arXiv subjects

Ankik Kumar Giri

Publications and source records attributed to Ankik Kumar Giri.

At least 19 recordsLinked to original sources

arXiv2026

On the well-posedness and long-time dynamics of a discrete coagulation--annihilation system

The present article investigates the well--posedness of a discrete coagulation--annihilation model originally introduced by E. Ben-Naim and P. Krapivsky in Physical Review E (1995). We first establish the existence of solutions for a broad class of coagulation and annihilation rates, including rates that may be unbounded. These results are obtained by combining uniform estimates with Helly's selection theorem and the Rellich--Kondrachov compact embedding theorem. Under appropriate structural assumptions on these rates, we further prove uniqueness and continuous dependence of solutions on the initial data, thereby establishing well--posedness of the model. Finally, for a specific class of coagulation and annihilation rates, we additionally study the differentiability of solutions and characterize their long--time asymptotic behaviour. These results provide a rigorous mathematical framework for the analysis of the model and contribute to a deeper understanding of its well--posedness, regularity, and long-time dynamics.

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arXiv2026

Graph Laplacian assisted regularization method under noise level free heuristic and statistical stopping rule

In this work, we address the solution of both linear and nonlinear ill-posed inverse problems by developing a novel graph-based regularization framework, where the regularization term is formulated through an iteratively updated graph Laplacian. The proposed approach operates without prior knowledge of the noise level and employs two distinct stopping criteria namely, the heuristic rule and the statistical discrepancy principle. To facilitate the latter, we utilize averaged measurements derived from multiple repeated observations. We provide a detailed convergence analysis of the method in statistical prospective, establishing its stability and regularization properties under both stopping strategies. The algorithm begins with the computation of an initial reconstruction using any suitable techniques like Tikhonov regularization (Tik), filtered back projection (FBP) or total variation (TV), which is used as the foundation for generating the initial graph Laplacian. The reconstruction is made better step by step using an iterative process, during which the graph Laplacian is dynamically re-calibrated to reflect how the solution's structure is changing. Finally, we present numerical experiments on X-ray Computed Tomography (CT) and phase retrieval CT, demonstrating the effectiveness and robustness of the proposed method and comparing its reconstruction performance under both stopping rules.

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arXiv2026

Randomized Krylov-Projected Iterated Tikhonov Regularization for Large-Scale Ill-posed Problems Under A Posteriori Stopping Rule

We introduce two novel randomized iterative regularization frameworks, termed \texttt{RIGKT} and \texttt{RIAT}, for solving large-scale linear ill-posed inverse problems governed by systems of equations. The proposed methods combine randomized iterated Tikhonov regularization with Krylov subspace projection techniques, utilizing Golub--Kahan bidiagonalization for general rectangular systems (\texttt{RIGKT}) and Arnoldi decomposition for square systems (\texttt{RIAT}). Unlike existing deterministic schemes that rely on fixed iteration counts, our framework incorporates randomized equation selection, an adaptive step-size strategy, and a global, discrepancy-based a posteriori early-stopping rule tailored specifically to the stochastic setting. We present a comprehensive regularization analysis establishing Bregman-distance monotonicity, finite termination, exact-data convergence, and pathwise stability under noise. Furthermore, we prove that the stopped iterates converge almost surely and in the mean-square sense to the true solution, establishing a rigorous regularization property. To the best of our knowledge, this is the first theoretical framework to simultaneously account for randomization, Krylov-subspace dimension reduction, and implementable early stopping. Numerical experiments involving two-dimensional X-ray computed tomography (CT) and image deblurring demonstrate that \texttt{RIGKT} and \texttt{RIAT} reliably reconstruct structural features across various noise regimes.

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arXiv2026

On the convergence of an adaptive denoiser driven iterative regularization with early stopping

Solving inverse problems requires appropriate regularization techniques to ensure well-posedness and stability. In recent years, denoiser-driven methods have emerged as effective regularization strategies, achieving state-of-the-art performance in various imaging applications. However, their stability and convergence within iterative regularization frameworks remain largely unexplored. In this work, we extend the framework of Regularization by Denoising (RED) by introducing a novel denoiser-driven iterative regularization scheme, referred to as \texttt{DDIR}, that incorporates a new regularization functional based on averaged denoisers. The proposed approach employs an adaptive step-size strategy together with an \emph{a posteriori} stopping rule to ensure stability while alleviating oscillatory behavior and semi-convergence effects induced by noise. As our main theoretical contribution, we prove that the resulting reconstruction method constitutes a stable and convergent regularization scheme in the classical sense. To the best of our knowledge, this provides the first rigorous justification of \texttt{DDIR} within the framework of regularization theory. Finally, we demonstrate the performance of the proposed method through numerical experiments on image deblurring and phase retrieval Computed Tomography (CT) using three denoisers, namely median, TNRD, and TV proximal. The results highlight the effectiveness of the method in terms of reconstruction accuracy and computational efficiency.

