arXiv subjects

Andreea Grecu

Publications and source records attributed to Andreea Grecu.

6 recordsLinked to original sources

arXiv2026

Extended Walk-on-Spheres Algorithm for Linear and Nonlinear Elliptic Problems of Divergence-type

The Walk-on-Spheres algorithm, introduced by M. E. Muller in 1956, is a well known Monte Carlo method that leverages Brownian exit distributions from spheres to solve the Laplace equation with Dirichlet boundary conditions. Its mesh-free nature, robustness on complex geometries, favorable scaling with dimension, and intrinsic parallelism distinguish it from mesh-based solvers. However, its efficient applicability has been essentially limited to operators that admit explicit probabilistic exit laws, excluding most variable-coefficient and nonlinear elliptic operators. We propose a general framework that aims to overcome this limitation by using the classical Dirichlet Laplacian and harmonic extension as universal building blocks. Rather than seeking a custom stochastic representation for each operator, we employ Walk-on-Spheres to precompute a reusable numerical operator toolbox that approximates the inverse Dirichlet Laplacian, the harmonic extension operator, and their gradients. These precomputed operators are then used to represent candidate solutions and to transform arbitrary Dirichlet boundary value problems into a finite-dimensional algebraic system/optimization problem for an unknown source term. Solving the resulting algebraic system/optimization problem and substituting back yields an approximate solution to the original PDE. Even more, for a general linear second order elliptic operator, the above mentioned precomputed toolbox can be directly used to obtain not just an approximation of a certain solution of the corresponding generalized Dirichlet problem, but an estimator of both the Green's integral operator and the elliptic measure operator. Numerical experiments on a range of benchmarks, including non-symmetric and anisotropic linear elliptic equations, semilinear and quasilinear problems, demonstrate the method's flexibility and efficiency.

math.NA↗

arXiv2025

A probabilistic approach to spectral analysis of Cauchy-type inverse problems: Convergence and stability analysis

A comprehensive convergence and stability analysis of some probabilistic numerical methods designed to solve Cauchy-type inverse problems is performed in this study. Such inverse problems aim at solving an elliptic partial differential equation (PDE) or a system of elliptic PDEs in a bounded Euclidean domain, subject to incomplete boundary and/or internal conditions, and are usually severely ill-posed. In a very recent paper \cite{CiGrMaI}, a probabilistic numerical framework has been developed by the authors, wherein such inverse problems could be analysed thoroughly by simulating the spectrum of some corresponding direct problem and its singular value decomposition based on stochastic representations and Monte Carlo simulations. Herein a full probabilistic error analysis of the aforementioned methods is provided, whereas the convergence of the corresponding approximations is proved and explicit error bounds are provided. This is achieved by employing tools from several areas such as spectral theory, regularity theory for elliptic measures, stochastic representations, and concentration inequalities.

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arXiv2025

Numerical spectral analysis of Cauchy-type inverse problems: A probabilistic approach

We investigate the inverse Cauchy and data completion problems for elliptic partial differential equations in a bounded domain $D \subset \mathbb{R}^d$, $d \ge 2$, with a special emphasis on the steady-state heat conduction in anisotropic media. More precisely, boundary conditions are prescribed on an accessible part of the boundary $\varnothing \neq Γ_0 \subsetneqq \partial{D}$ and/or internal conditions are available inside the domain $D$ and the aim is to reconstruct the solution to these inverse problems in the domain and on the inaccessible remaining boundary $Γ_1 := \partial{D} \setminus Γ_0$. Although such severely ill-posed problems have been studied intensively in the past decades, deriving efficient methods for approximating their solution still remains challenging in the general setting, e.g., in high dimensions, for solutions and/or domains with singularities, in complex geometries, etc. Herein, we derive a fundamental probabilistic framework for the stable reconstruction of the solution to the Cauchy and data completion problems in steady-state anisotropic heat conduction, as well as enhancing the knowledge on the impact of the geometry of the domain $D$ and the structure of the conductivity tensor $\mathbf{K}$ on the stability of these inverse problems. This is achieved in three steps: ({\it i}) the spectrum of the direct problem is simulated using stochastic estimators; ({\it ii}) the singular value decomposition of the corresponding direct operator is performed; and ({\it iii}) for the prescribed measurements, a natural subspace of approximate solutions is constructed. This approach is based on elliptic measures, in conjunction with probabilistic representations and parallel Monte Carlo simulations. Thorough numerical simulations performed on GPU, for various two- and three-dimensional geometries, are also provided.

math.NA↗

arXiv2020

Hardy uniqueness principle for the linear Schrodinger equation on quantum regular trees

In this paper we consider the linear Schrodinger equation (LSE) on a regular tree with the last generation of edges of infinite length and analyze some unique continuation properties. The first part of the paper deals with the LSE on the real line with a piece-wise constant coefficient and uses this result in the context of regular trees. The second part treats the case of a LSE with a real potential in the framework of a star-shaped graph.

math.AP↗

arXiv2019

The nonlinear Schrödinger equation with white noise dispersion on quantum graphs

We show that the nonlinear Schrödinger equation (NLSE) with white noise dispersion on quantum graphs is globally well-posed in $L^2$ once the free deterministic Schrödinger group satisfies a natural $L^1-L^{\infty}$ decay, which is verified in many examples. Also, we investigate the well-posedness in the energy domain in general and in concrete situations, as well as the fact that the solution with white noise dispersion is the scaling limit of the solution to the NLSE with random dispersion.

math.AP↗

arXiv2018

The Schrodinger Equation on a Star-Shaped Graph under General Coupling Conditions

We investigate dispersive and Strichartz estimates for the Schrödinger time evolution propagator $\mathrm{e}^{-\mathrm{i}tH}$ on a star-shaped metric graph. The linear operator, $H$, taken into consideration is the self-adjoint extension of the Laplacian, subject to a wide class of coupling conditions. The study relies on an explicit spectral representation of the solution in terms of the resolvent kernel which is further analyzed using results from oscillatory integrals. As an application, we obtain the global well-posedness for a class of semilinear Schrödinger equations.

math.AP↗