Index of Supplementary Materials

Title of paper: Nonparametric Uncertainty Quantification for Stochastic Gradient Flows

Authors: Tyrus Berry and John Harlim

File: fig2.mov

Type: Video

Contents: Comparison of the reconstructed posterior density evolution from the non-parametric filter in Section 5.1.2 with the posterior of the Kalman filter.

Justification: Illustrates the agreement of the filter posteriors which validates our filtering algorithm on this example.

File: fig5a.mov

Type: Video

Contents: Comparison of the reconstructed forecast density from the non-parametric forecast in Section 5.2.1 with the Monte-Carlo forecast up to time 10.

Justification: Illustrates the agreement of the reconstructed forecast with the Monte-Carlo histogram during the short time scale characterized by the equilibration of initial condition with respect to the right potential well.

File: fig5b.mov

Type: Video

Contents: Comparison of the reconstructed forecast density from the non-parametric forecast in Section 5.2.1 with the Monte-Carlo forecast up to time 1000.

Justification: Illustrates the agreement of the reconstructed forecast with the Monte-Carlo histogram during the long time scale characterized by the equilibration of the two potential wells.

File: fig6.mov

Type: Video

Contents: Comparison of the reconstructed posterior density evolution from the non-parametric filter in Section 5.2.2 with the posterior of the ensemble Kalman filter for a linear observation.

Justification:  Illustrates the difference between the non-linear filter posterior and the Gaussian posterior of the EnKF.

File: fig7a.mov

Type: Video

Contents: Comparison of the reconstructed posterior density evolution from the non-parametric filter in Section 5.2.2 with the posterior of the ensemble Kalman filter for a nonlinear and pathological observation function.

Justification:  Illustrates the symmetric bi-modal structure of the non-linear posterior due to a pathological observation.

File: fig7b.mov

Type: Video

Contents: Comparison of the reconstructed posterior density evolution from the non-parametric filter in Section 5.2.2 with the posterior of the ensemble Kalman filter for a nonlinear and nearly pathological observation function.

Justification:  Illustrates how the non-linear posterior can switch between bi-modal and unimodal structures in response to information from a nonlinear and nearly pathological observation.

File: fig9.mov

Type: Video

Contents: Comparison of the reconstructed density evolution from the non-parametric response estimate in Section 5.2.3 with the histogram of a Monte-Carlo simulation of the chaotically forced system (5.2) with the perturbed potential (5.4).

Justification:  Illustrates the close agreement on a short time scale between the response of the full chaotically forced system (5.2) to the perturbed potential (5.4) and the non-parametric model which approximates the system as a stochastically forced gradient flow.