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arXiv2025

On the convergence of iterative regularization method assisted by the graph Laplacian with early stopping

We present a data-assisted iterative regularization method for solving ill-posed inverse problems. The proposed approach, termed \texttt{IRMGL+$Ψ$}, integrates classical iterative techniques with a data-driven regularization term realized through an iteratively updated graph Laplacian. Our method commences by computing a preliminary solution using any suitable reconstruction method, which then serves as the basis for constructing the initial graph Laplacian. The solution is subsequently refined through an iterative process, where the graph Laplacian is simultaneously recalibrated at each step to effectively capture the evolving structure of the solution. A key innovation of this work lies in the formulation of this iterative scheme and the rigorous justification of the classical discrepancy principle as a reliable early stopping criterion specifically tailored to the proposed method. Under standard assumptions, we establish stability and convergence results for the scheme when the discrepancy principle is applied. Furthermore, we demonstrate the robustness and effectiveness of our method through numerical experiments utilizing four distinct initial reconstructors $Ψ$: the adjoint operator (Adj), filtered back projection (FBP), total variation (TV) denoising, and standard Tikhonov regularization (Tik). It is observed that \texttt{IRMGL+Adj} demonstrates a distinct advantage over the other initializers, producing a robust and stable approximate solution directly from a basic initial reconstruction.

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arXiv2025

Hanke-Raus heuristic rule for iteratively regularized stochastic gradient descent

Over the past decade, stochastic algorithms have emerged as scalable and efficient tools for solving large-scale ill-posed inverse problems by randomly selecting subsets of equations at each iteration. However, due to the ill-posedness and measurement noise, these methods often suffer from oscillations and semi-convergence behavior, posing challenges in achieving stable and accurate reconstructions. This study proposes a novel variant of the stochastic gradient descent (SGD) approach, the iteratively regularized stochastic gradient descent (IRSGD) to address nonlinear ill-posed problems in Hilbert spaces. Under standard assumptions, we demonstrate that the mean square iteration error of the method tends to zero for exact data. In the presence of noisy data, we first propose a heuristic parameter choice rule (HPCR) and then apply the IRSGD method in combination with HPCR. Precisely, HPCR selects the regularization parameter without requiring any a-priori information of the noise level. We show that the method terminatesinfinitelymanystepsincaseofnoisydataandhasregularizingfeatures. Finally, some numerical experiments are performed to demonstrate the practical efficacy of the method.

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arXiv2025

Well-posedness and large time behavior of a size-structured growth-coagulation-fragmentation model

The existence and uniqueness of weak solutions to a size-structured growth-coagulation-fragmentation (GCF) equation with a renewal boundary condition are shown for a class of unbounded coagulation and fragmentation kernels. The existence proof is based on a weak compactness framework in the weighted $L^1$-space. This result extends the existence results of Banasiak and Lamb [14] and Ackleh et al. [2,4]. Furthermore, we establish a stability result and derive uniqueness as a direct consequence of it. Moreover, this study explores the large time behavior of weak solutions.

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arXiv2025

Mass-conserving weak solutions to the continuous nonlinear fragmentation equation in the presence of mass transfer

A mathematical model for the continuous nonlinear fragmentation equation is considered in the presence of mass transfer. In this paper, we demonstrate the existence of mass-conserving weak solutions to the nonlinear fragmentation equation with mass transfer for collision kernels of the form $Φ(x,y) = κ(x^{σ_1} y^{σ_2} + y^{σ_1} x^{σ_2})$, $κ>0$, $0 \leq {σ_1} \leq {σ_2} \leq 1$, and ${σ_1} \neq 1$ for $(x, y) \in \mathbb{R}_+^2$, with integrable daughter distribution functions, thereby extending previous results obtained by Giri \& Lauren\c cot (2021). In particular, the existence of at least one global weak solution is shown when the collision kernel exhibits at least linear growth, and one local weak solution when the collision kernel exhibits sublinear growth. In both cases, finite superlinear moment bounds are obtained for positive times without requiring the finiteness of initial superlinear moments. Additionally, the uniqueness of solutions is confirmed in both cases.

