, distinguishing established singular-value results from conjectural eigenvalue behavior for variable coefficients.

Structure-preserving preconditioning for variable-coefficient space-fractional diffusion equations and θ-Method

  • The proposed preconditioners address dense, nonsymmetric, ill-conditioned systems, with analysis covering nonconstant coefficients in one and two dimensions and experiments confirming faster GMRES convergence.The chapter’s stated novelty is the variable-coefficient coverage in both spatial dimensions.Source: Structure-preserving preconditioning… · p. 110

8.1.1 Discretization

  • Shifted Grünwald differences discretize the fractional derivatives, while the θ-method advances time and yields all-at-once linear systems in both spatial settings.In one dimension, θ=0, 1, and 1/2 correspond to Forward Euler, Backward Euler, and Crank–Nicolson, respectively.Source: One-dimensional · pp. 111–112, Two-dimensional · p. 115

Two-dimensional

GLT analysis of the space matrix sequences

  • For α∈(1,2), the fractional matrix sequence has a Wiener-class symbol fα, and variable-coefficient sampling yields GLT symbol a(x)fα with the associated singular-value distribution.The GLT analysis applies when a is continuous almost everywhere.Source: GLT analysis of the space matrix… · pp. 117–118
  • For non-Hermitian variable-coefficient matrices, the GLT symbol establishes singular-value distribution but not eigenvalue distribution; numerical support exists, while a proof is available only for constant coefficients.The variable-coefficient eigenvalue claim is therefore conjectural rather than established.Source: GLT analysis of the space matrix… · pp. 119–120
  • For spatial discretizations, GLT analysis is accompanied by numerical eigenvalue comparisons, but the 2D variable-coefficient eigenvalue distribution remains open.The reported 2D comparisons examine constant-coefficient and variable-coefficient matrices against their sampled symbols; the authors note that GLT does not automatically establish eigenvalue distributions for non-Hermitian matrices.Source: GLT analysis of the space matrix… · p. 123, Figure 8.3 · p. 123, Figure 8.4 · p. 124

GLT analysis of the space-time matrix sequences

  • The space-time matrices have GLT analyses in 1D and 2D, combining spatial and θ-method temporal contributions through their Kronecker-sum structure.For non-Hermitian space-time matrices, numerical results suggest eigenvalue distributions may not follow the GLT symbol, even for constant coefficients; GLT rigorously describes singular-value distributions, while eigenvalue behavior requires additional analysis.Source: GLT analysis of the space-time matrix… · pp. 124–126

8.2 Preconditioning strategy: 1D space setting

  • For 1D systems, the proposed block circulant preconditioners replace the spatial Toeplitz block while retaining the θ-method temporal block and averaging the diffusion coefficients.Theorem 8.5 gives a unitary block diagonalization with invertible lower triangular Toeplitz blocks; Theorem 8.6 analyzes the preconditioned sequences and reports clustering at 1 for constant coefficients.Source: §8.2 · pp. 126, 127, 129

8.3 Preconditioning strategy: 2D space setting

  • For 2D systems, the preconditioner uses a two-level circulant approximation of the spatial operator, retains the θ-method temporal block, and replaces sampled coefficients with their spatial average.Theorem 8.7 states that the preconditioner is unitarily similar to a block diagonal matrix whose blocks are invertible lower triangular Toeplitz matrices.Source: §8.3 · pp. 129–130

8.4 Numerical experiments

  • The numerical evaluation examines eigenvalue clustering, condition numbers, GMRES iteration counts, and CPU time using MATLAB R2022b on a laptop.The reported machine has an 11th Gen Intel Core i5-1155G7 CPU and 16 GB RAM, running Windows 11 Pro.Source: §8.4 · p. 130

8.4.1 Eigenvalue distribution

  • Preconditioning clusters eigenvalues around 1 for a(x) = 1, while for nonconstant coefficients they cluster around the range of the function in Theorem 8.6.Unpreconditioned eigenvalues spread along the positive real axis, and their 2-norm condition numbers grow rapidly; preconditioned condition numbers remain bounded and stable, though not substantially lower.Source: §8.4.1 · p. 131, Figure 8.6 · p. 133, Figure 8.7 · p. 134

