Bayesian Inverse Problems and Model Uncertainties

贝叶斯逆问题和模型不确定性

基本信息

  • 批准号:
    1714617
  • 负责人:
  • 金额:
    $ 21.66万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2017
  • 资助国家:
    美国
  • 起止时间:
    2017-08-01 至 2021-07-31
  • 项目状态:
    已结题

项目摘要

The traditional and very natural paradigm in science is to build predictive mathematical models that move from causes to consequences. However, it often happens that observations of consequences are available, and one needs to identify the causes that made the observations possible. The latter type of problems are known as inverse problems. Inverse problems are characterized by their high sensitivity to errors in the measurements and the models used, the existence of not just one but several possible solutions, and their computational complexity. This project focuses on one particular but central aspect in inverse problems: Assume that a very detailed and complex predictive model exists, known to be able to produce predictions that match well with observations. Furthermore, assume that the model is computationally very demanding, and it contains numerous parameters whose values are unknown or poorly known. To solve the inverse problem in the required time frame, it may be that a simplified, or reduced model, needs to be used. Given that inverse problems are sensitive to errors in the model, it typically happens that the model reduction introduces an uncontrolled error, or discrepancy between the model and reality, that may render the solution of the inverse problem completely useless. The investigator and his colleagues have proposed a general methodology to handle the modeling error problem in the statistical framework, and in this project, the aim is to develop the methodology further so that it allows a reliable way to find a useful solution with limited computational resources, and to quantify the reliability of such solution. The main applications in this project are in the field of medicine, including mapping of the brain activity, identification and localization of stroke using a portable equipment, and development of fast and portable computing tools to model blood flow, but the results also have applications beyond medical applications. The technical difficulty in handling the modeling error in an inverse problem is that it depends on the unknown cause that the inverse problem is seeking. However, the Bayesian statistical paradigm provides a very natural solution to this problem. In the Bayesian context, the unknown of primary interest is described as a random variable that has an a priori probability distribution, and therefore, it is possible to estimate a probability distribution of the modeling error and include it as part of the likelihood model. This basic observation has been shown to lead to algorithms that dramatically improve the estimates compared to results that ignore the modeling error. In this project, the methodology will be developed further, by carefully following how the inclusion of the modeling error distribution affects the Bayesian posterior distribution of the unknown, and conversely, how the modeling error distribution can be updated after the data is used to update the prior density of the unknown. Such tracking will hopefully lead to a computationally efficient way of quantifying uncertainties in the inverse solutions in the presence of modeling errors. One family of problems the project addresses is multi-scale inverse problems, in which the unknowns of primary interest are describing fine-scale behavior of the system, while the observation represents a macroscopic, coarse scale quantity. These types of problems often appear in biological applications, where the high-fidelity microscopic models are often stochastic in nature, and cannot be handled directly in the standard Bayesian framework.
科学中的传统和非常自然的范式是建立从原因到后果的预测数学模型。然而,经常发生的情况是,对后果的观察是可用的,人们需要确定使观察成为可能的原因。后一种类型的问题被称为逆问题。反问题的特点是对测量和模型误差的敏感性高,存在不止一个而是多个可能的解,以及计算复杂性。本项目关注反问题中一个特殊但重要的方面:假设存在一个非常详细和复杂的预测模型,已知能够产生与观察结果匹配良好的预测。此外,假设该模型在计算上要求很高,并且它包含许多参数,这些参数的值是未知的或知之甚少。为了在所需的时间范围内解决逆问题,可能需要使用简化或缩减的模型。假设逆问题对模型中的误差敏感,通常会发生模型简化引入不受控制的误差或模型与现实之间的差异,这可能使逆问题的解决方案完全无用。研究者和他的同事们提出了一种通用的方法来处理统计框架中的建模误差问题,在这个项目中,目的是进一步发展这种方法,以便它可以用有限的计算资源找到一个有用的解决方案,并量化这种解决方案的可靠性。该项目的主要应用是在医学领域,包括绘制大脑活动图,使用便携式设备识别和定位中风,以及开发快速便携的计算工具来模拟血流,但其结果也有超出医疗应用的应用。在逆问题中处理建模误差的技术困难在于,它取决于逆问题所寻求的未知原因。然而,贝叶斯统计范式为这个问题提供了一个非常自然的解决方案。在贝叶斯上下文中,主要关注的未知数被描述为具有先验概率分布的随机变量,因此,可以估计建模误差的概率分布并将其包括为似然模型的一部分。与忽略建模误差的结果相比,这一基本观察结果已被证明会导致算法显着改善估计。在这个项目中,该方法将进一步发展,通过仔细遵循建模误差分布的包含如何影响未知的贝叶斯后验分布,以及相反,如何在数据用于更新未知的先验密度后更新建模误差分布。这样的跟踪将有望导致一个计算有效的方式量化的不确定性的逆解决方案中的建模误差的存在。 该项目解决的一系列问题是多尺度逆问题,其中主要感兴趣的未知数描述系统的细尺度行为,而观测代表宏观的粗尺度量。这些类型的问题经常出现在生物学应用中,其中高保真微观模型通常在本质上是随机的,并且不能在标准贝叶斯框架中直接处理。