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arXiv2024

Existence and Non-existence for Exchange-Driven Growth Model

The exchange-driven growth (EDG) model describes the evolution of clusters through the exchange of single monomers between pairs of interacting clusters. The dynamics of this process are primarily influenced by the interaction kernel $K_{j,k}$. In this paper, the global existence of classical solutions to the EDG equations is established for non-negative, symmetric interaction kernels satisfying $K_{j,k} \leq C(j^μk^ν + j^νk^μ) $, where $μ, ν\leq 2$, $μ+ ν\leq 3$, and $C>0$, with a broader class of initial data. This result extends the previous existence results obtained by Esenturk [10], Schlichting [23], and Eichenberg \& Schlichting [7]. Furthermore, the local existence of classical solutions to the EDG equations is demonstrated for symmetric interaction kernels that satisfy $K_{j,k} \leq C j^{2} k^{2}$ with $C > 0$, considering a broader class of initial data. In the intermediate regime $3 < μ+ ν\leq 4$, the occurrence of finite-time gelation is established for symmetric interaction kernels satisfying $C_{1}\left(j^{2}k^α+j^αk^{2}\right)\leq K_{j,k}\leq Cj^{2}k^{2}$, where $1 < α\leq 2$, $C>0$, and $C_{1} > 0$, as conjectured in [10]. In this case, the non-existence of the global solutions is ensured by the occurrence of finite-time gelation. Finally, the occurrence of instantaneous gelation of the solutions to EDG equations for symmetric interaction kernels satisfying $K_{j,k}\geq C\left(j^β+k^β\right)$ ($β>2, C>0)$ is shown, which also implies the non-existence of solutions in this case.

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arXiv2024

Well-posedness of the growth-coagulation equation with singular kernels

The well-posedness of the growth-coagulation equation is established for coagulation kernels having singularity near the origin and growing atmost linearly at infinity. The existence of weak solutions is shown by means of the method of the characteristics and a weak $L_1$-compactness argument. For the existence result, we also show our gratitude to Banach fixed point theorem and a refined version of the Arzelá-Ascoli theorem. In addition, the continuous dependence of solutions upon the initial data is shown with the help of the DiPerna-Lions theory, Gronwall's inequality and moment estimates. Moreover, the uniqueness of solution follows from the continuous dependence. The results presented in this article extend the contributions made in earlier literature.

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arXiv2024

Mass-Conserving Self-Similar Solutions to Collision-Induced Breakage Equations

Existence of mass-conserving self-similar solutions to collision-induced breakage equation is shown for a specific class of homogeneous collision kernels and breakage functions. The proof mainly relies on a dynamical approach and compactness method to constructing mass-conserving stationary solutions for an evolution problem, which induces mass-conserving self-similar solutions to collision-induced breakage equation. Furthermore, we also determine lower and upper bound of the scaling profile.

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arXiv2024

Stochastic Data-Driven Bouligand Landweber Method for Solving Non-smooth Inverse Problems

In this study, we present and analyze a novel variant of the stochastic gradient descent method, referred as Stochastic data-driven Bouligand Landweber iteration tailored for addressing the system of non-smooth ill-posed inverse problems. Our method incorporates the utilization of training data, using a bounded linear operator, which guides the iterative procedure. At each iteration step, the method randomly chooses one equation from the nonlinear system with data-driven term. When dealing with the precise or exact data, it has been established that mean square iteration error converges to zero. However, when confronted with the noisy data, we employ our approach in conjunction with a predefined stopping criterion, which we refer to as an \textit{a-priori} stopping rule. We provide a comprehensive theoretical foundation, establishing convergence and stability for this scheme within the realm of infinite-dimensional Hilbert spaces. These theoretical underpinnings are further bolstered by discussing an example that fulfills assumptions of the paper.

