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A Tutorial on Inverse Problems for Anomalous Diffusion Processes

Bangti Jin, William Rundell

arXiv:1501.00251v1math.NA

TL;DR

The paper asks how anomalous diffusion changes the ill-posedness of classical inverse problems and examines this question through formal analysis and numerical comparisons. It finds that fractional effects can either improve or worsen conditioning depending on the data and quantity of interest, while emphasizing that the underlying theory and practical reconstruction methods remain limited.

  • Problem

    The paper addresses limited understanding of how fractional nonlocal physics affects uniqueness, stability, and ill-posedness in inverse problems for anomalous diffusion.

  • Method

    The authors formally analyze and numerically compare several time- and space-fractional inverse problems with their classical Gaussian-diffusion counterparts.

  • Results

    Fractional diffusion can greatly improve or worsen inverse-problem ill-posedness, depending crucially on the unknown and the supplied data.

  • Takeaways & Limitations

    The effect of anomalous diffusion on reconstruction conditioning is problem-specific rather than uniformly beneficial or detrimental.

  • Takeaways & Limitations

    Rigorous theory remains sparse, especially for space-fractional equations, and stable efficient reconstruction remains challenging because of nonlocality.

Abstract

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Over the last two decades, anomalous diffusion processes in which the mean squares variance grows slower or faster than that in a Gaussian process have found many applications. At a macroscopic level, these processes are adequately described by fractional differential equations, which involves fractional derivatives in time or/and space. The fractional derivatives describe either history mechanism or long range interactions of particle motions at a microscopic level. The new physics can change dramatically the behavior of the forward problems. Naturally one expects that the new physics will impact related inverse problems in terms of uniqueness, stability, and degree of ill-posedness. The last aspect is especially important from a practical point of view, i.e., stably reconstructing the quantities of interest. In this paper, we employ a formal analytic and numerical way to examine the degree of ill-posedness of several "classical" inverse problems for fractional differential equations involving a Djrbashian-Caputo fractional derivative in either time or space, which represent the fractional analogues of that for classical integral order differential equations. We discuss four inverse problems, i.e., backward fractional diffusion, sideways problem, inverse source problem and inverse potential problem for time fractional diffusion, and inverse Sturm-Liouville problem, Cauchy problem, backward fractional diffusion and sideways problem for space fractional diffusion. It is found that contrary to the wide belief, the influence of anomalous diffusion on the degree of ill-posedness is not definitive: it can either significantly improve or worsen the conditioning of related inverse problems, depending crucially on the specific type of given data and quantity of interest. Further, the study exhibits distinct new features of "fractional" inverse problems.

1. Introduction

Anomalous diffusion models use fractional derivatives to represent non-Gaussian particle dynamics, creating inverse problems whose conditioning can differ substantially from classical diffusion. This paper formally and numerically examines that dependence across several fractional inverse problems.

  • 1. Introduction: Anomalous diffusion can be subdiffusive or superdiffusive, with applications including viscoelastic materials and heterogeneous media.These processes are associated microscopically with heavy-tailed waiting times or long-range particle jumps.
  • 1. Introduction: Fractional derivatives introduce nonlocal operators, so standard PDE tools such as product rules and integration by parts may fail or require modification.This nonlocality complicates both analysis and numerical treatment of the forward model.
  • 1. Introduction: Fractional model parameters and other quantities often must be inferred indirectly from boundary or interior measurements, producing diverse inverse problems.Unknowns may include fractional orders, coefficients, initial conditions, source terms, boundary conditions, and domain geometry.
  • 1. Introduction: The study examines time- and space-fractional analogues of classical inverse problems using formal analytic and numerical comparisons with Gaussian diffusion.The examples include backward diffusion, sideways diffusion, inverse source and potential problems, inverse Sturm-Liouville problems, and Cauchy problems.
  • 1. Introduction: Fractional nonlocality can either greatly improve or worsen ill-posedness, depending crucially on the unknown and the data supplied.The paper therefore rejects a definitive uniform effect of anomalous diffusion on inverse-problem conditioning.

2. Preliminaries

The preliminaries develop the special functions and spectral tools used to understand fractional diffusion and its inverse problems. They emphasize slower-than-exponential fractional decay, long-tailed fundamental solutions, and singular-value diagnostics for conditioning.