项目成果

期刊论文数量(16)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Sparsity Promoting Hybrid Solvers for Hierarchical Bayesian Inverse Problems
稀疏性促进分层贝叶斯逆问题的混合求解器
  • DOI:
    10.1137/20m1326246
  • 发表时间:
    2020
  • 期刊:
  • 影响因子:
    3.1
  • 作者:
    Calvetti, Daniela;Pragliola, Monica;Somersalo, Erkki
  • 通讯作者:
    Somersalo, Erkki
A Bayesian filtering approach to layer stripping for electrical impedance tomography
  • DOI:
    10.1088/1361-6420/ab6f9e
  • 发表时间:
    2020-01
  • 期刊:
  • 影响因子:
    2.1
  • 作者:
    D. Calvetti;S. Nakkireddy;E. Somersalo
  • 通讯作者:
    D. Calvetti;S. Nakkireddy;E. Somersalo
Iterative updating of model error for Bayesian inversion
  • DOI:
    10.1088/1361-6420/aaa34d
  • 发表时间:
    2018-02-01
  • 期刊:
  • 影响因子:
    2.1
  • 作者:
    Calvetti, Daniela;Dunlop, Matthew;Stuart, Andrew
  • 通讯作者:
    Stuart, Andrew
Metapopulation Network Models for Understanding, Predicting, and Managing the Coronavirus Disease COVID-19
  • DOI:
    10.3389/fphy.2020.00261
  • 发表时间:
    2020-06-19
  • 期刊:
  • 影响因子:
    3.1
  • 作者:
    Calvetti, Daniela;Hoover, Alexander P.;Somersalo, Erkki
  • 通讯作者:
    Somersalo, Erkki
Bayesian Mesh Adaptation for Estimating Distributed Parameters
  • DOI:
    10.1137/20m1326222
  • 发表时间:
    2020-12
  • 期刊:
  • 影响因子:
    0
  • 作者:
    D. Calvetti;A. Cosmo;S. Perotto;E. Somersalo
  • 通讯作者:
    D. Calvetti;A. Cosmo;S. Perotto;E. Somersalo
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Erkki Somersalo其他文献

The uniqueness of the one-dimensional electromagnetic inversion with bounded potentials
  • DOI:
    10.1016/0022-247x(87)90112-0
  • 发表时间:
    1987-11-01
  • 期刊:
  • 影响因子:
  • 作者:
    Lassi Päivärinta;Erkki Somersalo
  • 通讯作者:
    Erkki Somersalo
Perspectives in Numerical Analysis 2008
  • DOI:
    10.1007/s10543-008-0186-8
  • 发表时间:
    2008-08-05
  • 期刊:
  • 影响因子:
    1.700
  • 作者:
    Timo Eirola;Rolf Jeltsch;Claes Johnson;Erkki Somersalo;Rolf Stenberg
  • 通讯作者:
    Rolf Stenberg

Erkki Somersalo的其他文献

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{{ truncateString('Erkki Somersalo', 18)}}的其他基金

