arXiv subjects

Dakang Cen

Publications and source records attributed to Dakang Cen.

6 recordsLinked to original sources

arXiv2026

A data-driven model reduction approach for backward fractional diffusion-wave equations

In this work, we propose an observation system based on the available data which solution is one-be-one mapping to the forward problem(with the unknown initial function) solution. It implies their solutions share the same linear structure in the finite dimensional space. Theoretical results show model reduction approaches constructed for the observation system also work well for the forward problem, which significantly improve the efficiency of solving the inverse problem. Several numerical examples are presented to support our finding.

math.NA↗

arXiv2026

Mathematical analysis and symmetric fractional-order reduction method for diffusion-wave equations

In this work, our aim is to introduce a symmetric fractional-order reduction (SFOR) method to develop numerical algorithms on nonuniform temporal meshes for fractional wave equations under lower regularity assumptions. The $L$-type methods--including $L1$ and $L2$-$1_\sigma$ schemes--are specifically designed for diffusion-wave equations, and we propose novel optimal parameter selections tailored to nonuniform meshes. Finally, several numerical experiments are conducted to validate the efficiency and accuracy of the algorithms.

math.NA↗

arXiv2025

An efficient iteration method to reconstruct the drift term from the final measurement

This work investigates the inverse drift problem in the one-dimensional parabolic equation with the final time data. The authors construct an operator first, whose fixed points are the unknown drift, and then apply it to prove the uniqueness. The proof of uniqueness contains an iteration converging to the drift, which inspires the numerical algorithm. To handle the ill-posedness of the inverse problem, the authors add the mollification on the data first in the iterative algorithm, and then provide some numerical results.

math.NA↗

arXiv2025

Numerical analysis of scattered point measurement-based regularization for backward problems for fractional wave equations

In this work, our aim is to reconstruct the unknown initial value from terminal data. We develop a numerical framework on nonuniform time grids for fractional wave equations under the lower regularity assumptions. Then, we introduce a regularization method that effectively handles scattered point measurements contaminated with stochastic noise. The optimal error estimates of stochastic convergence not only balance discretization errors, the noise, and the number of observation points, but also propose an a priori choice of regularization parameters. Finally, several numerical experiments are presented to demonstrate the efficiency and accuracy of the algorithm.

math.NA↗

arXiv2025

Scattered point measurement-based regularization for backward problems for fractional wave equations

In this work, we are devoted to the reconstruction of an unknown initial value from the terminal data. The asymptotic and root-distribution properties of Mittag-Leffler functions are used to establish stability of the backward problem. Furthermore, we introduce a regularization method that effectively handles scattered point measurements contaminated with stochastic noise. Furthermore, we prove the stochastic convergence of our proposed regularization and provide an iterative algorithm to find the optimal regularization parameter. Finally, several numerical experiments are presented to demonstrate the efficiency and accuracy of the algorithm.

math.NA↗

arXiv2025

A randomized progressive iterative regularization method for data fitting problems

In this work, we investigate data fitting problems with random noises. A randomized progressive iterative regularization method is proposed. It works well for large-scale matrix computations and converges in expectation to the least-squares solution. Furthermore, we present an optimal estimation for the regularization parameter, which inspires the construction of self-consistent algorithms without prior information. The numerical results confirm the theoretical analysis and show the performance in curve and surface fittings.

math.NA↗