  • 2. Preliminaries: The paper introduces the Mittag-Leffler and Wright functions together with singular value decomposition for analyzing discrete ill-posed problems.These tools support both fractional diffusion solution representations and numerical conditioning analysis.
  • 2.1. Mittag-Leffler function.: The Mittag-Leffler function generalizes the exponential, with E1,1(z) = e^z, and appears in fractional diffusion solution representations.The functions Eα,1(−λtα) and tα−1Eα,α(−λtα) are specifically identified as solution components.
  • 2.1. Mittag-Leffler function.: Mittag-Leffler decay on the negative real axis is only linear, much slower than exponential decay, while positive-axis growth is exponential.The paper plots Eα,1(−π^2t^α) and t^α−1Eα,α(−π^2t^α) using the first Dirichlet eigenvalue π^2.
  • 2.1. Mittag-Leffler function.: At t = 1, Eα,1(−π^2t) remains distinctly nonzero for 0 < α < 1, whereas e^−π^2t nearly vanishes; near t = 0, the ordering reverses.This demonstrates the markedly different temporal behavior of the Mittag-Leffler and exponential functions.
  • 2.2. Wright function.: For 0 < α < 1, the time-fractional fundamental solution has polynomial long tails and is continuous but not differentiable at x = 0.Compared with the Gaussian heat kernel, these properties imply only limited smoothing by the time-fractional solution operator.
  • 2.3. Singular value decomposition.: Singular values quantify discrete conditioning: a gradually decaying spectrum toward zero is characteristic of many discrete ill-posed problems.The paper uses singular value decomposition and related regularization tools to analyze inverse-problem behavior.

3. Inverse problems for time fractional diffusion

Time-fractional inverse problems exhibit problem-dependent conditioning: fractional backward diffusion can be mildly ill-posed, sideways recovery remains theoretically severe but may be numerically easier, and inverse-source recovery shows mild ill-posedness. Numerical behavior depends strongly on fractional order, terminal time, frequency cutoff, and data type.

  • Backward fractional diffusion: The classical backward diffusion problem amplifies the jth Fourier-mode perturbation by e^(λ_jT), whereas fractional diffusion amplifies it only linearly in λ_j.The fractional multiplier behaves as 1/E_α,1(-λ_jT^α) ~ λ_j, corresponding to a two-derivative loss.
  • Backward fractional diffusion: Numerically, the backward fractional diffusion condition number stays near O(10^4) across broad α ranges, then blows up as α approaches one.For fixed T, the condition number is nonmonotone in α, and smaller terminal times cause quicker blowup.
  • Backward fractional diffusion: For T = 0.001, the first twenty classical singular values exceed the fractional counterparts, making those modes more stable to reconstruct.Thus theoretical mild ill-posedness does not guarantee better finite-dimensional reconstruction; spectral cutoff and singular-value distribution matter.
  • Sideways fractional diffusion: The sideways problem is severely ill-posed theoretically, yet decreasing α substantially slows numerical deterioration and can make the discretized problem nearly well-posed.At T = 1, a transition occurs around α = 1/2: conditioning deteriorates sharply above it but is nearly well-posed below it.
  • Sideways fractional diffusion: The sideways forward map has a triangular, causal Volterra structure that can guide reconstruction methods such as Lavrentiev regularization.The map relates the Dirichlet boundary condition at x = 1 to the flux boundary condition at x = 0.

4. Inverse problems for space fractional diffusion

Space fractional inverse problems display markedly different uniqueness and conditioning behavior across derivative choices, data configurations, and problem types. Numerical and analytical results show both improved and worsened ill-posedness relative to classical counterparts.

  • 4.1. Inverse Sturm-Liouville problem: A single complex Dirichlet spectrum can numerically determine a general potential in the Djrbashian-Caputo inverse Sturm-Liouville problem.The reconstruction uses a frozen Newton method and Fourier representation of the potential.
  • 4.1. Inverse Sturm-Liouville problem: The Riemann-Liouville spectrum reconstructs only the symmetric part of a general potential, unless the potential is known on the left half interval.With left-half knowledge, the remaining half can be uniquely reconstructed; thus the complex spectrum is not more informative than in the classical problem.
  • 4.2. Cauchy problem for fractional elliptic equation: For the fractional elliptic Cauchy problem, exponentially decaying boundary data correspond to exponentially growing solutions, indicating severe exponential ill-posedness.The comparison with the classical degree of ill-posedness remains unresolved, and efficient rigorous numerical solvers are lacking.
  • 4.3. Backward problem: The space fractional backward problem remains exponentially ill-posed, but slower eigenvalue growth makes it less ill-posed than the classical problem.Numerically, singular-value decay worsens as fractional order and terminal time increase, while more Fourier modes can nevertheless be recovered than in the classical setting.
  • 4.4. Sideways problem: In another sideways configuration, anomalous space diffusion severely worsens an already very ill-posed problem as β decreases toward one.Numerical singular-value spectra show slower decay for larger β but additional tiny singular values and rank deficiency as β approaches one.
  • 4.4. Sideways problem: For one sideways configuration, fractional order near one produces nearly well-posed behavior, whereas increasing β toward two sharply increases the condition number.For β ≤ 7/4, the singular values span only a narrow interval; the onset of blowup also depends on terminal time.

