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Maximum entropy algorithm with inexact upper entropy bound based on Fup basis functions with compact support
KTH, School of Architecture and the Built Environment (ABE), Land and Water Resources Engineering.
2009 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 228, no 24, 9079-9091 p.Article in journal (Refereed) Published
Abstract [en]

The maximum entropy (MaxEnt) principle is a versatile tool for statistical inference of the probability density function (pdf) from its moments as a least-biased estimation among all other possible pdfs. It maximizes Shannon entropy, satisfying the moment constraints. Thus, the MaxEnt algorithm transforms the original constrained optimization problem to the unconstrained dual optimization problem using Lagrangian multipliers. The Classic Moment Problem (CMP) uses algebraic power moments, causing typical conventional numerical methods to fail for higher-order moments (m > 5-10) due to different sensitivities of Lagrangian multipliers and unbalanced nonlinearities. Classic MaxEnt algorithms overcome these difficulties by using orthogonal polynomials, which enable roughly the same sensitivity for all Lagrangian multipliers. In this paper, we employ an idea based on different principles, using Fup(n) basis functions with compact support, which can exactly describe algebraic polynomials, but only if the Fup order-n is greater than or equal to the polynomial's order. Our algorithm solves the CMP with respect to the moments of only low order Fup(2) basis functions, finding a Fup(2) optimal pdf with better balanced Lagrangian multipliers. The algorithm is numerically very efficient due to localized properties of Fup(2) basis functions implying a weaker dependence between Lagrangian multipliers and faster convergence. Only consequences are an iterative scheme of the algorithm where power moments are a sum of Fup(2) and residual moments and an inexact entropy upper bound. However, due to small residual moments, the algorithm converges very quickly as demonstrated on two continuous pdf examples - the beta distribution and a bi-modal pdf, and two discontinuous pdf examples - the step and double Dirac pdf. Finally, these pdf examples present that Fup MaxEnt algorithm yields smaller entropy value than classic MaxEnt algorithm, but differences are very small for all practical engineering purposes.

Place, publisher, year, edition, pages
2009. Vol. 228, no 24, 9079-9091 p.
Keyword [en]
Maximum entropy algorithm; Classic Moment Problem; Fup basis functions; Compact support
National Category
Water Engineering
Identifiers
URN: urn:nbn:se:kth:diva-10653DOI: 10.1016/j.jcp.2009.09.011ISI: 000271790600008Scopus ID: 2-s2.0-70350026655OAI: oai:DiVA.org:kth-10653DiVA: diva2:222647
Note
Tidigare titel: Inexact Maximum Entropy algorithm based on Fup basis functions with compact support. QC 20100714. Updated from submitted to published 20120326Available from: 2009-06-09 Created: 2009-06-09 Last updated: 2017-12-13Bibliographically approved
In thesis
1. A multi-resolution approach for modeling flow and solute transport in heterogeneous porous media
Open this publication in new window or tab >>A multi-resolution approach for modeling flow and solute transport in heterogeneous porous media
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Subsurface processes are usually characterized by rare field experiments, sparse measurements,multi-resolution interpretations, stochastic description, related uncertainties and computational complexity. Over the last few decades, different computational techniques and strategies have become indispensable tools for flow and solute transport prediction in heterogeneous porousmedia. This thesis develops a multi-resolution approach based on Fup basis functions with compactsupport, enabling the use of an efficient and adaptive procedure, closely related to currentunderstood physical interpretation. All flow and transport variables, as well as intrinsic heterogeneity,are described in a multi-resolution representation, in the form of a linear combination ofFup basis functions. Each variable is represented on a particular adaptive grid with a prescribedaccuracy. The methodology is applied to solving problems with sharp fronts, and to solving flowand advective transport in highly heterogeneous porous media, under mean uniform flow conditions.The adaptive Fup collocation method, through the well known method of lines, efficientlytracks solutions with sharp fronts, resolving locations and frequencies at all spatial and/or temporalscales. The methodology yields continuous velocity fields and fluxes, enabling accurate andreliable transport analysis. Analysis of the advective transport proves the robustness of the firstordertheory for low and mild heterogeneity. Moreover, due to the accuracy of the improved Monte-Carlo methodology, this thesis presents the effects of high heterogeneity on ensembleflow and travel time statistics. The difference between Eulerian and Lagrangian velocity statisticsand the importance of higher travel time moments are indicative of high heterogeneity. The thirdtravel time moment mostly describes a peak and late arrivals, while higher moments are requiredfor early arrivals which are linked with the largest uncertainty. A particular finding is the linearityof all travel time moments, which implies that in the limit an advective transport in multi-Gaussian field becomes Fickian. By comparison, the transverse displacement pdf converges to aGaussian distribution around 20 integral scales after injection, even for high heterogeneity. Thecapabilities of the presented multi-resolution approach, and the quality of the obtained results,open new areas for further research.

