Related Experiment Video
Updated: May 11, 2026
07:46
Data Acquisition Protocol for Determining Embedded Sensitivity Functions
Published on: April 20, 2016
A COMPUTATIONAL MEASURE THEORETIC APPROACH TO INVERSE SENSITIVITY PROBLEMS II: A POSTERIORI ERROR ANALYSIS.
T Butler1, D Estep, J Sandelin
1Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, TX 78712 ( tbutler@ices.utexas.edu ).
SIAM Journal on Numerical Analysis
|May 14, 2013
Summary
This study develops a numerical method for probabilistic inverse sensitivity analysis when map outputs are implicitly defined and numerically approximated. It provides an a posteriori error estimate accounting for all statistical and numerical errors.
Area of Science:
- Numerical Analysis
- Computational Mathematics
- Sensitivity Analysis
Background:
- Previous work developed methods for exactly evaluated maps.
- Implicitly defined maps, often requiring differential equation solutions, present computational challenges.
- Numerical approximation of map outputs introduces deterministic errors.
Purpose of the Study:
- To develop and analyze a numerical method for probabilistic inverse sensitivity analysis with implicitly defined maps.
- To derive an a posteriori error estimate for the computed distribution.
- To account for both statistical and numerical errors in the analysis.
Main Methods:
- Numerical method development for inverse problems.
- A posteriori error estimation techniques.
- Analysis applied to maps defined by initial value problems.
Main Results:
- A general analysis of the numerical method for implicitly defined maps.
- An a posteriori error estimate incorporating statistical and numerical errors.
- Demonstration of the method's application to differential equation-based maps.
Conclusions:
- The developed method provides a robust way to assess the accuracy of inverse problem solutions.
- The a posteriori error estimate is crucial for understanding the reliability of results from numerically approximated maps.
- This work extends sensitivity analysis to more complex, computationally intensive scenarios.
Related Concept Videos
01:30Derivatives of Inverse Trigonometric Functions
A ship tracking an approaching aircraft relies on geometric measurements to find out the aircraft’s position relative to the observer. By measuring the slant distance to the aircraft and the angle of elevation, the horizontal and vertical components of the distance can be obtained using trigonometric relationships. This geometric approach provides a basis for analyzing how the observed angle changes as the aircraft moves closer to the ship.To examine the mathematical behavior of the angle of...
01:11Statistical Analysis: Overview
When we take repeated measurements on the same or replicated samples, we will observe inconsistencies in the magnitude. These inconsistencies are called errors. To categorize and characterize these results and their errors, the researcher can use statistical analysis to determine the quality of the measurements and/or suitability of the methods.
One of the most commonly used statistical quantifiers is the mean, which is the ratio between the sum of the numerical values of all results and the...
One of the most commonly used statistical quantifiers is the mean, which is the ratio between the sum of the numerical values of all results and the...
01:29Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving
Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
01:13Accuracy and Errors in Hypothesis Testing
Hypothesis testing is a fundamental statistical tool that begins with the assumption that the null hypothesis H0 is true. During this process, two types of errors can occur: Type I and Type II. A Type I error refers to the incorrect rejection of a true null hypothesis, while a Type II error involves the failure to reject a false null hypothesis.
In hypothesis testing, the probability of making a Type I error, denoted as α, is commonly set at 0.05. This significance level indicates a 5% chance...
In hypothesis testing, the probability of making a Type I error, denoted as α, is commonly set at 0.05. This significance level indicates a 5% chance...
01:24Approximate Integration
In many practical and theoretical contexts, the exact value of a definite integral may be inaccessible. This limitation typically arises when the antiderivative of a function is either unknown or cannot be expressed in a closed mathematical form. Alternatively, it can occur when a function is defined not by a formula but by a finite set of empirical data points, such as those collected during experiments. In these cases, approximate integration techniques provide a valuable solution.One of the...
01:10Propagation of Uncertainty from Systematic Error
The atomic mass of an element varies due to the relative ratio of its isotopes. A sample's relative proportion of oxygen isotopes influences its average atomic mass. For instance, if we were to measure the atomic mass of oxygen from a sample, the mass would be a weighted average of the isotopic masses of oxygen in that sample. Since a single sample is not likely to perfectly reflect the true atomic mass of oxygen for all the molecules of oxygen on Earth, the mass we obtain from this particular...