8.4.2 GMRES performance in the d-dimensional space setting, d = 1, 2

  • With preconditioning, GMRES iterations remain nearly constant as N and M grow, while CPU times increase mildly; without it, both rise substantially in the one-dimensional tests.The tests use θ = 0.5, tolerance 10^-8, and a maximum iteration count equal to the matrix dimension; they cover a(x) = 1, x^2 + 1, and x^2.Source: §8.4.2 · p. 141, Table 8.3 · p. 142, Table 8.4 · p. 142

GMRES performance in the two-dimensional setting

  • In two dimensions, tests with constant and variable coefficients show no relevant difference in convergence properties from the one-dimensional setting, despite smaller discretization parameters.No tests with α1 ≠ α2 are reported; the text attributes the similar or slightly better practical behavior to the smaller-parameter matrix sequence being spectrally negligible as matrix sizes increase.Source: GMRES performance in the… · p. 145, Table 8.8 · p. 145, Table 8.9 · p. 146

Maximal results for geometric means of GLT sequences

This section examines when geometric-mean sequences remain GLT, including scalar and commuting block symbols without invertibility and the degenerate noncommuting block case. It also specifies the numerical-experiment setup, including rank-one symbols, sampled piecewise-linear functions, and GLT-symbol notation.

9.1 Geometric mean of GLT matrix sequences

  • For degenerate, noncommuting block symbols, the geometric-mean sequence may be GLT with a symbol different from the classical mean, which may itself be undefined.The classical symbol formula is not guaranteed under these conditions.Source: Maximal results for geometric means of… · p. 147, §9.1 · p. 149
  • When the symbols commute, the geometric-mean sequence is GLT with symbol (κξ)^1/2 without invertibility for scalar symbols and for block symbols with r > 1.The scalar result applies for d ≥ 1; the block result assumes commuting symbols.Source: §9.1 · pp. 148, 150, 151

Eigenvalue distribution

  • In Example 1, increasing matrix dimensions show the geometric-mean eigenvalue distribution approaching the GLT symbol (κξ)^1/2.The numerical comparison supports the distribution result in Theorem 9.2.Source: Eigenvalue distribution · p. 152, Figure 9.1 · p. 152
  • For the zero-symbol example, elevated eigenvalues reach approximately 10^-3 at n = 40 and 10^-4 at n = 80, forming a secondary distribution attributed to momentary symbols.The main eigenvalue distribution concentrates along zero, as predicted by Theorem 9.2.Source: Eigenvalue distribution · p. 153, Figure 9.2 · p. 154

9.2.3 Minimal eigenvalues and conditioning

  • The minimal-eigenvalue decay indicator stabilizes near 2 in Example 1 and is reported to converge to 4 in Example 2; these numerical observations are not established by the theoretical derivations.For Example 1, the minimal eigenvalue of A_n decays as n^-4, while that of B_n approaches min ξ = 1; the geometric-mean minimal eigenvalue behaves asymptotically like the geometric mean of the component minima.Source: Table 9.1 · p. 154, Example 1: minimal eigenvalue · p. 155, Table 9.2 · p. 155

9.2.4 Numerical study: noncommuting, rank-deficient symbols

  • Experiments with rank-deficient, noncommuting symbols indicate convergence to a rank-deficient candidate symbol that is generally distinct from the classical geometric mean.The experiments test cases outside the commuting or almost-everywhere-invertible settings.Source: §9.2.4 · pp. 155–156

Setup of the numerical experiments

A candidate symbol for the geometric mean

The section defines a candidate symbol for geometric means of positive-semidefinite GLT symbols and formulates a conjecture about GLT closure. A positive shift supplies a heuristic route to the conjecture, but a full proof for pointwise noncommuting symbols remains open.

