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Ill-Posedness in Limited Discrete Fourier Inversion and Regularization for Quasi Distributions in LaMET
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abstract
We systematically investigated the limited inverse discrete Fourier transform of the quasi distributions from the perspective of inverse problem theory. This transformation satisfies two of Hadamard's well-posedness criteria, existence and uniqueness of solutions, but critically violates the stability requirement, exhibiting exponential sensitivity to input perturbations. To address this instability, we implemented Tikhonov regularization with L-curve optimized parameters, demonstrating its validity for controlled toy model studies and real lattice QCD results of quasi distribution amplitudes. The reconstructed solutions is consistent with the physics-driven $\lambda$-extrapolation method. Our analysis demonstrates that the inverse Fourier problem within the large-momentum effective theory (LaMET) framework belongs to a class of moderately tractable ill-posed problems, characterized by distinct spectral properties that differ from those of more severely unstable inverse problems encountered in other lattice QCD applications. Tikhonov regularization establishes a rigorous mathematical framework for addressing the underlying instability, enabling first-principles uncertainty quantification without relying on ansatz-based assumptions.
Forward citations
Cited by 7 Pith papers
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Reconstructing the full kinematic dependence of GPDs from pseudo-distributions
Lattice QCD pseudo-distributions at m_π=358 MeV are inverted via multidimensional Gaussian process regression to reconstruct the full kinematic dependence of GPDs H^{u-d} and E^{u-d} while directly extracting double d...
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Revisiting Quark Confinement in the Proton through the Force on Quarks
Using light-cone sum-rule input and Tikhonov-regularized inversion, the paper reconstructs the quark confining force in the proton and confirms an attractive, approximately linear-potential force at intermediate distances.
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Solving the Inverse Source Problem in Femtoscopy with a Toy Model
Tikhonov regularization reconstructs the input Gaussian source function from correlation functions generated by a square-well toy model in femtoscopy.
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Extracting Mellin moments of double parton distributions from lattice data
The skewness dependence of hadronic correlation functions affects Mellin moment extraction for double parton distributions from existing lattice data, as quantified through several models.
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Extracting Mellin moments of double parton distributions from lattice data
Skewness dependence in hadronic matrix elements must be accounted for when extracting Mellin moments of double parton distributions from lattice QCD, and model studies show its quantitative impact on current data.
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Kinematic enhancement for nucleon interpolators
Kinematically enhanced nucleon interpolators improve precision of renormalized quark matrix elements by an order of magnitude at 2.5 GeV with no observed lattice spacing dependence on CLS ensembles.
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Comment on "LaMET's Asymptotic Extrapolation vs. Inverse Problem"
The paper defends the inverse-problem view of LaMET reconstructions and argues that rigid parametric extrapolations underestimate PDF uncertainties when lattice data are noisy.
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