arXiv subjects

A. Ratnani

Publications and source records attributed to A. Ratnani.

7 recordsLinked to original sources

arXiv2026

Backward problem for a degenerate viscous Hamilton-Jacobi equation: stability and numerical identification

This work is devoted to the analysis of the backward problem for a viscous Hamilton-Jacobi equation with degenerate diffusion and a general Hamiltonian that is not necessarily quadratic. First, we focus on linear degenerate parabolic equations in the nondivergence setting. We prove the conditional stability of the backward problem using Carleman estimates. Then, by a linearization technique, we prove similar results for the nonlinear viscous Hamilton-Jacobi equation. Regarding numerical identification, we first investigate the linear degenerate equation with noisy data using the adjoint state method, combined with a Conjugate Gradient algorithm, to solve the associated minimization problem. Finally, the numerical identification for the nonlinear viscous Hamilton-Jacobi equation is investigated by the Van Cittert iteration. Numerical tests are presented to show the performance of the proposed algorithms.

math.AP↗

arXiv2024

Stability of backward inverse problems for degenerate mean-field game systems

We investigate inverse backward-in-time problems for a class of second-order degenerate Mean-Field Game (MFG) systems. More precisely, given the final datum $(u(\cdot, T),m(\cdot, T))$ of a solution to the one-dimensional mean-field game system with a degenerate diffusion coefficient, we aim to determine the intermediate states $(u(\cdot,t_{0}),m(\cdot,t_{0}))$ for any $t_{0} \in [0, T)$, i.e., the value function and the mean distribution at intermediate times, respectively. We prove conditional stability estimates under suitable assumptions on the diffusion coefficient and the initial state $(u(\cdot,0),m(\cdot,0))$. The proofs are based on Carleman's estimates with a simple weight function. We first prove a Carleman estimate for the Hamilton-Jacobi-Bellman (HJB) equation. A second Carleman estimate will be derived for the Fokker-Planck (FP) equation. Then, by combining the two estimates, we obtain a Carleman estimate for the mean-field game system, leading to the stability of the backward problems.

math.AP↗

arXiv2024

Novel Approach for solving the discrete Stokes problems based on Augmented Lagrangian and Global Techniques: Application to Saddle-Point Linear Systems from Incompressible flow

In this paper, a novel augmented Lagrangian preconditioner based on global Arnoldi for accelerating the convergence of Krylov subspace methods applied to linear systems of equations with a block three-by-three structure, these systems typically arise from discretizing the Stokes equations using mixed-finite element methods. In practice, the components of velocity are always approximated using a single finite element space. More precisely, in two dimensions, our new approach based on standard space of scalar finite element basis functions to discretize the velocity space. This componentwise splitting can be shown to induce a natural block three-by-three structure. Spectral analyses is established for the exact versions of these preconditioners. Finally, the obtained numerical results claim that our novel approach is more efficient and robust for solving the discrete Stokes problems. The efficiency of our new approach is evaluated by measuring computational time.

math.NA↗

arXiv2024

Eigenvalues Distributions and Control Theory

This work deals with the isogeometric Galerkin discretization of the eigenvalue problem related to the Laplace operator subject to homogeneous Dirichlet boundary conditions on bounded intervals. This paper uses GLT theory to study the behavior of the gap of discrete spectra toward the uniform gap condition needed for the uniform boundary observability/controllability problems. The analysis refers to a regular $B$-spline basis and concave or convex reparametrizations. Under suitable assumptions on the reparametrization transformation, we prove that structure emerges within the distribution of the eigenvalues once we reframe the problem into GLT-symbol analysis. We also demonstrate numerically, that the necessary average gap condition proposed in \cite{bianchi2018spectral} is not equivalent to the uniform gap condition. However, by improving the result in \cite{bianchi2021analysis} we construct sufficient criteria that guarantee the uniform gap property.

math.NA↗

arXiv2023

A model reduction method for large-scale linear multidimensional dynamical systems

In this work, we explore the application of multilinear algebra in reducing the order of multidimentional linear time-invariant (MLTI) systems. We use tensor Krylov subspace methods as key tools, which involve approximating the system solution within a low-dimensional subspace. We introduce the tensor extended block and global Krylov subspaces and the corresponding Arnoldi based processes. Using these methods, we develop a model reduction using projection techniques. We also show how these methods could be used to solve large-scale Lyapunov tensor equations that are needed in the balanced truncation method which is a technique for order reduction. We demonstrate how to extract approximate solutions via the Einstein product using the tensor extended block Arnoldi and the extended global Arnoldi processes.

math.NA↗

arXiv2021

RBF approximation of three dimensional PDEs using Tensor Krylov subspace methods

In this paper, we propose different algorithms for the solution of a tensor linear discrete ill-posed problem arising in the application of the meshless method for solving PDEs in three-dimensional space using multiquadric radial basis functions. It is well known that the truncated singular value decomposition (TSVD) is the most common effective solver for ill-conditioned systems, but unfortunately the operation count for solving a linear system with the TSVD is computationally expensive for large-scale matrices. In the present work, we propose algorithms based on the use of the well known Einstein product for two tensors to define the tensor global Arnoldi and the tensor Gloub Kahan bidiagonalization algorithms. Using the so-called Tikhonov regularization technique, we will be able to provide computable approximate regularized solutions in a few iterations.

math.NA↗

arXiv2021

On the tensor nuclear norm and the total variation regularization for image and video completion

In the present paper we propose two new algorithms of tensor completion for three-order tensors. The proposed methods consist in minimizing the average rank of the underlying tensor using its approximate function namely the tensor nuclear norm and then the recovered data will be obtained by using the total variation regularisation technique. We will adopt the Alternating Direction Method of Multipliers (ADM), using the tensor T-product, to solve the main optimization problems associated to the two algorithms. In the last section, we present some numerical experiments and comparisons with the most known image completion methods.

math.NA↗