A candidate symbol for the geometric mean

  • For positive-semidefinite symbols, shifting each by εI gives strictly positive symbols whose geometric mean has a pointwise limit as ε tends to zero.Existence and uniqueness of the limit follow from monotone continuity of the matrix geometric mean.Source: A candidate symbol for the geometric… · p. 157
  • The candidate symbol is essentially positive semidefinite and satisfies an essential-support identity.The supplied passage does not include the identity itself.Source: A candidate symbol for the geometric… · pp. 157–158
  • The conjecture proposes GLT closure for geometric means of HPD GLT matrix sequences under stated symbol conditions.Its displayed conclusion is truncated in the supplied passage.Source: A candidate symbol for the geometric… · p. 158
  • The conjecture’s heuristic shifts both sequences by εI, applies GLT closure under a.c.s. convergence, and takes ε to zero.The passage says this supports the GLT-algebraic part of the conjecture; a full proof when the symbols do not commute pointwise remains open.Source: A candidate symbol for the geometric… · p. 158
  • The numerical tests are organized into four test pairs grouped as Cases 1 and 2, each with two examples.The candidate symbols for these pairs are stated to be predictions of Conjecture 9.2.1.Source: Candidate symbols predicted by… · pp. 158–159

Numerical verification of the Conjecture

The numerical test compares sorted eigenvalues of the geometric-mean matrices with the rearranged values of the candidate symbol. Quantile convergence also gives a way to estimate the measure of the symbol’s zero set.

Numerical verification of the Conjecture

  • The test samples about 2,000 points of the symbol, computes its r eigenvalues at each point, then sorts the pooled values into an empirical quantile.The sampling domain is [0,1] × [−π, π].Source: Numerical verification of the Conjecture · pp. 159–160
  • For n ∈ {40, 80, 160, 320}, the test sorts the eigenvalues of Gn and records its minimum, maximum, and 2-norm condition number.The condition measures are recorded for later stability analysis.Source: Numerical verification of the Conjecture · p. 159
  • The numerical comparison overlays sorted matrix eigenvalues with rearranged sampled-symbol eigenvalues across increasing n to assess asymptotic distribution matching.The test also counts eigenvalues below 0.1 as an estimate of the measure of the symbol’s zero set.Source: Numerical verification of the Conjecture · pp. 159–160
  • When the rearranged symbol quantile is continuous at t, the sorted eigenvalue λ⌈tdn⌉(Gn) converges to that quantile as n grows.The comparison uses sorted eigenvalues because spectral distributions are invariant under measure-preserving rearrangements.Source: Numerical verification of the Conjecture · p. 160

Numerical results

The eigenvalue distributions generally follow the rearranged symbols, including zero clusters whose proportions approach the symbols’ zero-set measures. Momentary perturbations and symbol overlap shape convergence and conditioning across the four examples.

Example 1.

  • For Case 1, Example 2, the symbol has a zero region and a positive value on the support overlap; the measured zero-cluster proportion approaches its target with error below 10^-3 by n = 160.The error appears to decay at O(n^-1).Source: Example 1 · pp. 160, 165, Figure 9.4 · p. 161, Table 9.8 · p. 165
  • In Case 2, Example 1, the zero-symbol eigenvalues converge to zero without proper outliers, although a momentary perturbation produces an apparent 0.5 plateau through n = 320.The measured proportion then reaches 1; the text anticipates faster convergence than in Case 1, Example 2.Source: Example 1 · pp. 162, 165, Figure 9.5 · p. 162, Table 9.9 · p. 165
  • Case 1, Example 2 reaches cond2(Gn) = O(10^8) at n = 320 as λmin decays exponentially while λmax converges rapidly to 2.9939.The reported λmax limit is eGmax = 2.9939.Source: Example 1 · p. 163, Table 9.4 · p. 164
  • In Case 2, Example 1, λmin appears to decay quadratically and λmax linearly to zero, yielding moderate condition-number growth; Case 2, Example 2 instead has λmax converging to about 1.22 and linearly decaying λmin.The text describes Case 2, Example 2 as better conditioned than Case 1, Example 2, and relates this to the slower norm decay of its positive perturbations.Source: Example 1 · p. 164, Table 9.5 · p. 164, Table 9.6 · p. 164
  • For Case 1, Example 1, the zero symbol predicts a zero-distribution; the measured fraction below 0.1 rises toward 1, with only a small number of positive outliers.The number of outliers is at most o(n), and the empirical proportion remains within a small error of its target.Source: Example 1 · pp. 160, 164, Figure 9.3 · p. 161, Table 9.7 · p. 165
  • In Case 2, Example 2, the zero-cluster proportion converges to 17/18 ≈ 0.9444, while eigenvalues form a zero plateau followed by two shorter positive clusters.The plateau length matches the measure of the complement of the rearranged symbol’s support.Source: Example 1 · pp. 161, 165, Figure 9.6 · p. 163, Table 9.10 · p. 166

