Standard error computations for uncertainty quantification in inverse problems: Asymptotic theory vs. bootstrapping

Banks, Harvey Thomas; Holm, Kathleen; Robbins, Danielle

doi:10.1016/j.mcm.2010.06.026

Cited by 38 publications

(46 citation statements)

References 22 publications

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“…To this end, we will compare two techniques for determining confidence intervals, specifically the asymptotic theory discussed in [5,16] versus using bootstrapping as discussed in [5]. In reality, one will obtain a set of experimental data and then one needs to determine how many (if any) relaxation times are required to represent well the data.…”

Section: Estimation Of Materials Parametersmentioning

confidence: 99%

“…Most asymptotic error theory [5,16] is described in the context of an ODE model exampleż(t) = f (z(t; θ); θ). However, we can use the PDE sensitivities of the model output with respect to each parameter in θ, namely ∂u(L, t; 10 θ ) ∂θ i , in a similar manner to the ODE sensitivities in the asymptotic theory.…”

Section: Asymptotic Error Analysismentioning

confidence: 99%

“…However, we can use the PDE sensitivities of the model output with respect to each parameter in θ, namely ∂u(L, t; 10 θ ) ∂θ i , in a similar manner to the ODE sensitivities in the asymptotic theory. The steps of the asymptotic theory error analysis are as follows (the theory for the following steps is described in [5,16]). …”

Section: Asymptotic Error Analysismentioning

confidence: 99%

“…For ease of presentation, we reiterate here the algorithm described in [5], in the context of the current viscoelastic model under study.…”

Section: Bootstrapping Error Analysismentioning

confidence: 99%

“…Increase the index m by 1 and repeat steps 3-5. This iterative process should be carried out for M times where M is large (we used M = 1000, as suggested in [5]). This will give M estimates…”

Section: Bootstrapping Error Analysismentioning

confidence: 99%

See 4 more Smart Citations

Material parameter estimation and hypothesis testing on a 1D viscoelastic stenosis model: Methodology

Banks

Kenz

et al. 2013

Journal of Inverse and Ill-Posed Problems

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Non-invasive detection, localization and characterization of an arterial stenosis (a blockage or partial blockage in the artery) continues to be an important problem in medicine. Partial blockage stenoses are known to generate disturbances in blood flow which generate shear waves in the chest cavity. We examine a one-dimensional viscoelastic model that incorporates Kelvin-Voigt damping and internal variables, and develop a proof-of-concept methodology using simulated data. We first develop an estimation procedure for the material parameters. We use this procedure to determine confidence intervals for the estimated parameters, which indicates the efficacy of finding parameter estimates in practice. Confidence intervals are computed using asymptotic error theory as well as bootstrapping. We then develop a model comparison test to be used in determining if a particular data set came from a low input amplitude or a high input amplitude; this we anticipate will aid in determining when stenosis is present. These two thrusts together will serve as the methodological basis for our continuing analysis using experimental data currently being collected.Mathematics Subject Classification: 62F12; 62F40; 65M32; 74D05.

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Section: Estimation Of Materials Parametersmentioning

confidence: 99%

Section: Asymptotic Error Analysismentioning

confidence: 99%

Section: Asymptotic Error Analysismentioning

confidence: 99%

“…For ease of presentation, we reiterate here the algorithm described in [5], in the context of the current viscoelastic model under study.…”

Section: Bootstrapping Error Analysismentioning

confidence: 99%

Section: Bootstrapping Error Analysismentioning

confidence: 99%

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Material parameter estimation and hypothesis testing on a 1D viscoelastic stenosis model: Methodology

Banks

Kenz

et al. 2013

Journal of Inverse and Ill-Posed Problems

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A bootstrap‐based method for optimal design of experiments

Emire

Zettel

Hitzmann

2016

Journal of Chemometrics

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Bootstrapping can be used for the estimation of parameter variances, and it is straightforward to be implemented but computationally demanding compared with other methods for parameter error estimation. It is not bound to any restrictions such as the distribution of measurement errors. And because of the possible asymmetry of the probability densities of the parameters, the parameter estimation errors acquired by bootstrapping are likely to be more accurate. In this work the feasibility of a bootstrap‐based method for optimal experimental design was evaluated for the Peleg model. The optimal design was performed, based on the Cramér‐Rao lower bound as a benchmark. Afterwards, the optimal design was calculated based on the bootstrap method. It is demonstrated that a bootstrap‐based optimal design of experiments will give comparable results with the Cramér‐Rao lower bound optimal designs, however with slightly different measurement points in time. If the parameter errors obtained from both optimal experimental designs are compared, they deviate for the 2 methods on average by 1.5%. Bootstrapping can be used for problems, which cannot be solved using Cramér‐Rao lower bound because of necessary but invalid assumptions. However, the benefits of the bootstrap method come at the cost of a significant increase in computational effort. Under similar conditions, the computation time for a bootstrap‐based optimal design was 25 minutes compared with 5 seconds when using the Cramér‐Rao lower bound method. As computers get faster and faster over time, the increase in computational demand will probably become less relevant in the future.

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Optimal Design of Experiments Based on Bootstrapping

Emire

Hitzmann

2017

Chemie Ingenieur Technik

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In diesem Beitrag wird gezeigt, dass eine optimale Versuchsplanung zur Bestimmung kinetischer Parameter mittels Bootstrapping in moderater Zeit durchgeführt werden kann. Als Beispielsystem dient hier eine Folgereaktion erster Ordnung. Bootstrapping zur Parameterfehlerschätzung ist zwar signifikant rechenaufwändiger, bietet aber gegenüber der klassischen Methode, basieren auf der Fisher-Information, den Vorteil, dass die Wahrscheinlichkeitsdichteverteilungen der Parameter nicht als normalverteilt angenommen werden müssen. Auch die Messfehler müssen für Bootstrapping nicht bekannt sein.This contribution shows, that an optimal experimental design for an accurate determination of kinetic parameters based on bootstrapping can be calculated in reasonable time. As example system, a consecutive reaction with kinetics of first order was used. Bootstrapping for estimation of parameter errors has a significantly higher computational demand than the classical method, based on the Fisher information. But bootstrapping has the big advantage of not requiring the probability densities of the parameters or measurement errors to follow any distribution. They don't even have to be known.

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Standard error computations for uncertainty quantification in inverse problems: Asymptotic theory vs. bootstrapping

Cited by 38 publications

References 22 publications

Material parameter estimation and hypothesis testing on a 1D viscoelastic stenosis model: Methodology

Material parameter estimation and hypothesis testing on a 1D viscoelastic stenosis model: Methodology

A bootstrap‐based method for optimal design of experiments

Optimal Design of Experiments Based on Bootstrapping

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