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Pasetto, Damiano (2013) Reduced Order Models and Data Assimilation for Hydrological Applications. [Tesi di dottorato]

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Abstract (inglese)

The present thesis work concerns the study of Monte Carlo (MC)-based data assimilation methods applied to the numerical simulation of complex hydrological models with stochastic parameters. The ensemble Kalman filter (EnKF) and the sequential importance resampling (SIR) are implemented in the CATHY model, a solver that couples the subsurface water flow in porous media with the surface water dynamics. A detailed comparison of the results given by the two filters in a synthetic test case highlights the main benefits and drawbacks associated to these techniques. A modification of the SIR update is suggested to improve the performance of the filter in case of small ensemble sizes and small variances of the measurement errors. With this modification, both filters are able to assimilate pressure head and streamflow measurements and correct model errors, such as biased initial and boundary conditions. SIR technique seems to be better suited for the simulations at hand as they do not make use of the Gaussian approximation inherent the EnKF method. Further research is needed, however, to assess the robustness of the particle filters methods in particular to ensure accuracy of the results even when relatively small ensemble sizes are employed. In the second part of the thesis the focus is shifted to reducing the computational burden associated with the construction of the MC realizations (which constitutes the core of the EnKF and SIR). With this goal, we analyze the computational saving associated to the use of reduced order models (RM) for the generation of the ensemble of solutions. The proper orthogonal decomposition (POD) is applied to the linear equations of the groundwater flow in saturated porous media with a randomly distributed recharge and random heterogeneous hydraulic conductivity. Several test cases are used to assess the errors on the ensemble statistics caused by the RM approximation. Particular attention is given to the efficient computation of the principal components that are needed to project the model equations in the reduced space. The greedy algorithm selects the snapshots in the set of the MC realizations in such a way that the final principal components are parameter independent. An innovative residual-based estimation of the error associated to the RM solution is used to assess the precision of the RM and to stop the iterations of the greedy algorithm. By way of numerical applications in synthetic and real scenarios, we demonstrate that this modified greedy algorithm determines the minimum number of principal components to use in the reduction and, thus, leads to important computational savings.

Abstract (italiano)

Questo lavoro di tesi riguarda lo studio di tecniche di assimilazione di dati basate sul metodo di Monte Carlo (MC) per la simulazione numerica di modelli idrologici in presenza di parametri stocastici. I metodi ensemble Kalman filter (EnKF) e sequential importance resampling (SIR) sono implementati nel modello CATHY, un modello idrologico che accoppia il flusso d'acqua sotterraneo in mezzi porosi con la dinamica del flusso d’acqua superficiale. Il confronto dettagliato dei risultati ottenuti con i due filtri in un caso test sintetico evidenzia i principali vantaggi e inconvenienti associati a queste tecniche. Per migliorare le prestazioni del metodo SIR, in questa tesi è proposta una modifica del passo di update che risulta fondamentale nei casi in cui si usi un ensemble di dimensioni ridotte e la varianza associata all'errore di misura sia piccola. Grazie a questa modifica, entrambi i filtri sono in grado di assimilare misure di carico piezometrico e portata, riducendo la propagazione temporale di errori di modellizzazione dovuti, ad esempio, all'utilizzo di condizioni iniziali o al contorno distorte. La tecnica SIR sembra essere più adeguata dell'EnKF per l’applicazione ai casi test presentati. Si dimostra infatti che l'ipotesi di Gaussianità, che contraddistingue il metodo EnKF, non è soddisfatta in questi casi test, rendendo preferibili metodi più generali come il SIR. Ulteriori approfondimenti sono comunque necessari per stabilire l'affidabilità dei metodi di tipo particle filter, in particolare per garantire l'accuratezza del filtro SIR anche quando viene usato un numero relativamente piccolo di realizzazioni. Siccome il passo di previsione dei metodi SIR ed EnKF è basato sul metodo di MC, la seconda parte della tesi riguarda il problema di ridurre gli onerosi tempi di calcolo associati alla costruzione delle realizzazioni di MC. Con questo obbiettivo, si analizza il risparmio in tempo computazione ottenuto dall'uso di modelli di ordine ridotto (RM) per la generazione dell'ensemble delle soluzioni. La tecnica proper orthogonal decomposition (POD) è applicata alle equazioni lineari del flusso d’acqua sotterraneo in mezzi porosi saturi con ricarica stocastica e distribuita spazialmente, oppure con conducibilità idraulica stocastica e descritta per zone. Gli errori di approssimazione introdotti dal modello ridotto sul calcolo delle singole realizzazioni di MC e sulle corrispondenti statistiche sono analizzati in diversi casi test al variare della distribuzione probabilistica dei parametri stocastici. Particolare attenzione è dedicata alla procedura di calcolo delle principal components che sono necessarie per la proiezione delle equazioni del modello nello spazio ridotto. Il greedy algorithm seleziona gli snapshots tra le realizzazioni di MC considerate, facendo in modo che le principal components finali siano indipendenti dalla particolare realizzazione dei parametri stocastici. Infine, viene introdotta una stima innovativa della norma dell'errore associato alla soluzione del modello ridotto. Tale stima, basata sul calcolo del residuo, è di fondamentale importanza per stimare la precisione del RM e, quindi, inferire sul numero di principal components da usare nella riduzione. Le applicazioni numeriche effettuate su casi test sintetici e reali dimostrano che il greedy algorithm così modificato determina un numero minore di principal components rispetto al metodo tradizionale, pur mantenendo la medesima accuratezza.

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Tipo di EPrint:Tesi di dottorato
Relatore:Putti, Mario
Correlatore:Guadagnini, Alberto - Yeh, William W-G.
Dottorato (corsi e scuole):Ciclo 25 > Scuole 25 > SCIENZE DELL'INGEGNERIA CIVILE E AMBIENTALE,
Data di deposito della tesi:30 Gennaio 2013
Anno di Pubblicazione:Gennaio 2013
Parole chiave (italiano / inglese):Data Assimilation; Reduced Order Model; Groundwater Hydrology; CATHY
Settori scientifico-disciplinari MIUR:Area 01 - Scienze matematiche e informatiche > MAT/08 Analisi numerica
Area 08 - Ingegneria civile e Architettura > ICAR/01 Idraulica
Struttura di riferimento:Dipartimenti > Dipartimento di Matematica
Codice ID:5762
Depositato il:21 Ott 2013 10:07
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