Kinetic methods for inverse problems

Herty, Michaël; Visconti, Giuseppe

doi:10.3934/krm.2019042

Cited by 38 publications

(74 citation statements)

References 44 publications

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“…The numerical experiment considered here is the example originally presented in (Ernst, Sprungk and Starkloff, 2015) and also used in (Herty and Visconti, 2018). We start by defining the forward map which is given by the one-dimensional elliptic boundary value problem…”

Section: Numerical Results: Low Dimensional Parameter Spacementioning

confidence: 99%

“…with boundary conditions p(0) = 0 and p(1) = u 2 . The explicit solution for this problem, (see Herty and Visconti, 2018), is given by…”

Section: Numerical Results: Low Dimensional Parameter Spacementioning

confidence: 99%

“…The resulting Bayesian inverse problem is then solved, approximately, by the algorithms we now outline and with with observed data y = (27.5, 79.7) . Following (Herty and Visconti, 2018), we consider an initial ensemble drawn from N(0, 1) × U(90, 110). Figure 1 shows the results for the solution of the Bayesian inverse problem considered above.…”

Section: Numerical Results: Low Dimensional Parameter Spacementioning

confidence: 99%

“…Furthermore, the papers (Schillings and Stuart, 2017; study continuous time limits of EKI algorithms and, in the case of linear inverse problems, exhibit a gradient flow structure for the standard least squares loss function, preconditioned by the empirical covariance of the particles; a related structure was highlighted in (Bergemann and Reich, 2010a). The paper (Herty and Visconti, 2018), which has inspired aspects of our work, builds on the paper (Schillings and Stuart, 2017) to study the same problem in the mean-field limit; their mean-field perspective brings considerable insight which we build upon in this paper. Recent work (Ding and Li, 2019) has studied the approach to the mean-field limit for linear inverse problems, together with making connection to the appropriate nonlinear Fokker-Planck equation whose solution characterizes the distribution in the mean-field limit.…”

Section: Literature Reviewmentioning

confidence: 99%

“…In order to frame the analysis of our methods, we introduce a new metric, named the Kalman-Wasserstein metric, defined through both the covariance matrix of the mean field limit and the Wasserstein metric. The work builds on the novel perspectives introduced in (Herty and Visconti, 2018) and leads to new algorithms that will be useful within large-scale parameter learning and uncertainty quantification studies, such as those proposed in (Schneider et al, 2017).…”

Section: Literature Reviewmentioning

confidence: 99%

See 4 more Smart Citations

Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler

Garbuno-Iñigo¹,

Hoffmann²,

Li³

et al. 2020

SIAM J. Appl. Dyn. Syst.

142

253

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Solving inverse problems without the use of derivatives or adjoints of the forward model is highly desirable in many applications arising in science and engineering. In this paper we propose a new version of such a methodology, a framework for its analysis, and numerical evidence of the practicality of the method proposed. Our starting point is an ensemble of over-damped Langevin diffusions which interact through a single preconditioner computed as the empirical ensemble covariance. We demonstrate that the nonlinear Fokker-Planck equation arising from the mean-field limit of the associated stochastic differential equation (SDE) has a novel gradient flow structure, built on the Wasserstein metric and the covariance matrix of the noisy flow. Using this structure, we investigate large time properties of the Fokker-Planck equation, showing that its invariant measure coincides with that of a single Langevin diffusion, and demonstrating exponential convergence to the invariant measure in a number of settings. We introduce a new noisy variant on ensemble Kalman inversion (EKI) algorithms found from the original SDE by replacing exact gradients with ensemble differences; this defines the ensemble Kalman sampler (EKS). Numerical results are presented which demonstrate its efficacy as a derivative-free approximate sampler for the Bayesian posterior arising from inverse problems.

