Searcharxiv⌕ Search

arXiv subjects

Tram Thi Ngoc Nguyen

Publications and source records attributed to Tram Thi Ngoc Nguyen.

17 recordsLinked to original sources
arXiv2026

Data assimilation via model reference adaptation for linear and nonlinear dynamical systems

We address data assimilation for linear and nonlinear dynamical systems via the so-called model reference adaptive system. Continuing our theoretical developments, we deliver the first practical implementation of this approach for online parameter identification with time series data. Our semi-implicit scheme couples a modified state equation with a parameter evolution law that is driven by model-data residuals. We demonstrate four benchmark problems of increasing complexity: the Darcy flow, the Fisher-KPP equation, a nonlinear potential equation and finally, an Allen-Cahn type equation. Across all cases, explicit model reference adaptive system construction, verified assumptions and numerically stable reconstructions underline our proposed method as a reliable, versatile tool for data assimilation and real-time inversion.

math.OC↗
arXiv2026

Linear toroidal-inertial waves on a differentially rotating sphere with application to helioseismology: Modeling, forward and inverse problems

This paper develops a mathematical framework for interpreting observations of solar inertial waves in an idealized setting. Under the assumption of purely toroidal linear waves on the sphere, the stream function of the flow satisfies a fourth-order scalar equation. We prove well-posedness of wave solutions under explicit conditions on differential rotation. Moreover, we study the inverse problem of simultaneously reconstructing viscosity and differential rotation parameters from either complete or partial surface data. We establish convergence guarantee of iterative regularization methods by verifying the tangential cone condition, and prove local unique identifiability of the unknown parameters. Numerical experiments with Nesterov-Landweber iteration confirm reconstruction robustness across different observation strategies and noise levels.

math.AP↗
arXiv2026

The extended adjoint state and nonlinearity in correlation-based passive imaging

This articles investigates physics-based passive imaging problem, wherein one infers an unknown medium using ambient noise and correlation of the noise signal. We develop a general backpropagation framework via the so-called extended adjoint state, suitable for any elliptic PDE; crucially, this approach reduces by half the number of required PDE solves. Applications to several different PDE models demonstrate the universality of our method. In addition, we analyze the nonlinearity of the correlated model, revealing a surprising tangential cone condition-like structure, thereby advancing the state of the art towards a convergence guarantee for regularized reconstruction in passive imaging.

math.NA↗
arXiv2026

Determination of active forces in actomyosin systems as inverse source problems for the Stokes equation

The identification of forces and stresses is a central task in biophysics research: Knowledge on forces is key to understanding dynamic processes in active biological systems that are able to self-organize and display emergent properties by converting energy into mechanical work. The aim of this paper is to identify forces generated by a filament-motor network of F-actin and myosin -- actomyosin -- and exerted on the surrounding fluid, therefore causing a fluid flow. In particular, we evaluate optical microscopy data stemming from two different physical settings, confined and non-confined active gels. As a theoretical model, we use the Stokes equation together with an incompressibility condition and suitable boundary conditions reflecting the physical settings. The problem of determining the forces from knowledge on the fluid flow is formulated as an inverse source problem. Due to experimental limitations, only incomplete data are available. We provide a rigorous analysis of the forward problems and the impact of missing data, derive the adjoints of the forward operators needed for regularization, and demonstrate our methods on both synthetic and experimentally measured data.

physics.flu-dyn↗
arXiv2025

FEM-based A-optimal sensor placement for heat source inversion from final time measurement

Within the field of optimal experimental design, \emph{sensor placement} refers to the act of finding the optimal locations of data collecting sensors, with the aim to optimise reconstruction of an unknown parameter from finite data. In this work, we investigate sensor placement for the inverse problem of reconstructing a heat source given final time measurements. Employing forward and adjoint analysis of this PDE-driven model, we show how one can leverage the first author's recently invented \emph{redundant-dominant $p$-continuation} algorithm to obtain binary A-optimal sensor placements also for this time-dependent model.

math.OC↗
arXiv2025

Sequential bi-level regularized inversion with application to hidden reaction law discovery

