Inverse problems play a central role in many applications, from imaging to earthquake science, but still present many challenges due to their intrinsic ill-posedness. Approaching inverse problems within the Bayesian framework allows us to address the ill-posedness, such as the non-uniqueness and sensitivity to noise of the solution. In this work, the focus is on hierarchical Bayesian methods, and in particular on a computationally efficient iterative algorithm that produces an approximation of the Maximum A Posteriori (MAP) estimate, referred to as a quasi-MAP (qMAP) estimate. The novelty in this work is to employ the hierarchical approach to mesh adaptation: as the iterative solver progresses, the hierarchical information about the solution is used to adapt the underlying mesh that is used for representing and computing the discretized solution, thus increasing in a selective way the accuracy to capture singularities in the solution. The viability of the method is demonstrated with three different computed examples: one-dimensional deconvolution problem, two-dimensional limited angle tomography problem, and a PDE based two dimensional inverse aquifer source problem.
I problemi inversi occupano un ruolo centrale in molte applicazioni, dallo studio di immagini ai fenomeni sismici; tuttavia, a causa della loro intrinseca al posizione, essi risultano tutt'ora di difficile risoluzione. Affrontare i problemi inversi in un contesto Bayesiano ci permette di ovviare a questioni dovute alla mal posizione, come la non unicità e la sensitività al rumore della soluzione. In questo lavoro, ci concentriamo sui modelli gerarchici Bayesiani, in particolare su un algoritmo computazionalmente effciente che produce un'approssimazione della stima MAP (Maximum A Posteriori), defi nita come la stima quasi-MAP (qMAP). La maggiore novità di questa tesi consiste nell'impiego del modello gerarchico per l'adattazione di griglia: con la progressione del solutore iterativo, le informazioni sulla soluzione derivanti dal modello gerarchico vengono utilizzate per adattare la mesh su cui essa viene approssimata, di conseguenza incrementando selettivamente l'accuratezza per cogliere le singolarità della soluzione. La validità del metodo è dimostrata tramite tre diversi test: il problema della deconvoluzione unidimensionale, la tomografi a bidimensionale con angolo di rotazione limitato, e il problema inverso due-dimensionale basato su EDP per la stima di sorgenti o pozzi acquiferi.
Mesh adaptation techniques driven by hierarchical Bayesian models
COSMO, ANNA
2017/2018
Abstract
Inverse problems play a central role in many applications, from imaging to earthquake science, but still present many challenges due to their intrinsic ill-posedness. Approaching inverse problems within the Bayesian framework allows us to address the ill-posedness, such as the non-uniqueness and sensitivity to noise of the solution. In this work, the focus is on hierarchical Bayesian methods, and in particular on a computationally efficient iterative algorithm that produces an approximation of the Maximum A Posteriori (MAP) estimate, referred to as a quasi-MAP (qMAP) estimate. The novelty in this work is to employ the hierarchical approach to mesh adaptation: as the iterative solver progresses, the hierarchical information about the solution is used to adapt the underlying mesh that is used for representing and computing the discretized solution, thus increasing in a selective way the accuracy to capture singularities in the solution. The viability of the method is demonstrated with three different computed examples: one-dimensional deconvolution problem, two-dimensional limited angle tomography problem, and a PDE based two dimensional inverse aquifer source problem.File | Dimensione | Formato | |
---|---|---|---|
2018_07_Cosmo.pdf
non accessibile
Descrizione: Testo della tesi
Dimensione
3.8 MB
Formato
Adobe PDF
|
3.8 MB | Adobe PDF | Visualizza/Apri |
I documenti in POLITesi sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
https://hdl.handle.net/10589/142340