5. Concluding remarks

The study finds that anomalous diffusion does not uniformly improve inverse-problem conditioning: effects depend on the data and unknown, while rigorous analysis remains limited.

  • Anomalous diffusion can either improve or worsen inverse-problem ill-posedness, depending on the given data and unknown quantity.The paper examines sideways, backward, inverse source, inverse Sturm-Liouville, and Cauchy problems.
  • Rigorous inverse-problem theory remains sparse, focusing mainly on one-dimensional uniqueness, existence, and stability.Many open problems remain, and stable, efficient reconstruction is still an active research topic.
  • Nonlocal forward models make efficient schemes and rigorous numerical analysis especially challenging for space fractional equations.The paper notes that theoretical or rigorous numerical studies of space fractional FDEs are almost nonexistent.

A.1. Computation of the Mittag-Leffler and Wright functions.

The appendix discusses numerical evaluation of the Mittag-Leffler and Wright functions, emphasizing region-dependent approximations and transformations that reduce kernel singularity.

  • Mittag-Leffler evaluation partitions the complex plane and uses power series, integral representations, or exponential asymptotics by argument size.The approximations target small, intermediate, and large argument regions, respectively.
  • Wright-function evaluation similarly combines power series, integral representations, and asymptotic formulas across argument regimes.The intermediate regime uses an integral representation for the parameter range relevant to the one-dimensional fundamental solution.
  • The Wright-function kernel has successive singular terms, making direct numerical quadrature inefficient.A change of variables s = r^-ρ transforms the kernel before quadrature.
  • For the time-fractional diffusion fundamental solution, the Wright-function representation uses ρ = −α/2 and µ = 1 + ρ.With these parameters, the resulting kernel simplifies.
  • The transformed kernel is free from the severe singularity, enabling effective Gauss-Jacobi quadrature.A rigorous whole-plane algorithm for the Wright function remains unavailable.

A.2. Time fractional diffusion.

The time-fractional diffusion appendix formulates the initial-boundary problem and discretizes it with L1 time approximation and central spatial differences, while accounting for history dependence.

  • The model uses a Djrbashian-Caputo time derivative with reaction-potential term q, source f, initial value v, and Dirichlet boundary data g and h.The spatial domain is the unit interval and the evolution runs on (0,T].
  • The adopted discretization combines L1 approximation in time with the standard central difference scheme in space.The interval [0,T] is divided into uniform time subintervals and the unit spatial interval into a uniform mesh.
  • The L1 local truncation error is bounded by cτ^(2−α) for solutions twice continuously differentiable in time, while general accuracy is first order.The constant c depends only on the solution u.
  • The fully discrete method requires solving a tridiagonal linear system at each time step.The current right-hand side includes all previous time steps, creating computational expense for small step sizes.
  • The fractional history term is a central computational challenge because each time step depends on all preceding steps.Higher-order alternatives such as convolution quadrature are also noted.

A.3. Space fractional diffusion.

The space-fractional diffusion appendix uses a variational formulation and Galerkin finite elements in space, combined with backward Euler time stepping for time-dependent problems.

  • For the time-dependent problem, backward Euler time stepping is combined with finite elements in space after splitting off a boundary-data particular solution.The approximate solution is decomposed into particular and homogeneous components.
  • The space-fractional model uses a fractional spatial derivative of order β, potential q, source f, initial condition v, and Dirichlet boundary data g and h.The problem is posed on the unit interval over times up to T.
  • The Galerkin finite element method is based on a variational formulation with homogeneous Dirichlet boundary conditions.The trial space is U ≡ H^(β/2)_1, while the test space is constructed separately.
  • The formulation uses left-sided and right-sided Riemann-Liouville derivatives of order γ ∈ (0,1).These derivatives enter the fractional elliptic variational framework.
  • The finite element discretization uses a uniform mesh and continuous piecewise linear basis functions, with modified test functions satisfying an integral condition.The modification constructs V_h from U_h through constants γ_i determined by that condition.
  • The resulting linear system has lower Hessenberg structure because of fractional-derivative nonlocality, but its coefficient matrix remains unchanged during time stepping.An LU factorization can therefore accelerate the computation.
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