Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2607.10100

Regularity and high-order time stepping for semilinear subdiffusion equations with singular initial data beyond the $L^\infty$ framework

Runjie Zhang·Dongling Wang
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper aims to analyze a numerical scheme for semilinear subdiffusion problems with singular initial data beyond the $L^\infty$ framework. The main difficulty lies in the stronger singular behavior of the nonlinear term compared with previous analyses. Since the singular initial datum is too rough to guarantee a uniform $L^\infty$ bound for the solution, the usual Lipschitz framework in the base space is no longer sufficient. The analysis must instead be carried out in weaker fractional Sobolev-type spaces, where nonlinear composition is more delicate and the term $f(u(t))$ may exhibit an amplified singularity relative to that of $u(t)$. To overcome this difficulty, we exploit the smoothing properties of the subdiffusion solution operators and formulate suitable nonlinear assumptions in fractional operator spaces. These smoothing estimates allow part of the singularity to be transferred from the nonlinear term to the solution operators, where it can be controlled. Under these assumptions, we establish well-posedness and regularity results for the mild solution and derive a pointwise-in-time error estimate for the exponential convolution quadrature method. Numerical experiments confirm the predicted convergence rates.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Runjie Zhang, Dongling Wang. 2026-07-11. Regularity and high-order time stepping for semilinear subdiffusion equations with singular initial data beyond the $L^\infty$ framework. https://arxiv.org/abs/2607.10100

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

IterativeCUR: Large Rank-Adaptive Approximation From a Small Recycled Sketch

The computation of accurate low-rank matrix approximations is central to improving the scalability of various techniques in machine learning, uncertainty quantification, and control. Traditionally, low-rank approximations are constructed using SVD-based approaches such as truncated SVD or Randomized SVD. Although these SVD approaches---especially Randomized SVD---have proven to be very computationally efficient, other low-rank approximation methods can offer even greater performance. One such approach is the CUR decomposition, which forms a low-rank approximation using direct row and column subsets of a matrix. Because CUR uses direct matrix subsets, it is also often better able to preserve native matrix structures like sparsity or non-negativity than SVD-based approaches and can facilitate data interpretation in many contexts. This paper introduces IterativeCUR, which draws on previous work in randomized numerical linear algebra to build a new algorithm that is highly competitive compared to prior work. IterativeCUR is adaptive in the sense that it takes as an input parameter the desired tolerance $ε$ and outputs (with arbitrarily high probability) an approximation of error bounded by $ε$, rather than requiring an a priori guess of the numerical rank. IterativeCUR typically runs significantly faster than both existing CUR algorithms and techniques such as Randomized SVD. Its asymptotic complexity is $\mathcal{O}(mn + (m+n)r^2)$ for an $m\times n$ matrix of output rank $r$. IterativeCUR relies on a single small sketch from the matrix that is successively downdated as the algorithm proceeds. We demonstrate through extensive experiments that IterativeCUR achieves up to $4\times$ speed-up over state-of-the-art pivoting-on-sketch approaches with no loss of accuracy, and up to $40\times$ speed-up over rank-adaptive randomized SVD approaches.

math.NA↗
arXiv2026

Multigrid with Linear Storage Complexity

As the discretization error for the solution of a partial differential equation (PDE) decreases, the precision required to store the corresponding coefficients naturally increases. Storing the solution's finite element coefficients explicitly requires $\mathcal O(n \log n)$ bits of storage, where $n$ is the number of degrees of freedom (DoFs). This paper presents a full multigrid method to compute the solution in a compressed format that reduces the storage complexity of the solution and intermediate vectors to $\mathcal O(n)$ bits. This reduction allows a matrix-free implementation to solve elliptic PDEs with an overall linear space complexity. For problems limited by the memory capacity of current supercomputers, we expect a memory footprint reduction of about an order of magnitude compared to state-of-the-art mixed-precision methods. We demonstrate the applicability of our algorithm by solving two model problems. Depending on the PDE and polynomial degree, but irrespective of the problem size, the solution vector on the finest grid requires between 4 and 12 bits per DoF, and the residual and correction require 3 to 6 bits each. Additional data is stored on the coarse grids with modestly increasing bit widths toward coarser grids.

math.NA↗
arXiv2026

ELIPPS: Exact Learning for Inverse Problems from Partial Self-supervision

In undersampled inverse problems (such as sparse-view computed tomography), only a small number of measurements are collected, which reduces radiation exposure and acquisition time and cost, and can also address the inaccessibility of certain acquisition arrangements. Most learning-based methods for such problems require supervision in the form of fully sampled measurements and ground-truth images, which are costly or even infeasible to acquire. To overcome this issue, we propose \emph{Exact Learning for Inverse Problems from Partial Self-supervision} (ELIPPS), an incomplete self-supervised training paradigm for undersampled inverse problems that requires neither ground-truth images nor fully sampled measurements. ELIPPS learns solely from incomplete forward measurements on a fixed incomplete supervision set. Our theory shows that when the data distribution is invariant under certain transformations, minimizing a masked empirical risk is equivalent to minimizing the full self-supervised risk, i.e. the risk against the complete, noise-free measurement. The equivalence is exact for arbitrary equivariant hypothesis classes and holds for signal-dependent noise such as the pre-log Poisson statistics of low-dose tomography, and for the log-transformed count model up to a quantifiable bias. When the underlying coverage condition is only approximately satisfied on a discrete grid, we give a stability estimate in terms of the associated frame constants and the irreducible error of the inverse problem. We realize ELIPPS for computed tomography by exploiting rotation and reflection invariance. In our experiments, ELIPPS substantially outperforms naive masked supervision and reaches the same order of accuracy as a reference model trained with full clean measurements.

math.NA↗