Abstract [sv]

Markprocesser karakteriseras ofta av fåtaliga fältexperiment, glesa mätningar, heterogenitet påolika skalor, slumpmässighet och relaterade osäkerheter, samt beräkningsmässiga svårigheter.Under de senaste årtiondena har olika beräkningstekniker och strategier blivit ovärderliga verktygför att förutspå vattenflöde och ämnestransport i heterogena porösa medier. Denna doktorsavhandling utvecklar ett angreppssätt med flerskaliga upplösningar baserat på Fup basis funktionermed kompakt stöd, som möjliggör en effektiv och anpassningsbar procedur, nära relaterad tillrådande fysiska tolkningar. Alla flödes- och transportvariabler, så väl som heterogeniteten, beskrivsav en flerskaligt upplöst representation, i form av linjära kombinationer av Fup basis funktioner.Varje variabel representeras på ett speciellt anpassningsbar gridnät med given noggrannhet.Metoden appliceras för att lösa problem med skarpa fronter, samt vattenflöde och advektivämnestransport i starkt heterogena porösa medier. Adaptive Fup collocation metoden tillsammansmed den välkända Method of lines, spårar effektivt lösningar med skarpa fronter och löserupp positioner och frekvenser på alla rums- och/eller tidsskalor. Metoden ger kontinuerliga hastighetsfältoch flöden, och möjliggör noggrann och tillförlitlig transportanalys. Analys av advektivtransport understöder stabiliteten i första-ordningens transport teori för låg och mild heterogenitet.Utöver detta, som resultat av noggrannheten i den förbättrade Monte-Carlo metodiken, visardenna avhandling effekten av hög heterogenitet på ensemble statistiken för flöden och transporttider.Skillnaden mellan Eulerisk och Lagrangian hastighetsstatistik och betydelsen av högrestatistiska moment för transporttider, indikerar hög heterogenitet. Det tredje transporttidsmomentetbeskriver huvudsakligen sannolikhetspiken och de långa transporttiderna, medan högremoment behövs för de korta transporttiderna, som har den största osäkerheten. En speciell upptäcktär linjäariteten i transporttidsmoment, som indikerar att advektiv transport i multi-Gaussiska fält blir Gaussisk i gränsen. Som jämförelse konvergerar sannolikhetsfunktioner förden transversella transportförflyttningen mot en Gaussisk fördelning vid runt 20 korrelationslängder efter injektion, även för hög heterogenitet. Förmågan i det presenterade angreppssättet med flerskalig upplösning, och resultatens noggrannhet, öppnar nya områden för fortsatt forskning.

Place, publisher, year, edition, pages
Stockholm: KTH, 2009. xiv, 51 p.
Series
Trita-LWR. PHD, ISSN 1650-8602 ; 1051
Keyword
Multi-resolution adaptive approach; Atomic and Fup basis functions; Monte- Carlo method; Heterogeneous porous media; Flow; Transport; Travel time
National Category
Water Engineering
Identifiers
urn:nbn:se:kth:diva-10655 (URN)978-91-7415-377-4 (ISBN)
Public defence
2009-06-18, M3, Entreplan, KTH, Brinellvägen 64, Stockholm, 10:00 (English)
Opponent
Supervisors
Note
QC 20100714Available from: 2009-06-09 Created: 2009-06-09 Last updated: 2011-08-30Bibliographically approved

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