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arXiv2024

The continuous collision-induced nonlinear fragmentation equation with non-integrable fragment daughter distributions

Existence, non-existence, and uniqueness of mass-conserving weak solutions to the continuous collision-induced nonlinear fragmentation equations are established for the collision kernels $Φ$ satisfying $Φ(x_1,x_2)={x_1}^{λ_1} {x_2}^{λ_2}+{x_2}^{λ_1} {x_1}^{λ_2}$, $(x_1,x_2)\in(0,\infty)^2$, with ${λ_1} \leq {λ_2}\leq 1$, and non-integrable fragment daughter distributions. In particular, global existence of mass-conserving weak solutions is shown when $1\leλ:={λ_1}+{λ_2}\le2$ with $λ_1\ge k_0$, the parameter $k_0\in(0,1)$ being related to the non-integrability of the fragment daughter distribution. The existence of at least one mass-conserving weak solution is also demonstrated when $2k_0 \le λ< 1$ with $λ_1\ge k_0$ but its maximal existence time is shown to be finite. Uniqueness is also established in both cases. The last result deals with the non-existence of mass-conserving weak solutions, even on a small time interval, for power law fragment daughter distribution when $λ_1<k_0$. It is worth mentioning that the previous literature on the nonlinear fragmentation equation does not treat non-integrable fragment daughter distribution functions.

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arXiv2023

The discrete collision-induced breakage equation with mass transfer: well-posedness and stationary solutions

The discrete collisional breakage equation, which captures the dynamics of cluster growth when clusters encounter binary collisions with possible matter transfer, is discussed in this article. The existence of global mass-conserving solutions is investigated for the collision kernels $a_{i,j}=A(i^α j^β + i^βj^α)$, $i, j \ge 1$, with $α\in (-\infty,1)$, $β\in [α,1]\cap (0,1]$, and $A>0$ and for a large class of possibly unbounded daughter distribution functions. All algebraic superlinear moments of these solutions are bounded on time intervals $[T,\infty)$ for any $T>0$. The uniqueness issue is further handled under additional restrictions on the initial data. Finally, non-trivial stationary solutions are constructed by a dynamical approach.

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arXiv2023

Existence of solutions to the continuous RednerBen-AvrahamKahng coagulation equation

We take into account a coagulation model that simulates a distinct kind of dynamics. In this model, two particles collide to produce a single particle, but the resulting particle decreases in size, allowing each particle to be fully identified by its size. It is demonstrated that the corresponding evolving integral partial differential equation has solutions for product-type coagulation kernels i.e. $0 \le \mathfrak{K}(\varrho, ς)= \mathfrak{K}(ς, \varrho)=r(ς) r(\varrho)+α(ς, \varrho), (ς, \varrho)\in \mathbb{R}_+^2$ and $\sup_{ς\in [0,R]} \frac{\mathfrak{K}(ς, \varrho)}{\varrho} \to 0 \ \mbox{as} \ \varrho \to \infty$.

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arXiv2023

On the discrete Safronov-Dubovskii coagulation equation: well-posedness, mass-conservation and asymptotic behaviour

The global existence of mass-conserving weak solutions to the Safronov-Dubovskii coagulation equation is shown for the coagulation kernels satisfying the at most linear growth for large sizes. In contrast to previous works, the proof mainly relies on the de la Vallee-Poussin theorem [8, Theorem 7.1.6], which only requires the finiteness of the first moment of the initial condition. By showing the necessary regularity of solutions, it is shown that the weak solutions con-structed herein are indeed classical solutions. Under additional restrictions on the initial data, the uniqueness of solutions is also shown. Finally, the continuous dependence on the initial data and the large-time behaviour of solutions are also addressed.

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arXiv2023

Well-posedness to the discrete collision-induced breakage equation and various properties of solutions

A discrete version of the nonlinear collision-induced breakage equation is studied. Existence of solutions is investigated for a broad class of unbounded collision kernels and daughter distribution functions, the collision kernel $a_{i,j}$ satisfiying $a_{i,j} \leq A i j$ for some $A>0$. More precisely, it is proved that given suitable conditions, there exists at least one mass-conserving solution for all times. A result on the uniqueness of solutions is also demonstrated under reasonably general conditions. Furthermore, the propagation of moments, differentiability, and the continuous dependence of solutions are established, along with some invariance properties and the large-time behaviour of solutions.

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arXiv2022

Convergence rates of nonlinear inverse problems in Banach spaces: conditional stability and weaker norms

In this short note, we formulate the convergence rates of the well known Tikhonov regularization scheme for solving the nonlinear ill-posed problems in Banach spaces. For deriving the convergence rates, we employ the novel smoothness concept of conditional stability estimates in terms of weaker norms. Moreover, we show that our results are applicable on two ill-posed inverse problems.

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