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Section: Numerical Results: Low Dimensional Parameter Spacementioning

confidence: 99%

“…with boundary conditions p(0) = 0 and p(1) = u 2 . The explicit solution for this problem, (see Herty and Visconti, 2018), is given by…”

Section: Numerical Results: Low Dimensional Parameter Spacementioning

confidence: 99%

Section: Numerical Results: Low Dimensional Parameter Spacementioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

See 3 more Smart Citations

Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler

Garbuno-Iñigo¹,

Hoffmann²,

Li³

et al. 2020

SIAM J. Appl. Dyn. Syst.

142

253

View full text Add to dashboard Cite

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Actionable Artificial Intelligence for the Future of Production

Behery

Brauner

Zhou

et al. 2023

Interdisciplinary Excellence Accelerator Series

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The Internet of Production (IoP) promises to be the answer to major challenges facing the Industrial Internet of Things (IIoT) and Industry 4.0. The lack of inter-company communication channels and standards, the need for heightened safety in Human Robot Collaboration (HRC) scenarios, and the opacity of data-driven decision support systems are only a few of the challenges we tackle in this chapter. We outline the communication and data exchange within the World Wide Lab (WWL) and autonomous agents that query the WWL which is built on the Digital Shadows (DS). We categorize our approaches intomachine level, process level, and overarching principles. This chapter surveys the interdisciplinary work done in each category, presents different applications of the different approaches, and offers actionable items and guidelines for future work.The machine level handles the robots and machines used for production and their interactions with the human workers. It covers low-level robot control and optimization through gray-box models, task-specific motion planning, and optimization through reinforcement learning. In this level, we also examine quality assurance through nonintrusive real-time quality monitoring, defect recognition, and quality prediction. Work on this level also handles confidence, verification, and validation of re-configurable processes and reactive, modular, transparent process models. The process level handles the product life cycle, interoperability, and analysis and optimization of production processes, which is overall attained by analyzing process data and event logs to detect and eliminate bottlenecks and learn new process models. Moreover, this level presents a communication channel between human workers and processes by extracting and formalizing human knowledge into ontology and providing a decision support by reasoning over this information. Overarching principles present a toolbox of omnipresent approaches for data collection, analysis, augmentation, and management, as well as the visualization and explanation of black-box models.

show abstract

Actionable Artificial Intelligence for the Future of Production

Behery,

Brauner,

Zhou

et al. 2023

Interdisciplinary Excellence Accelerator Series

View full text Add to dashboard Cite

The Internet of Production (IoP) promises to be the answer to major challenges facing the Industrial Internet of Things (IIoT) and Industry 4.0. The lack of inter-company communication channels and standards, the need for heightened safety in Human Robot Collaboration (HRC) scenarios, and the opacity of data-driven decision support systems are only a few of the challenges we tackle in this chapter. We outline the communication and data exchange within the World Wide Lab (WWL) and autonomous agents that query the WWL which is built on the Digital Shadows (DS). We categorize our approaches into machine level, process level, and overarching principles. This chapter surveys the interdisciplinary work done in each category, presents different applications of the different approaches, and offers actionable items and guidelines for future work.The machine level handles the robots and machines used for production and their interactions with the human workers. It covers low-level robot control and optimization through gray-box models, task-specific motion planning, and optimization through reinforcement learning. In this level, we also examine quality assurance through nonintrusive real-time quality monitoring, defect recognition, and quality prediction. Work on this level also handles confidence, verification, and validation of re-configurable processes and reactive, modular, transparent process models. The process level handles the product life cycle, interoperability, and analysis and optimization of production processes, which is overall attained by analyzing process data and event logs to detect and eliminate bottlenecks and learn new process models. Moreover, this level presents a communication channel between human workers and processes by extracting and formalizing human knowledge into ontology and providing a decision support by reasoning over this information. Overarching principles present a toolbox of omnipresent approaches for data collection, analysis, augmentation, and management, as well as the visualization and explanation of black-box models.

show abstract

Kinetic methods for inverse problems

Cited by 38 publications

References 44 publications

Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler

Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler

Actionable Artificial Intelligence for the Future of Production

Actionable Artificial Intelligence for the Future of Production

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