arXiv subjects

Phuoc-Truong Huynh

Publications and source records attributed to Phuoc-Truong Huynh.

5 recordsLinked to original sources

arXiv2025

A Bayesian approach to inverse problems in spaces of measures

In this work, we develop a Bayesian framework for solving inverse problems in which the unknown parameter belongs to a space of Radon measures taking values in a separable Hilbert space. The inherent ill-posedness of such problems is addressed by introducing suitable measure-valued priors that encode prior information and promote desired sparsity properties of the parameter. Under appropriate assumptions on the forward operator and noise model, we establish the well-posedness of the Bayesian formulation by proving the existence, uniqueness, and stability of the posterior with respect to perturbations in the observed data. In addition, we also discuss computational strategies for approximating the posterior distribution. Finally, we present some examples that demonstrate the effectiveness of the proposed approach.

math.ST↗

arXiv2025

On the optimal choice of the illumination function in photoacoustic tomography

This work studies the inverse problem of photoacoustic tomography (more precisely, the acoustic subproblem) as the identification of a space-dependent source parameter. The model consists of a wave equation involving a time-fractional damping term to account for power law frequency dependence of the attenuation, as relevant in ultrasonics. We solve the inverse problem in a Bayesian framework using a Maximum A Posteriori (MAP) estimate, and for this purpose derive an explicit expression for the adjoint operator. On top of this, we consider optimization of the choice of the laser excitation function, which is the time-dependent part of the source in this model, to enhance the reconstruction result. The method employs the $A$-optimality criterion for Bayesian optimal experimental design with Gaussian prior and Gaussian noise. To efficiently approximate the cost functional, we introduce an approximation scheme based on projection onto finite-dimensional subspaces. Finally, we present numerical results that illustrate the theory.

math.NA↗

arXiv2024

Compactness of Green operators with applications to semilinear nonlocal elliptic equations

In this paper, we consider a class of integro-differential operators $\mathbb{L}$ posed on a $C^2$ bounded domain $Ω\subset \mathbb{R}^N$ with appropriate homogeneous Dirichlet conditions where each of which admits an inverse operator commonly known as the Green operator $\mathbb{G}^Ω$. Under mild conditions on $\mathbb{L}$ and its Green operator, we establish various sharp compactness of $\mathbb{G}^Ω$ involving weighted Lebesgue spaces and weighted measure spaces. These results are then employed to prove the solvability for semilinear elliptic equation $\mathbb{L} u + g(u) = μ$ in $Ω$ with boundary condition $u=0$ on $\partial Ω$ or exterior condition $u=0$ in $\mathbb{R}^N \setminus Ω$ if applicable, where $μ$ is a Radon measure on $Ω$ and $g: \mathbb{R} \to \mathbb{R}$ is a nondecreasing continuous function satisfying a subcriticality integral condition. When $g(t)=|t|^{p-1}t$ with $p>1$, we provide a sharp sufficient condition expressed in terms of suitable Bessel capacities for the existence of a solution. The contribution of the paper consists of (i) developing novel unified techniques which allow to treat various types of fractional operators and (ii) obtaining sharp compactness and existence results in weighted spaces, which refine and extend several related results in the literature.

math.AP↗

arXiv2024

Towards optimal sensor placement for inverse problems in spaces of measures

The objective of this work is to quantify the reconstruction error in sparse inverse problems with measures and stochastic noise, motivated by optimal sensor placement. To be useful in this context, the error quantities must be explicit in the sensor configuration and robust with respect to the source, yet relatively easy to compute in practice, compared to a direct evaluation of the error by a large number of samples. In particular, we consider the identification of a measure consisting of an unknown linear combination of point sources from a finite number of measurements contaminated by Gaussian noise. The statistical framework for recovery relies on two main ingredients: first, a convex but non-smooth variational Tikhonov point estimator over the space of Radon measures and, second, a suitable mean-squared error based on its Hellinger-Kantorovich distance to the ground truth. To quantify the error, we employ a non-degenerate source condition as well as careful linearization arguments to derive a computable upper bound. This leads to asymptotically sharp error estimates in expectation that are explicit in the sensor configuration. Thus they can be used to estimate the expected reconstruction error for a given sensor configuration and guide the placement of sensors in sparse inverse problems.

math.NA↗

arXiv2022

Semilinear nonlocal elliptic equations with source term and measure data

Recently, several works have been carried out in attempt to develop a theory for linear or sublinear elliptic equations involving a general class of nonlocal operators characterized by mild assumptions on the associated Green kernel. In this paper, we study the Dirichlet problem for superlinear equation (E) ${\mathbb L} u = u^p +λμ$ in a bounded domain $Ω$ with homogeneous boundary or exterior Dirichlet condition, where $p>1$ and $λ>0$. The operator ${\mathbb L}$ belongs to a class of nonlocal operators including typical types of fractional Laplacians and the datum $μ$ is taken in the optimal weighted measure space. The interplay between the operator ${\mathbb L}$, the source term $u^p$ and the datum $μ$ yields substantial difficulties and reveals the distinctive feature of the problem. We develop a new unifying technique based on a fine analysis on the Green kernel, which enables us to construct a theory for semilinear equation (E) in measure frameworks. A main thrust of the paper is to provide a fairly complete description of positive solutions to the Dirichlet problem for (E). In particular, we show that there exist a critical exponent $p^*$ and a threshold value $λ^*$ such that the multiplicity holds for $1<p<p^*$ and $0<λ<λ^*$, the uniqueness holds for $1<p<p^*$ and $λ=λ^*$, and the nonexistence holds in other cases. Various types of nonlocal operator are discussed to exemplify the wide applicability of our theory.

math.AP↗