Source-linked AI summary
Inverse Source Problem for a Time-Fractional Diffusion-Wave Equation with a Singular Inverse-Square Potential
Zewen Wang, Bin Wu, Yikan Liu, Liangwei Jin, Shufang Qiu
TL;DR
The paper studies recovery of an unknown spatial source from terminal measurements in a time-fractional diffusion-wave equation with a singular inverse-square potential. It establishes the forward and identifiability theory, then combines Tikhonov regularization with an adjoint-based conjugate gradient method; numerical experiments show accurate and stable reconstructions across the tested settings.
Problem
The work addresses recovery of a spatial source from terminal data when fractional diffusion-wave dynamics include a singular inverse-square potential, a setting less developed than regular-coefficient or subdiffusive cases.
Method
The paper uses Hardy inequalities and spectral analysis for the forward problem, then applies Tikhonov regularization with an adjoint involving a right-sided Riemann–Liouville derivative and conjugate-gradient reconstruction.
Results
The terminal observation operator is compact, the spatial source is unique under a modal nondegeneracy condition, and numerical tests produce accurate and stable reconstructions in one- and two-dimensional settings.
Takeaways & Limitations
The proposed framework supports source identification for smooth and nonsmooth profiles across different fractional orders and singular potentials, including exact and noisy terminal data.
Takeaways & Limitations
Future work is needed for more general geometries, variable coefficients, other incomplete measurements, and more efficient large-scale numerical schemes.
Abstract
from arXiv · showhide
This paper investigates an inverse source problem for a time-fractional diffusion-wave equation with a singular inverse-square potential. The source term is assumed to consist of a known temporal factor and an unknown spatial component, which is to be recovered from terminal-state measurements. The well-posedness and regularity of the forward problem are established within an appropriate energy framework by exploiting Hardy-type inequalities and the spectral properties of the associated singular elliptic operator. The terminal observation operator is then shown to be compact, and uniqueness of the spatial source is established under a suitable nondegeneracy condition on the temporal factor. To stabilize the resulting ill-posed inverse problem, a Tikhonov regularization approach is introduced. The gradient of the regularized functional is derived through an adjoint problem involving a right-sided fractional derivative, leading to an adjoint-based conjugate gradient method with an exact line search for the numerical reconstruction of the unknown source. Numerical experiments are conducted on both one-and two-dimensional spatial domains, using both exact and noisy terminal data, to demonstrate the effectiveness and stability of the proposed source reconstruction method.
1. Introduction
The paper addresses recovery of a spatial source in a time-fractional diffusion-wave equation with a singular inverse-square potential, extending analysis beyond settings with regular coefficients or subdiffusion. It establishes the model, identifies the diffusion-wave-specific analytical challenges, and outlines the proposed reconstruction framework.
- Inverse source problems recover unobserved forcing terms from indirect measurements but are typically ill-posed because forward evolution smooths information.
- Prior terminal-time source-recovery studies used spectral regularization, Tikhonov minimization, quasi-boundary methods, and iterative algorithms.
- Inverse-square potentials form a critical singular class whose analysis relies on Hardy inequalities and threshold conditions.
- Most related fractional inverse-source results concern regular spatial operators, classical parabolic equations, or subdiffusion rather than the diffusion-wave regime.
- For fractional orders between one and two, the problem requires two initial conditions, an initial-velocity term, and a different fractional adjoint structure.
- The paper recovers a separable source's spatial factor from terminal-state data while restricting the singular coefficient to the subcritical Hardy regime.
2. Model and Preliminaries
The model uses a time-fractional diffusion-wave equation with known temporal source factor and unknown spatial factor observed through the terminal state. Hardy inequalities define the energy framework and support the associated singular elliptic operator's spectral analysis.
- 2.1. Problem setting: The forward model is posed on Ω = (0, 1) over Ω × (0, T), with a left-sided Caputo derivative of order α.
- 2.1. Problem setting: The inverse problem assumes q ∈ L∞(0, T) is known, p ∈ L2(Ω) is unknown, and p is recovered from the terminal state.