Bridging the Gap between Discrete and Continuous Partial Differential Equations in Medical imaging
弥合医学成像中离散和连续偏微分方程之间的差距
  • 批准号:
    2204618
  • 财政年份:
    2022
  • 资助金额:
    $ 21.66万
  • 项目类别:
    Standard Grant
Computational Model-based Statistical Methods in Biomedicine
生物医学中基于计算模型的统计方法
  • 批准号:
    1312424
  • 财政年份:
    2013
  • 资助金额:
    $ 21.66万
  • 项目类别:
    Standard Grant
New statistical approaches to inverse problems in biomedicine
生物医学逆问题的新统计方法
  • 批准号:
    1016183
  • 财政年份:
    2010
  • 资助金额:
    $ 21.66万
  • 项目类别:
    Standard Grant

相似国自然基金

新型简化Inverse Lax-Wendroff方法的发展与应用
  • 批准号:
  • 批准年份:
    2022
  • 资助金额:
    30 万元
  • 项目类别:
    青年科学基金项目
基于高阶格式的Inverse Lax-Wendroff方法及其稳定性分析
  • 批准号:
    11801143
  • 批准年份:
    2018
  • 资助金额:
    25.0 万元
  • 项目类别:
    青年科学基金项目

相似海外基金

Gaussian process regression for Bayesian inverse problems
贝叶斯逆问题的高斯过程回归
  • 批准号:
    EP/X01259X/1
  • 财政年份:
    2023
  • 资助金额:
    $ 21.66万
  • 项目类别:
    Research Grant
CAREER: Ensemble Kalman Methods and Bayesian Optimization in Inverse Problems and Data Assimilation
职业:反问题和数据同化中的集成卡尔曼方法和贝叶斯优化
  • 批准号:
    2237628
  • 财政年份:
    2023
  • 资助金额:
    $ 21.66万
  • 项目类别:
    Continuing Grant
Machine Learning for Bayesian Inverse Problems
贝叶斯逆问题的机器学习
  • 批准号:
    2208535
  • 财政年份:
    2022
  • 资助金额:
    $ 21.66万
  • 项目类别:
    Standard Grant
Bayesian inverse problems for soft tissue mechanics
软组织力学的贝叶斯反问题
  • 批准号:
    2596737
  • 财政年份:
    2021
  • 资助金额:
    $ 21.66万
  • 项目类别:
    Studentship
Tensor decomposition sampling algorithms for Bayesian inverse problems
贝叶斯逆问题的张量分解采样算法
  • 批准号:
    EP/T031255/1
  • 财政年份:
    2021
  • 资助金额:
    $ 21.66万
  • 项目类别:
    Research Grant
Design and Sensitivity Analysis of Infinite-Dimensional Bayesian Inverse Problems
无限维贝叶斯反问题的设计与敏感性分析
  • 批准号:
    2111044
  • 财政年份:
    2021
  • 资助金额:
    $ 21.66万
  • 项目类别:
    Continuing Grant
CAREER: Scalable Approaches for Large-Scale Data-driven Bayesian Inverse Problems in High Dimensional Parameter Spaces
职业:高维参数空间中大规模数据驱动的贝叶斯逆问题的可扩展方法
  • 批准号:
    1845799
  • 财政年份:
    2019
  • 资助金额:
    $ 21.66万
  • 项目类别:
    Continuing Grant
Analysis of maximum a posteriori estimators: Common convergence theories for Bayesian and variational inverse problems
最大后验估计量分析:贝叶斯和变分逆问题的常见收敛理论
  • 批准号:
    415980428
  • 财政年份:
    2019
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    $ 21.66万
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    Research Grants
CAREER: Large-Scale Bayesian Inverse Problems Governed by Differential and Differential-Algebraic Equations
职业:微分方程和微分代数方程控制的大规模贝叶斯逆问题
  • 批准号:
    1654311
  • 财政年份:
    2017
  • 资助金额:
    $ 21.66万
  • 项目类别:
    Continuing Grant
Classification of Methods for Bayesian Inverse Problems Governed by Partial Differential Equations
偏微分方程治理贝叶斯反问题方法的分类
  • 批准号:
    1723211
  • 财政年份:
    2017
  • 资助金额:
    $ 21.66万
  • 项目类别:
    Continuing Grant
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