Conclusions and future work

The dissertation analyzes structured matrix sequences from discretized multidimensional fractional PDEs and applies GLT theory to fast solvers, particularly preconditioning. Its final chapter summarizes findings, similarities and differences, and future directions.

Conclusions and future work

  • The work analyzes structured matrix sequences arising from discretizations of multidimensional fractional partial differential equations.The matrix-sequence analyses support the dissertation’s numerical applications.Source: Conclusions and future work · p. 167
  • Dense, ill-conditioned coefficient matrices make the associated linear systems computationally challenging.The dissertation reports using GLT theory to support development of fast solvers, especially preconditioning strategies.Source: Conclusions and future work · p. 167
  • The final chapter synthesizes findings, compares studied scenarios, and suggests future research directions.Source: Conclusions and future work · p. 167

Summary of research contributions

The research spans inverse problems for fractional equations and GLT-based spectral analysis and preconditioning. Its final theoretical contribution removes symbol invertibility under commutation for geometric means and establishes maximality in a noncommuting degenerate case.

Summary of research contributions

  • Chapter 5 derives solution representations and existence and uniqueness results for regular solutions, while establishing that both inverse problems are Hadamard ill-posed.Fast regularization algorithms remain a significant challenge.Source: Summary of research contributions · p. 167
  • For the Chapter 6 inverse problem, quasi-boundary regularization and a block triangular preconditioner address a structured two-by-two system, with GLT analysis and numerical reconstruction results.Future work targets a more accurate preconditioner that accounts for remaining outliers.Source: Summary of research contributions · p. 167
  • Chapter 7 extends variable-coefficient inverse-problem analysis to two dimensions; its block triangular preconditioner is proved weakly clustered at one and supports fast GMRES convergence.The reported convergence behavior depends mainly on spatial discretization, with limited sensitivity to temporal refinement.Source: Summary of research contributions · p. 168
  • Chapter 8 analyzes variable-coefficient space-fractional systems in one and two dimensions and proposes circulant-based structure-preserving preconditioners, with experiments showing reduced GMRES iterations and computational cost.The analysis covers the nonconstant coefficient case.Source: Summary of research contributions · p. 168
  • For geometric means of Hermitian positive definite GLT sequences, commuting symbols allow removal of symbol invertibility, while simultaneous degeneracy with noncommuting symbols can break the classical relation.The result covers scalar symbols in any dimension and general block symbols under commutation; spectral experiments also reveal momentary-symbol phenomena.Source: Summary of research contributions · pp. 168–169

Future research directions

Future work extends GLT-based analysis to richer discretizations and focuses on optimized implementations that realize the theoretical complexity of the proposed preconditioners.

Future research directions

  • High-order finite elements in d dimensions, discontinuous Galerkin methods, and isogeometric analysis are proposed extensions of the theoretical framework.These settings lead to matrix-valued GLT symbols.Source: Future research directions · p. 169
  • The work proposes extending the approach to more complex cases using approximating classes of sequences and non-Hermitian perturbations of Hermitian matrix sequences.Source: Future research directions · p. 169
  • Optimized, structure-exploiting code, careful optimization, and parallelization are proposed to improve practical applicability and match the preconditioners’ derived theoretical complexity.Source: Future research directions · p. 169
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