- 2.2. Hardy energy space and the singular elliptic operator: The inverse-square coefficient is assumed below the critical Hardy constant, ensuring continuity and coercivity of the spatial bilinear form.
- 2.2. Hardy energy space and the singular elliptic operator: In the subcritical regime µ < 1/4, the Hardy energy space coincides with H1_0(Ω) as a set, with equivalent norms.
- 2.2. Hardy energy space and the singular elliptic operator: The coercive closed form generates a unique positive self-adjoint operator Lµ with homogeneous Dirichlet boundary condition.
- 2.2. Hardy energy space and the singular elliptic operator: Compact resolvent yields an orthonormal eigenfunction basis of L2(Ω), providing the spectral structure used in the analysis.
3. Forward Problem: Well-Posedness and Regularity
The forward equation is formulated through a spatially weak solution and solved spectrally using the singular operator's eigenfunctions and Mittag–Leffler functions. Under the stated assumptions, the solution exists uniquely with the regularity needed to define terminal observations.
- 3.1. Weak solution and spectral representation: A weak solution is defined using spatial testing, with u ∈ C([0,T];H1(Ω)) ∩ C1([0,T];H−1(Ω)) and the prescribed initial data.
- 3.1. Weak solution and spectral representation: The spatial weak form characterizes the evolution equation, while spectral representation establishes existence and regularity.
- 3.1. Weak solution and spectral representation: Expanding in Lµ eigenfunctions reduces each Fourier coefficient to a scalar fractional initial-value problem and yields operator-valued Mittag–Leffler representations.
- 3.2. Existence, uniqueness, and regularity: For 1 < α < 2, p ∈ L2(Ω), q ∈ L∞(0,T), u0 ∈ H1(Ω), and u1 ∈ L2(Ω), the problem has a unique weak solution.
- 3.2. Existence, uniqueness, and regularity: The kernel K belongs to L1(0,T; L(L2,H1)), supporting continuity arguments for the source contribution.
- 3.2. Existence, uniqueness, and regularity: The modal homogeneous problem has only the zero solution, and completeness of the eigenfunctions gives uniqueness of the full solution.
- 3.2. Existence, uniqueness, and regularity: The terminal state u(·,T) is well defined in H1(Ω) for every p ∈ L2(Ω), supplying the source-to-terminal map for inversion.
4. Inverse Source Problem and Uniqueness
The inverse problem recovers an unknown spatial source factor from a source-induced terminal response. Uniqueness follows when every spatial eigenmode remains observable through a nonzero temporal multiplier.
- Problem formulation: The source-to-terminal operator isolates the response generated by the unknown spatial factor after subtracting the zero-source solution.The forward map is affine because initial data may be nonzero; subtracting u[0] yields a linear operator A from L2(Ω) into L2(Ω).
- Spectral characterization: The terminal observation determines each Fourier coefficient of the source through the modal multiplier Kn(q).The multiplier represents the response of the nth spatial eigenmode to the prescribed temporal excitation at time T.
- Uniqueness: Under Kn(q) ≠ 0 for every n, known initial data and temporal factor imply at most one spatial source p ∈ L2(Ω).The proof uses equality of terminal observations, the spectral representation, and completeness of the eigenfunctions.
- Uniqueness: If one modal multiplier vanishes, the corresponding source component cannot be detected from the terminal measurement, so uniqueness fails in that mode.The nondegeneracy condition must therefore be checked for the prescribed temporal excitation rather than inferred solely from a sign assumption on q.
- Special case q ≡ 1: For q(t) ≡ 1, the exceptional set of observation times is at most countable, and uniqueness holds for every T outside that set.This conclusion follows from the isolated zeros of the relevant Mittag–Leffler expressions and the spectral condition.
5. Tikhonov regularization and conjugate gradient reconstruction
The paper formulates the ill-posed terminal-data inverse source problem as Tikhonov minimization and derives an adjoint-based conjugate gradient reconstruction method. The adjoint uses right-sided fractional operators, while exact line search and residual updates support the iterative algorithm.
