Scientific data interpolation with low dimensional manifold model

Zhu, Wei; Wang, Bao; Barnard, Richard C.; Hauck, Cory D.; Jenko, Frank; Osher, Stanley

doi:10.1016/j.jcp.2017.09.048

Title: Scientific data interpolation with low dimensional manifold model

Abstract

Here, we propose to apply a low dimensional manifold model to scientific data interpolation from regular and irregular samplings with a significant amount of missing information. The low dimensionality of the patch manifold for general scientific data sets has been used as a regularizer in a variational formulation. The problem is solved via alternating minimization with respect to the manifold and the data set, and the Laplace–Beltrami operator in the Euler–Lagrange equation is discretized using the weighted graph Laplacian. Various scientific data sets from different fields of study are used to illustrate the performance of the proposed algorithm on data compression and interpolation from both regular and irregular samplings.

Authors:

^[1]; Wang, Bao ^[1];

^[2];

^[3]; Jenko, Frank ^[1]; Osher, Stanley ^[1]

Univ. of California Los Angeles, Los Angeles, CA (United States)
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States); Univ. of Tennessee, Knoxville, TN (United States)

Publication Date:: Thu Sep 28 00:00:00 EDT 2017

Research Org.:: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

OSTI Identifier:: 1407782

Alternate Identifier(s):: OSTI ID: 1549289

Grant/Contract Number:: AC05-00OR22725; De-AC05-00OR22725; DOE-SC0013838

Resource Type:: Accepted Manuscript

Journal Name:: Journal of Computational Physics

Additional Journal Information:: Journal Volume: 352; Journal Issue: C; Journal ID: ISSN 0021-9991

Publisher:: Elsevier

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; Low dimensional manifold model (LDMM); Scientific data interpolation; Data compression; Regular and irregular sampling; Weighted graph Laplacian

Citation Formats


                    Zhu, Wei, Wang, Bao, Barnard, Richard C., Hauck, Cory D., Jenko, Frank, and Osher, Stanley. Scientific data interpolation with low dimensional manifold model.  United States: N. p., 2017. 
Web.  doi:10.1016/j.jcp.2017.09.048.

Copy to clipboard


                    Zhu, Wei, Wang, Bao, Barnard, Richard C., Hauck, Cory D., Jenko, Frank, & Osher, Stanley. Scientific data interpolation with low dimensional manifold model.  United States.  https://doi.org/10.1016/j.jcp.2017.09.048

Copy to clipboard


                    Zhu, Wei, Wang, Bao, Barnard, Richard C., Hauck, Cory D., Jenko, Frank, and Osher, Stanley. Thu .  
"Scientific data interpolation with low dimensional manifold model".  United States.  https://doi.org/10.1016/j.jcp.2017.09.048.  https://www.osti.gov/servlets/purl/1407782.

Copy to clipboard


                    
@article{osti_1407782,

  title        = {Scientific data interpolation with low dimensional manifold model},

  author       = {Zhu, Wei and Wang, Bao and Barnard, Richard C. and Hauck, Cory D. and Jenko, Frank and Osher, Stanley},

  abstractNote = {Here, we propose to apply a low dimensional manifold model to scientific data interpolation from regular and irregular samplings with a significant amount of missing information. The low dimensionality of the patch manifold for general scientific data sets has been used as a regularizer in a variational formulation. The problem is solved via alternating minimization with respect to the manifold and the data set, and the Laplace–Beltrami operator in the Euler–Lagrange equation is discretized using the weighted graph Laplacian. Various scientific data sets from different fields of study are used to illustrate the performance of the proposed algorithm on data compression and interpolation from both regular and irregular samplings.},

  doi          = {10.1016/j.jcp.2017.09.048},

  journal      = {Journal of Computational Physics},

  number       = C,

  volume       = 352,

  place        = {United States},

  year         = {Thu Sep 28 00:00:00 EDT 2017},

  month        = {Thu Sep 28 00:00:00 EDT 2017}

}

Copy to clipboard

Journal Article:

Free Publicly Available Full Text

Accepted Manuscript (Publisher)

Accepted Manuscript (DOE)

Publisher's Version of Record

https://doi.org/10.1016/j.jcp.2017.09.048

Other availability

Search WorldCat to find libraries that may hold this journal

Citation Metrics:

Cited by: 7 works

Citation information provided by
Web of Science

Save / Share:

Export Metadata

Save to My Library

Works referenced in this record:

Exemplar-based interpolation of sparsely sampled images
dataset, January 2013

Facciolo, Gabriele; Arias, Pablo
Figshare
DOI: 10.6084/m9.figshare.879640

Beyond alias hierarchical scale curvelet interpolation of regularly and irregularly sampled seismic data
journal, November 2010

