Error estimate and control in the MSFV method for multiphase flow in porous media

n this paper the iterative MSFV method is extended to include the sequential implicit simulation of time dependent problems involving the solution of a system of pressure-saturation equations. To control numerical errors in simulation results, an error estimate, based on the residual of the MSFV approximate pressure field, is introduced. In the initial time steps in simulation iterations are employed until a specified accuracy in pressure is achieved. This initial solution is then used to improve the localization assumption at later time steps. Additional iterations in pressure solution are employed only when the pressure residual becomes larger than a specified threshold value. Efficiency of the strategy and the error control criteria are numerically investigated. This paper also shows that it is possible to derive an a-priori estimate and control based on the allowed pressure-equation residual to guarantee the desired accuracy in saturation calculation.

Accurate and efficient simulation of multiphase flow in a heterogeneous reservoir by using error estimate and control in the multiscale finite-volume framework

The multiscale finite-volume (MSFV) method is designed to reduce the computational cost of elliptic and parabolic problems with highly heterogeneous anisotropic coefficients. The reduction is achieved by splitting the original global problem into a set of local problems (with approximate local boundary conditions) coupled by a coarse global problem. It has been shown recently that the numerical errors in MSFV results can be reduced systematically with an iterative procedure that provides a conservative velocity field after any iteration step. The iterative MSFV (i-MSFV) method can be obtained with an improved (smoothed) multiscale solution to enhance the localization conditions, with a Krylov subspace method [e.g., the generalized-minimal-residual (GMRES) algorithm] preconditioned by the MSFV system, or with a combination of both. In a multiphase-flow system, a balance between accuracy and computational efficiency should be achieved by finding a minimum number of i-MSFV iterations (on pressure), which is necessary to achieve the desired accuracy in the saturation solution. In this work, we extend the i-MSFV method to sequential implicit simulation of time-dependent problems. To control the error of the coupled saturation/pressure system, we analyze the transport error caused by an approximate velocity field. We then propose an error-control strategy on the basis of the residual of the pressure equation. At the beginning of simulation, the pressure solution is iterated until a specified accuracy is achieved. To minimize the number of iterations in a multiphase-flow problem, the solution at the previous timestep is used to improve the localization assumption at the current timestep. Additional iterations are used only when the residual becomes larger than a specified threshold value. Numerical results show that only a few iterations on average are necessary to improve the MSFV results significantly, even for very challenging problems. Therefore, the proposed adaptive strategy yields efficient and accurate simulation of multiphase flow in heterogeneous porous media.

Model selection and error estimation

We study model selection strategies based on penalized empirical loss minimization. We point out a tight relationship between error estimation and data-based complexity penalization: any good error estimate may be converted into a data-based penalty function and the performance of the estimate is governed by the quality of the error estimate. We consider several penalty functions, involving error estimates on independent test data, empirical {\sc vc} dimension, empirical {\sc vc} entropy, andmargin-based quantities. We also consider the maximal difference between the error on the first half of the training data and the second half, and the expected maximal discrepancy, a closely related capacity estimate that can be calculated by Monte Carlo integration. Maximal discrepancy penalty functions are appealing for pattern classification problems, since their computation is equivalent to empirical risk minimization over the training data with some labels flipped.

On aposteriori error analysis of dG schemes approximating hyperbolic conservation laws

This contribution is concerned with aposteriori error analysis of discontinuous Galerkin (dG) schemes approximating hyperbolic conservation laws. In the scalar case the aposteriori analysis is based on the L1 contraction property and the doubling of variables technique. In the system case the appropriate stability framework is in L2, based on relative entropies. It is only applicable if one of the solutions, which are compared to each other, is Lipschitz. For dG schemes approximating hyperbolic conservation laws neither the entropy solution nor the numerical solution need to be Lipschitz. We explain how this obstacle can be overcome using a reconstruction approach which leads to an aposteriori error estimate.

Error estimates for a neumann problem in highly oscillating thin domains

In this work we analyze the convergence of solutions of the Poisson equation with Neumann boundary conditions in a two-dimensional thin domain with highly oscillatory behavior. We consider the case where the height of the domain, amplitude and period of the oscillations are all of the same order, and given by a small parameter e > 0. Using an appropriate corrector approach, we show strong convergence and give error estimates when we replace the original solutions by the first-order expansion through the Multiple-Scale Method.