- Operator formulation: The inverse problem is recast through a linear source-to-terminal operator after subtracting the known zero-source terminal contribution.The resulting formulation isolates the unknown spatial source contribution in terminal observations.
- Compactness and ill-posedness: A is compact because it maps L2(Ω) boundedly into H1(Ω), whose embedding into L2(Ω) is compact.Consequently, uniqueness does not prevent instability when terminal data are perturbed.
- Tikhonov regularization: The Tikhonov functional has a unique minimizer because its quadratic penalty makes it coercive and strictly convex.The regularization parameter satisfies λ > 0.
- Adjoint problem and gradient: Right-sided Riemann–Liouville fractional operators arise through fractional integration by parts to formulate the adjoint and obtain an explicit L2(Ω)-gradient.The terminal residual enters the adjoint through fractional boundary terms, and the fractional terminal conditions should not be replaced without justification by classical ones.
- Conjugate gradient reconstruction: The reconstruction uses an adjoint-based conjugate gradient iteration with Fletcher–Reeves search directions and an exact line-search step length.Linearity allows terminal residual updates using the computed sensitivity terminal state, avoiding an additional forward solve solely for that update.
- Conjugate gradient reconstruction: The algorithm alternates forward and adjoint solves, computes the gradient and search direction, updates the source and residual, and stops by a gradient tolerance or discrepancy principle.The procedure outputs the reconstructed source after the stopping criterion is satisfied.
6. Numerical experiments
Numerical experiments test the reconstruction method on one- and two-dimensional problems with smooth and nonsmooth sources, exact and noisy terminal data, and varied fractional parameters. The results report stable, accurate recovery, including under noise and in the presence of singular potentials.
- Numerical method: The forward and adjoint problems are discretized using L1 time stepping, finite differences in space, time reversal, lifting, and LU-based linear solves.The adjoint's right-sided fractional derivative is converted into a left-sided Caputo derivative before applying the same L1 discretization.
- Numerical method: The reconstruction uses relative L2(Ω) source error, multiplicative random perturbations for noisy data, and discrepancy-based stopping to avoid overfitting.Exact-data iterations instead use a relative-gradient stopping criterion.
- Example 2: Example 2 tests a piecewise linear L2(Ω) source that is nonsmooth at x = 0.5, with q(t) = 1 satisfying the modal nondegeneracy condition.The source is reconstructed from numerically generated terminal data because no analytical closed-form solution is available.
- Two-dimensional experiment: The two-dimensional experiment produces good agreement with the true source for (α, µ) = (1.20, 0.15) and (1.60, 3.3), supporting the method's effectiveness in the spatially extended setting.The two-dimensional analysis relies on the higher-dimensional Hardy inequality under the stated subcritical condition µ < 4.
- Overall findings: Across all three examples, reconstructions are accurate and stable for one- and two-dimensional domains, smooth and nonsmooth sources, and different fractional orders and singular potentials.The two-dimensional extension is theoretically justified for rectangular settings under the specified Hardy-inequality condition.
7. Conclusion
The paper establishes a framework for identifying separable spatial sources in a time-fractional diffusion-wave equation with a singular inverse-square potential, then stabilizes reconstruction from terminal data. Numerical tests support accurate and stable recovery, while future work targets broader settings and more efficient schemes.
- Hardy-type inequalities and spectral properties of the singular elliptic operator establish forward well-posedness in an appropriate energy space.
- A modal nondegeneracy condition on the temporal factor guarantees uniqueness by ensuring each eigenmode is detectable from terminal measurements.
- Tikhonov regularization and an adjoint-based conjugate gradient method with exact line search address the ill-posed reconstruction problem.
- Experiments with exact and noisy terminal data across one- and two-dimensional domains demonstrate accurate and stable source recovery.
- Future work will consider more general geometries, variable coefficients, incomplete measurements, and more efficient large-scale numerical schemes.