arXiv subjects

Dmitrii Chaikovskii

Publications and source records attributed to Dmitrii Chaikovskii.

7 recordsLinked to original sources

arXiv2026

Hyb-Adam-UM: hybrid ultrametric-aware mtDNA phylogeny reconstruction

Motivation: mtDNA distance matrices are standard inputs for distance-based phylogeny, but computing all pairwise alignments is costly. Missing entries can degrade inferred topology and branch lengths, and generic matrix-completion methods may disrupt tree-like (ultrametric) structure. Results: We propose Hyb-Adam-UM, which starts from an alignment-limited Needleman-Wunsch distance backbone and completes the matrix by minimizing a robust triplet ultrametric-violation functional. An Adam-style finite-difference optimizer updates only missing entries while enforcing symmetry, non-negativity, and a zero diagonal. From one complete reference matrix, we generate 20 masked instances at 30%, 50%, 65%, and 85% missingness. Hyb-Adam-UM consistently reduces ultrametric violations and achieves competitive reconstruction error, with improved topological accuracy and branch-length agreement relative to MW*/NJ* projection baselines (which exactly preserve observed distances) and Soft-Impute; gains are most pronounced at 85% missingness. Availability and implementation: https://github.com/mitichya/hyb-adam-um/; Zenodo: https://doi.org/10.5281/zenodo.18609748 Supplementary information: Supplementary data available online.

q-bio.PE↗

arXiv2025

Deep asymptotic expansion method for solving singularly perturbed time-dependent reaction-advection-diffusion equations

Physics-informed neural network (PINN) has shown great potential in solving partial differential equations. However, it faces challenges when dealing with problems involving steep gradients. The solutions to singularly perturbed time-dependent reaction-advection-diffusion equations exhibit internal moving transition layers with sharp gradients, and thus the standard PINN becomes ineffective. In this work, we propose a deep asymptotic expansion (DAE) method, which is inspired by asymptotic analysis and leverages deep learning to approximate the smooth part of the expansion. We first derive the governing equations for transition layers, which are then solved using PINN. Numerical experiments show that the DAE outperforms the standard PINN, gPINN, and PINN with adaptive sampling. We also show its robustness with respect to training point distributions, network architectures, and random seeds.

math.NA↗

arXiv2025

Internal layer solutions and coefficient recovery in time-periodic reaction-diffusion-advection equations

This article investigates the non-stationary reaction-diffusion-advection equation, emphasizing solutions with internal layers and the associated inverse problems. We examine a nonlinear singularly perturbed partial differential equation (PDE) within a bounded spatial domain and an infinite temporal domain, subject to periodic temporal boundary conditions. A periodic asymptotic solution featuring an inner transition layer is proposed, advancing the mathematical modeling of reaction-diffusion-advection dynamics. Building on this asymptotic analysis, we develop a simple yet effective numerical algorithm to address ill-posed nonlinear inverse problems aimed at reconstructing coefficient functions that depend solely on spatial or temporal variables. Conditions ensuring the existence and uniqueness of solutions for both forward and inverse problems are established. The proposed method's effectiveness is validated through numerical experiments, demonstrating high accuracy in reconstructing coefficient functions under varying noise conditions.

math.NA↗

arXiv2025

Asymptotic solution for three-dimensional reaction-diffusion-advection equation with periodic boundary conditions

In this study, we investigate the dynamics of moving fronts in three-dimensional spaces, which form as a result of in-situ combustion during oil production. This phenomenon is also observed in other contexts, such as various autowave models and the propagation of acoustic waves. Our analysis involves a singularly perturbed reaction-diffusion-advection type initial-boundary value problem of a general form. We employ methods from asymptotic theory to develop an approximate smooth solution with an internal layer. Using local coordinates, we focus on the transition layer, where the solution undergoes rapid changes. Once the location of the transition layer is established, we can describe the solution across the full domain of the problem. Numerical examples are provided, demonstrating the high accuracy of the asymptotic method in predicting the behaviors of moving fronts.

math.AP↗

arXiv2023

Solving forward and inverse problems in a non-linear 3D PDE via an asymptotic expansion based approach

This paper concerns the use of asymptotic expansions for the efficient solving of forward and inverse problems involving a nonlinear singularly perturbed time-dependent reaction--diffusion--advection equation. By using an asymptotic expansion with the local coordinates in the transition-layer region, we prove the existence and uniqueness of a smooth solution with a sharp transition layer for a three-dimensional partial differential equation. Moreover, with the help of asymptotic expansion, a simplified model is derived for the corresponding inverse source problem, which is close to the original inverse problem over the entire region except for a narrow transition layer. We show that such simplification does not reduce the accuracy of the inversion results when the measurement data contain noise. Based on this simpler inversion model, an asymptotic-expansion regularization algorithm is proposed for efficiently solving the inverse source problem in the three-dimensional case. A model problem shows the feasibility of the proposed numerical approach.

math.NA↗

arXiv2022

Asymptotic expansion regularization for inverse source problems in two-dimensional singularly perturbed nonlinear parabolic PDEs

In this paper, we develop an asymptotic expansion-regularization (AER) method for inverse source problems in two-dimensional nonlinear and nonstationary singularly perturbed partial differential equations (PDEs). The key idea of this approach is the use of the asymptotic-expansion theory, which allows us to determine the conditions for the existence and uniqueness of a solution to a given PDE with a sharp transition layer. As a by-product, we derive a simpler link equation between the source function and first-order asymptotic approximation of the measurable quantities, and based on that equation we propose an efficient inversion algorithm, AER, for inverse source problems. We prove that this simplification will not decrease the accuracy of the inversion result, especially for inverse problems with noisy data. Various numerical examples are provided to demonstrate the efficiency of our new approach.

math.NA↗

arXiv2022

Convergence analysis for forward and inverse problems in singularly perturbed time-dependent reaction-advection-diffusion equations

In this paper, by employing the asymptotic expansion method, we prove the existence and uniqueness of a smoothing solution for a time-dependent nonlinear singularly perturbed partial differential equation (PDE) with a small-scale parameter. As a by-product, we obtain an approximate smooth solution, constructed from a sequence of reduced stationary PDEs with vanished high-order derivative terms. We prove that the accuracy of the constructed approximate solution can be in any order of this small-scale parameter in the whole domain, except a negligible transition layer. Furthermore, based on a simpler link equation between this approximate solution and the source function, we propose an efficient algorithm, called the asymptotic expansion regularization (AER), for solving nonlinear inverse source problems governed by the original PDE. The convergence-rate results of AER are proven, and the a posteriori error estimation of AER is also studied under some a priori assumptions of source functions. Various numerical examples are provided to demonstrate the efficiency of our new approach.

math.NA↗