Selection bias in gene extraction on the basis of microarray gene-expression data

In the context of cancer diagnosis and treatment, we consider the problem of constructing an accurate prediction rule on the basis of a relatively small number of tumor tissue samples of known type containing the expression data on very many (possibly thousands) genes. Recently, results have been presented in the literature suggesting that it is possible to construct a prediction rule from only a few genes such that it has a negligible prediction error rate. However, in these results the test error or the leave-one-out cross-validated error is calculated without allowance for the selection bias. There is no allowance because the rule is either tested on tissue samples that were used in the first instance to select the genes being used in the rule or because the cross-validation of the rule is not external to the selection process; that is, gene selection is not performed in training the rule at each stage of the cross-validation process. We describe how in practice the selection bias can be assessed and corrected for by either performing a cross-validation or applying the bootstrap external to the selection process. We recommend using 10-fold rather than leave-one-out cross-validation, and concerning the bootstrap, we suggest using the so-called. 632+ bootstrap error estimate designed to handle overfitted prediction rules. Using two published data sets, we demonstrate that when correction is made for the selection bias, the cross-validated error is no longer zero for a subset of only a few genes.

2D PCA-Based localization for mobile robots in unstructured environments

In this paper a new PCA-based positioning sensor and localization system for mobile robots to operate in unstructured environments (e. g. industry, services, domestic ...) is proposed and experimentally validated. The inexpensive positioning system resorts to principal component analysis (PCA) of images acquired by a video camera installed onboard, looking upwards to the ceiling. This solution has the advantage of avoiding the need of selecting and extracting features. The principal components of the acquired images are compared with previously registered images, stored in a reduced onboard image database, and the position measured is fused with odometry data. The optimal estimates of position and slippage are provided by Kalman filters, with global stable error dynamics. The experimental validation reported in this work focuses on the results of a set of experiments carried out in a real environment, where the robot travels along a lawn-mower trajectory. A small position error estimate with bounded co-variance was always observed, for arbitrarily long experiments, and slippage was estimated accurately in real time.

Mortar-Techniken zur Behandlung von Grenzschichtproblemen

Elliptic differential equations, finite element method, mortar element method, streamline diffusion FEM, upwind method, numerical method, error estimate, interpolation operator, grid generation, adaptive refinement

Coifman-aallokkeet ja skaalaussuotimen muodostus differentiaalievoluut ion avulla

Ortogonaalisen M-kaistaisen moniresoluutioanalyysin matemaattiset perusteet esitetään yksityiskohtaisesti. Coifman-aallokkeiden määritelmä yleistetään dilaatiokertoimelle M ja nollasta poikkeavalle häviävien momenttien keskukselle.Funktion approksimointia näytepisteistä aallokkeiden avulla pohditaan ja erityisesti esitetään approksimaation asymptoottinen virhearvio Coifman-aallokkeille. Skaalaussuotimelle osoitetaan välttämättömät ja riittävät ehdot, jotka johtavat yleistettyihin Coifman-aallokkeisiin. Moniresoluutioanalyysin tiheys todistetaansuoraan Lebesguen integraalin määritelmään perustuen yksikön partitio-ominaisuutta käyttäen. Todistus on riittävä sellaisenaan avaruudessa L2(Wd) käyttämättä Fourier-tason ominaisuuksia tai ehtoja. Mallatin algoritmi johdetaan M-kaistaisille aallokkeille ja moniuloitteisille signaaleille. Algoritmille esitetään myös rekursiivinen muoto. Differentiaalievoluutioalgoritmin avulla ratkaistaan Coifman-aallokkeisiin liittyvien skaalaussuotimien kertoimien arvoja useille skaalausfunktiolle. Approksimaatio- ja kuvanpakkausesimerkkejä esitetään menetelmien havainnollistamiseksi. Differentiaalievoluutioalgoritmin avulla etsitään myös referenssikuville optimoitu skaalaussuodin. Löydetty suodin on regulaarinen ja erittäinsymmetrinen.

Boundary identification in the domain of a parabolic partial differential equation

