| 1. |
- Babuska, Ivo, et al.
(author)
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A stochastic collocation method for elliptic partial differential equations with random input data
- 2007
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In: SIAM Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 45:3, s. 1005-1034
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Journal article (peer-reviewed)abstract
- In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms ( input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in [I. Babuska, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 ( 2004), pp. 800-825] and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method.
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| 2. |
- Collier, Nathan, et al.
(author)
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A continuation multilevel Monte Carlo algorithm
- 2015
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In: BIT Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 55:2, s. 399-432
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Journal article (peer-reviewed)abstract
- We propose a novel Continuation Multi Level Monte Carlo (CMLMC) algorithm for weak approximation of stochastic models. The CMLMC algorithm solves the given approximation problem for a sequence of decreasing tolerances, ending when the required error tolerance is satisfied. CMLMC assumes discretization hierarchies that are defined a priori for each level and are geometrically refined across levels. The actual choice of computational work across levels is based on parametric models for the average cost per sample and the corresponding variance and weak error. These parameters are calibrated using Bayesian estimation, taking particular notice of the deepest levels of the discretization hierarchy, where only few realizations are available to produce the estimates. The resulting CMLMC estimator exhibits a non-trivial splitting between bias and statistical contributions. We also show the asymptotic normality of the statistical error in the MLMC estimator and justify in this way our error estimate that allows prescribing both required accuracy and confidence in the final result. Numerical results substantiate the above results and illustrate the corresponding computational savings in examples that are described in terms of differential equations either driven by random measures or with random coefficients.
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| 3. |
- Lundin, Andreas, et al.
(author)
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Veno-arterial CO2 difference and respiratory quotient after cardiac arrest: An observational cohort study.
- 2021
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In: Journal of critical care. - : Elsevier BV. - 1557-8615 .- 0883-9441. ; 62:April, s. 131-137
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Journal article (peer-reviewed)abstract
- To characterize venous-arterial CO2 difference (ΔpCO2) and the respiratory quotient (RQ) in post cardiac arrest patients and evaluate the association between these parameters and patient outcome.Data were obtained retrospectively from post cardiac arrest patients admitted between 2007 and 2016 to a medical intensive care unit. Comatose, adult patients in whom arterial and venous blood gas analyses were concomitantly performed in the first 24h were included. Patients were grouped according to the time-point of sampling; 0-6, 6-12 and 12-24h after admission.308 patients were included; 174 (56%) died before ICU discharge and 212 (69%) had an unfavorable neurologic outcome. RQ was associated with ICU mortality (OR:1.09 (95%CI: 1.04-1.14; p<0.01)), although not with neurological outcome. ΔpCO2 was negatively associated with both ICU mortality (OR: 0.92 (95%CI: 0.86-0.99; p=0.02)) and poor neurologic outcome (adjusted OR: 0.93 (95%CI: 0.87-0.99; p=0.02)). ΔpCO2 predicted an elevated RQ; a ΔpCO2 above 8.5mmHg identified a high RQ with reasonable sensitivity and specificity.RQ was associated with ICU mortality and ΔpCO2 identified elevated RQ in the early phase after cardiac arrest. However, ΔpCO2 were negatively associated with both ICU mortality and neurologic outcome.
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| 4. |
- Malenova, Gabriela
(author)
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Uncertainty quantification for high frequency waves
- 2018
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Doctoral thesis (other academic/artistic)abstract
- We consider high frequency waves satisfying the scalar wave equation with highly oscillatory initial data represented by a short wavelength ε. The speed of propagation of the medium as well as the phase and amplitude of the initial data is assumed to be uncertain, described by a finite number of independent random variables with known probability distributions. We introduce quantities of interest (QoIs) as spatial and/or temporal averages of the squared modulus of the wave solution, or its derivatives. The main focus of this work is on fast computation of the statistics of those QoIs in the form of moments like the average and variance. They are difficult to obtain numerically by standard methods, as the cost grows rapidly with ε−1 and the dimension of the stochastic space. We therefore propose a fast approximation method consisting of three techniques: the Gaussian beam method to approximate the wave solution, the numerical steepest descent method to compute the QoIs and the sparse stochastic collocation to evaluate the statistics.The Gaussian beam method is introduced to avoid the considerable cost of approximating the wave solution by direct methods. A Gaussian beam is an asymptotic solution to the wave equation localized around rays, bicharacteristics of the wave equation. This setup allows us to replace the PDE by a set of ODEs that can be solved independently of ε.The computation of QoIs includes evaluations of highly oscillatory integrals. The idea of the numerical steepest descent method is to change the integration path in the complex plane such that the integrand is non-oscillatory along it. Standard quadrature methods can be then utilized. We construct such paths for our case and show error estimates for the integral approximation by the Gauss-Laguerre and Gauss-Hermite quadrature.Finally, the evaluation of statistical moments of the QoI may suffer from the curse of dimensionality. The sparse grid collocation method introduces a framework where certain large group of points can be neglected while only slightly reducing the convergence rate. The regularity of the QoIs in terms of the input random parameters and the wavelength is important for the method to be efficient. In particular, the size of the derivatives should be bounded independently of ε. We show that the QoIs indeed have this property, despite the highly oscillatory character of the waves.
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