In this article, we develop and present a novel regularization scheme for ill-posed inverse problems governed by nonlinear time-dependent partial differential equations (PDEs). In our recent work, we introduced a bi-level regularization framework. This study significantly improves upon the bi-level algorithm by sequentially initializing the lower-level problem, yielding accelerated convergence and demonstrable multi-scale effect, while retaining regularizing effect and allows for the usage of inexact PDE solvers. Moreover, by collecting the lower-level trajectory, we uncover an interesting connection to the incremental load method. The sequential bi-level approach illustrates its universality through several reaction-diffusion applications, in which the nonlinear reaction law needs to be determined. We moreover prove that the proposed tangential cone condition is satisfied.

math.NA↗
arXiv2024

Traction force microscopy for linear and nonlinear elastic materials as a parameter identification inverse problem

Traction force microscopy is a method widely used in biophysics and cell biology to determine forces that biological cells apply to their environment. In the experiment, the cells adhere to a soft elastic substrate, which is then deformed in response to cellular traction forces. The inverse problem consists in computing the traction stress applied by the cell from microscopy measurements of the substrate deformations. In this work, we consider a linear model, in which 3D forces are applied at a 2D interface, called 2.5D traction force microscopy, and a nonlinear pure 2D model, from which we directly obtain a linear pure 2D model. All models lead to a linear resp. nonlinear parameter identification problem for a boundary value problem of elasticity. We analyze the respective forward operators and conclude with some numerical experiments for simulated and experimental data.

math.NA↗
arXiv2024

Bi-level regularization via iterative mesh refinement for aeroacoustics

In this work, we illustrate the connection between adaptive mesh refinement for finite element discretized PDEs and the recently developed \emph{bi-level regularization algorithm}. By adaptive mesh refinement according to data noise, regularization effect and convergence are immediate consequences. We moreover demonstrate its numerical advantages to the classical Landweber algorithm in term of time and reconstruction quality for the example of the Helmholtz equation in an aeroacoustic setting.

math.NA↗
arXiv2024

Inferring solar differential rotation and viscosity via passive imaging with inertial waves

The recent discovery of inertial waves on the surface of the Sun offers new possibilities to learn about the solar interior. These waves are long-lived with a period on the order of the Sun rotation period ($\sim$27 days) and are sensitive to parameters deep inside the Sun. They are excited by turbulent convection, leading to a passive imaging problem. In this work, we present the forward and inverse problem of reconstructing viscosity and differential rotation on the Sun from cross-covariance observations of these inertial waves.

astro-ph.SR↗
arXiv2024

Bi-level iterative regularization for inverse problems in nonlinear PDEs

We investigate the ill-posed inverse problem of recovering unknown spatially dependent parameters in nonlinear evolution PDEs. We propose a bi-level Landweber scheme, where the upper-level parameter reconstruction embeds a lower-level state approximation. This can be seen as combining the classical reduced setting and the newer all-at-once setting, allowing us to, respectively, utilize well-posedness of the parameter-to-state map, and to bypass having to solve nonlinear PDEs exactly. Using this, we derive stopping rules for lower- and upper-level iterations and convergence of the bi-level method. We discuss application to parameter identification for the Landau-Lifshitz-Gilbert equation in magnetic particle imaging.

math.NA↗
arXiv2023

Learning-informed parameter identification in nonlinear time-dependent PDEs

We introduce and analyze a method of learning-informed parameter identification for partial differential equations (PDEs) in an all-at-once framework. The underlying PDE model is formulated in a rather general setting with three unknowns: physical parameter, state and nonlinearity. Inspired by advances in machine learning, we approximate the nonlinearity via a neural network, whose parameters are learned from measurement data. The later is assumed to be given as noisy observations of the unknown state, and both the state and the physical parameters are identified simultaneously with the parameters of the neural network. Moreover, diverging from the classical approach, the proposed all-at-once setting avoids constructing the parameter-to-state map by explicitly handling the state as additional variable. The practical feasibility of the proposed method is confirmed with experiments using two different algorithmic settings: A function-space algorithm based on analytic adjoints as well as a purely discretized setting using standard machine learning algorithms.