Bakgrunden och inspirationen till föreliggande studie är tidigare forskning i tillämpningar på randidentifiering i metallindustrin. Effektiv randidentifiering möjliggör mindre säkerhetsmarginaler och längre serviceintervall för apparaturen i industriella högtemperaturprocesser, utan ökad risk för materielhaverier. I idealfallet vore en metod för randidentifiering baserad på uppföljning av någon indirekt variabel som kan mätas rutinmässigt eller till en ringa kostnad. En dylik variabel för smältugnar är temperaturen i olika positioner i väggen. Denna kan utnyttjas som insignal till en randidentifieringsmetod för att övervaka ugnens väggtjocklek. Vi ger en bakgrund och motivering till valet av den geometriskt endimensionella dynamiska modellen för randidentifiering, som diskuteras i arbetets senare del, framom en flerdimensionell geometrisk beskrivning. I de aktuella industriella tillämpningarna är dynamiken samt fördelarna med en enkel modellstruktur viktigare än exakt geometrisk beskrivning. Lösningsmetoder för den s.k. sidledes värmeledningsekvationen har många saker gemensamt med randidentifiering. Därför studerar vi egenskaper hos lösningarna till denna ekvation, inverkan av mätfel och något som brukar kallas förorening av mätbrus, regularisering och allmännare följder av icke-välställdheten hos sidledes värmeledningsekvationen. Vi studerar en uppsättning av tre olika metoder för randidentifiering, av vilka de två första är utvecklade från en strikt matematisk och den tredje från en mera tillämpad utgångspunkt. Metoderna har olika egenskaper med specifika fördelar och nackdelar. De rent matematiskt baserade metoderna karakteriseras av god noggrannhet och låg numerisk kostnad, dock till priset av låg flexibilitet i formuleringen av den modellbeskrivande partiella differentialekvationen. Den tredje, mera tillämpade, metoden kännetecknas av en sämre noggrannhet förorsakad av en högre grad av icke-välställdhet hos den mera flexibla modellen. För denna gjordes även en ansats till feluppskattning, som senare kunde observeras överensstämma med praktiska beräkningar med metoden. Studien kan anses vara en god startpunkt och matematisk bas för utveckling av industriella tillämpningar av randidentifiering, speciellt mot hantering av olinjära och diskontinuerliga materialegenskaper och plötsliga förändringar orsakade av “nedfallande” väggmaterial. Med de behandlade metoderna förefaller det möjligt att uppnå en robust, snabb och tillräckligt noggrann metod av begränsad komplexitet för randidentifiering.

The PML for rough surface scattering

In this paper we investigate the use of the perfectly matched layer (PML) to truncate a time harmonic rough surface scattering problem in the direction away from the scatterer. We prove existence and uniqueness of the solution of the truncated problem as well as an error estimate depending on the thickness and composition of the layer. This global error estimate predicts a linear rate of convergence (under some conditions on the relative size of the real and imaginary parts of the PML function) rather than the usual exponential rate. We then consider scattering by a half-space and show that the solution of the PML truncated problem converges globally at most quadratically (up to logarithmic factors), providing support for our general theory. However we also prove exponential convergence on compact subsets. We continue by proposing an iterative correction method for the PML truncated problem and, using our estimate for the PML approximation, prove convergence of this method. Finally we provide some numerical results in 2D.(C) 2008 IMACS. Published by Elsevier B.V. All rights reserved.

A wavenumber independent boundary element method for an acoustic scattering problem

In this paper we consider the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data, a problem which models, for example, outdoor sound propagation over inhomogeneous. at terrain. To achieve good approximation at high frequencies with a relatively low number of degrees of freedom, we propose a novel Galerkin boundary element method, using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that, on each element, the approximation space contains polynomials ( of degree.) multiplied by traces of plane waves on the boundary. We prove stability and convergence and show that the error in computing the total acoustic field is O( N-(v+1) log(1/2) N), where the number of degrees of freedom is proportional to N logN. This error estimate is independent of the wavenumber, and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity.

Morfometria e descrição de otólitos de dourada (Brachyplatystoma rosseauxii) (Castelnau, 1855) e de piramutaba (B. vaillantii) (Valenciennes, 1840) (Siluriformes: pimelodidae) e verificação de anéis de crescimento em otólitos de juvenis de dourada e de piramutaba na Ilha de Colares - Baía de Marajó-PA

A dourada (Brachyplatystoma rousseauxii) e a piramutaba (B. vaillantii), são dois bagres migradores de água doce, exploradas pela pesca comercial artesanal e industrial na Amazônia. São de grande importância para a pesca comercial nesta região e atualmente seus estoques estão em estado de sobrexplotação. A literatura cientifica não registra trabalhos sobre a biologia e ecologia referentes aos estágios juvenis destes bagres. Em estudos de dinâmica populacional, a correta estimação da idade é essencial na estimativa dos parâmetros de crescimento, que são essenciais para a avaliação dos estoques. Os estudos de determinação de idade geralmente são feitos em espécimes subadultos e adultos e a idade dos juvenis é retrocalculada. O que pode levar aos erros de avaliação. A presente pesquisa teve como objetivos: (1) descrever e estimar as relações entre as medidas morfométricas dos otólitos lapillus com as medidas corporais de dourada e de piramutaba, e (2) verificar o surgimento dos primeiros anéis de crescimento nos espécimes juvenis. Para o estudo da morfometria e descrição foram utilizados otólitos de dourada e de piramutaba juvenis, sub-adultos e adultos provenientes dos Municípios de Colares (PA), Almeirim (PA), Santarém (PA), Manaus (AM), do Distrito de Mosqueiro (PA), e do estuário amazônico. O período de coletas foi diferente para cada local, e de setembro de 1996 a janeiro de 2006. As seguintes relações foram estimadas: comprimento furcal dos espécimes (CF) x peso total dos espécimes (PT); CF x comprimento máximo do otólito (CMO); CF x largura máxima do otólito (LMO); CF x peso total do otólito (PTO); PT x CMO; PT x LMO e PT x PTO. E as equações estimadas foram: Dourada: PT = 9,5 x 10-6 x Cf 3,0279, CF = 25,259 * CMO1,1571, CF = 40,626 * LMO1,2127, CF = 840,419 * PTO0,4077, PT = 0,125 * CMO3,6547, PT = 0,568 * LMO3,8294, PT = 7549,98 * PTO1,2651; Piramutaba: PT = 5,4 x 10-6 x Cf 3,1441, CF = 31,871 * CMO1,3381, CF = 55,852 * LMO1,3722, CF = 1641,064 * PTO0,4686, PT = 0,3127 * CMO4,1857, PT = 1,823 * LMO4,2788, PT = 66675,977 * PTO1,4508. A descrição dos otólitos classificou os lapilli de dourada e de piramutaba em três categorias conforme seu desenvolvimento e CF dos espécimes: dourada I (60 a 80 mm), dourada II (80 a 110 mm), dourada III (110 a 210 mm), piramutaba I (65 a 80 mm), piramutaba II (80 a 110 mm), e piramutaba III (110 a 190 mm). E no estudo da verificação de anéis de crescimento em lapillus foram utilizados somente os juvenis, com CF £ 200 mm. Na baía de Marajó, a dourada e a piramutaba forma o primeiro anel de crescimento com CF médio de 175,03 mm e 171,03 mm, respectivamente. E as duas espécies têm entre nove e doze meses de idade quando formam o primeiro anel de crescimento. Os raios médios do primeiro anel hialino do corte transversal do lapillus de dourada e de piramutaba foram 0,36 mm (± 0,03) e 0,33 mm (± 0,01), respectivamente. O raio total médio do corte transversal do lapillus de dourada foi de 0,64 mm (± 0,14) e o da piramutaba foi de 0,53 mm (± 0,09). A equação obtida entre o raio total do corte transversal do lapillus de juvenis de dourada com seu CF foi CF = 248,589 * Rt1,1901. E a equação estimada para a piramutaba foi CF = 400,494 * Rt1,7178. Douradas e piramutabas com CF de 60-70 mm têm idade de três a seis meses, com CF de 70-110 mm têm de seis a nove meses e com CF de 110-200 mm têm de nove a doze meses.

Prevalence and geographical distribution of Toxoplasma gondii in dogs in the urban area of Botucatu, SP, Brazil

Toxoplasmosis is a worldwide zoonosis caused by Toxoplasma gondii, and can infect a wide variety of animals including humans. Domestic animals can be an important sentinel population for infection in the community. Occurrence of T. gondii infection was assessed in dogs in the urban area of Botucatu city, SP, Brazil. In the sample, 10% rate for error estimate, 95% confidence interval, and 5% significance level were established. Serum samples were collected from dogs during a rabies vaccination campaign, and were processed using modified agglutination test (MAT). Blood samples were collected from 670 dogs, with homogeneous distribution in five regions in the urban area, representing 3.74% of 17,910 animals vaccinated. In this sample, 17.3% (116/670) dogs (68 – 58.6% female and 48 – 41.4% male) were positive for T. gondii infection (p<0.03). Regarding age of the infected dogs 4.6% (4/88) were younger and 95.4% (84/88) were older than one year (p<0.01); the age of 28 positive animals were undetermined. The serum titers of anti-T. gondii antibodies were: 16 (69.8%; 81/116), 64 (13.8%; 16/116), 256 (15.5%; 18/116), and 1024 (0.9%; 1/116). Prevalence was distributed among the North 14.2% (19/134), South 18.0% (31/172), East 15.7% (19/121), West 21.6% (27/125), and Central 16.9% (20/118) regions of the municipality (p=0.5). In all these regions, females and dogs aged more than one year showed a higher occurrence of T. gondii infection (p<0.05).

Asymptotic integral kernel for ensembles of random normal matrices with radial potentials

The method of steepest descent is used to study the integral kernel of a family of normal random matrix ensembles with eigenvalue distribution P-N (z(1), ... , z(N)) = Z(N)(-1)e(-N)Sigma(N)(i=1) V-alpha(z(i)) Pi(1 <= i<j <= N) vertical bar z(i) - z(j)vertical bar(2), where V-alpha(z) = vertical bar z vertical bar(alpha), z epsilon C and alpha epsilon inverted left perpendicular0, infinity inverted right perpendicular. Asymptotic formulas with error estimate on sectors are obtained. A corollary of these expansions is a scaling limit for the n-point function in terms of the integral kernel for the classical Segal-Bargmann space. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3688293]