math.OC↗
arXiv2022

Discretization of parameter identification in PDEs using Neural Networks

We consider the ill-posed inverse problem of identifying a nonlinearity in a time-dependent PDE model. The nonlinearity is approximated by a neural network, and needs to be determined alongside other unknown physical parameters and the unknown state. Hence, it is not possible to construct input-output data pairs to perform a supervised training process. Proposing an all-at-once approach, we bypass the need for training data and recover all the unknowns simultaneously. In the general case, the approximation via a neural network can be realized as a discretization scheme, and the training with noisy data can be viewed as an ill-posed inverse problem. Therefore, we study discretization of regularization in terms of Tikhonov and projected Landweber methods for discretization of inverse problems, and prove convergence when the discretization error (network approximation error) and the noise level tend to zero.

math.NA↗
arXiv2021

On numerical aspects of parameter identification for the Landau-Lifshitz-Gilbert equation in Magnetic Particle Imaging

The Landau-Lifshitz-Gilbert equation yields a mathematical model to describe the evolution of the magnetization of a magnetic material, particularly in response to an external applied magnetic field. It allows one to take into account various physical effects, such as the exchange within the magnetic material itself. In particular, the Landau-Lifshitz-Gilbert equation encodes relaxation effects, i.e., it describes the time-delayed alignment of the magnetization field with an external magnetic field. These relaxation effects are an important aspect in magnetic particle imaging, particularly in the calibration process. In this article, we address the data-driven modeling of the system function in magnetic particle imaging, where the Landau-Lifshitz-Gilbert equation serves as the basic tool to include relaxation effects in the model. We formulate the respective parameter identification problem both in the all-at-once and the reduced setting, present reconstruction algorithms that yield a regularized solution and discuss numerical experiments. Apart from that, we propose a practical numerical solver to the nonlinear Landau-Lifshitz-Gilbert equation, not via the classical finite element method, but through solving only linear PDEs in an inverse problem framework.

math.NA↗
arXiv2021

A model reference adaptive system approach for nonlinear online parameter identification

Dynamical systems, for instance in model predictive control, often contain unknown parameters, which must be determined during system operation. Online or on-the-fly parameter identification methods are therefore necessary. The challenge of online methods is that one must continuously estimate parameters as experimental data becomes available. The existing techniques in the context of time-dependent partial differential equations exclude the case where the system depends nonlinearly on the parameters.Based on a model reference adaptive system approach, we present an online parameter identification method for nonlinear infinite-dimensional evolutionary system.

math.OC↗
arXiv2019

Parameter identification for the Landau-Lifshitz-Gilbert equation in Magnetic Particle Imaging

Magnetic particle imaging (MPI) is a tracer-based technique for medical imaging where the tracer consists of ironoxide nanoparticles. The key idea is to measure the particle response to a temporally changing external magnetic field to compute the spatial concentration of the tracer inside the object. A decent mathematical model demands for a data-driven computation of the system function which does not only describe the measurement geometry but also encodes the interaction of the particles with the external magnetic field. The physical model of this interaction is given by the Landau-Lifshitz-Gilbert (LLG) equation. The determination of the system function can be seen as an inverse problem of its own which can be interpreted as a calibration problem for MPI. In this contribution the calibration problem is formulated as an inverse parameter identification problem for the LLG equation. We give a detailed analysis of the direct as well as the inverse problem in an all-at-once as well as in a reduced setting. The analytical results yield a deeper understanding of inverse problems connected to the LLG equation and provide a starting point for the development of robust numerical solution methods in MPI.

math.AP↗
arXiv2019

Landweber-Kaczmarz for parameter identification in time-dependent inverse problems: All-at-once versus Reduced version

In this study, we consider a general time-space system, whose model operator and observation operator are locally Lipschitz continuous, over a finite time horizon and parameter identification by using Landweber-Kaczmarz regularization. The problem is investigated in two different modeling settings: An All-at-once and a Reduced version, together with two observation scenarios: continuous and discrete observations. Segmenting the time line into several subintervals leads to the idea of applying the Kaczmarz method. A loping strategy is incorporated into the method to yield the loping Landweber-Kaczmarz iteration.

math.AP↗