Naghizadeh, Mostafa; Sacchi, Mauricio D.
GEOPHYSICS, Vol. 75, Issue 6
DOI: 10.1190/1.3509468

Wave‐equation trace interpolation
journal, July 1987

Ronen, Joshua
GEOPHYSICS, Vol. 52, Issue 7
DOI: 10.1190/1.1442366

2-D continuation operators and their applications
journal, November 1996

Bagaini, Claudio; Spagnolini, Umberto
GEOPHYSICS, Vol. 61, Issue 6
DOI: 10.1190/1.1444100

Seismic data mapping and reconstruction
journal, May 2002

Stolt, Robert H.
GEOPHYSICS, Vol. 67, Issue 3
DOI: 10.1190/1.1484532

Seismic reflection data interpolation with differential offset and shot continuation
journal, March 2003

Fomel, Sergey
GEOPHYSICS, Vol. 68, Issue 2
DOI: 10.1190/1.1567243

Nonlinear total variation based noise removal algorithms
journal, November 1992

Rudin, Leonid I.; Osher, Stanley; Fatemi, Emad
Physica D: Nonlinear Phenomena, Vol. 60, Issue 1-4
DOI: 10.1016/0167-2789(92)90242-F

Nontexture Inpainting by Curvature-Driven Diffusions
journal, December 2001

Chan, Tony F.; Shen, Jianhong
Journal of Visual Communication and Image Representation, Vol. 12, Issue 4
DOI: 10.1006/jvci.2001.0487

Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA)
journal, November 2005

Elad, M.; Starck, J. -L.; Querre, P.
Applied and Computational Harmonic Analysis, Vol. 19, Issue 3
DOI: 10.1016/j.acha.2005.03.005

Inpainting and Zooming Using Sparse Representations
journal, February 2008

Fadili, M. J.; Starck, J. -L.; Murtagh, F.
The Computer Journal, Vol. 52, Issue 1
DOI: 10.1093/comjnl/bxm055

A Review of Image Denoising Algorithms, with a New One
journal, January 2005

Buades, A.; Coll, B.; Morel, J. M.
Multiscale Modeling & Simulation, Vol. 4, Issue 2
DOI: 10.1137/040616024

Nonlocal Operators with Applications to Image Processing
journal, January 2009

Gilboa, Guy; Osher, Stanley
Multiscale Modeling & Simulation, Vol. 7, Issue 3
DOI: 10.1137/070698592

Sparse Representation for Color Image Restoration
journal, January 2008

Mairal, Julien; Elad, Michael; Sapiro, Guillermo
IEEE Transactions on Image Processing, Vol. 17, Issue 1
DOI: 10.1109/TIP.2007.911828

Solving Inverse Problems With Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity
journal, May 2012

Guoshen Yu, ; Sapiro, G.; Mallat, S.
IEEE Transactions on Image Processing, Vol. 21, Issue 5
DOI: 10.1109/TIP.2011.2176743

Nonparametric Bayesian Dictionary Learning for Analysis of Noisy and Incomplete Images
journal, January 2012

Zhou, Mingyuan; Chen, Haojun; Paisley, John
IEEE Transactions on Image Processing, Vol. 21, Issue 1, p. 130-144
DOI: 10.1109/TIP.2011.2160072

Diffuse Interface Models on Graphs for Classification of High Dimensional Data
journal, January 2012

Bertozzi, Andrea L.; Flenner, Arjuna
Multiscale Modeling & Simulation, Vol. 10, Issue 3
DOI: 10.1137/11083109X

An Algorithm for Finding Best Matches in Logarithmic Expected Time
journal, September 1977

Friedman, Jerome H.; Bentley, Jon Louis; Finkel, Raphael Ari
ACM Transactions on Mathematical Software, Vol. 3, Issue 3
DOI: 10.1145/355744.355745

Unsupervised Classification in Hyperspectral Imagery With Nonlocal Total Variation and Primal-Dual Hybrid Gradient Algorithm
journal, May 2017

Zhu, Wei; Chayes, Victoria; Tiard, Alexandre
IEEE Transactions on Geoscience and Remote Sensing, Vol. 55, Issue 5
DOI: 10.1109/TGRS.2017.2654486

Algorithm 862: MATLAB tensor classes for fast algorithm prototyping
journal, December 2006

Bader, Brett W.; Kolda, Tamara G.
ACM Transactions on Mathematical Software, Vol. 32, Issue 4
DOI: 10.1145/1186785.1186794

Multiscale Nature of the Dissipation Range in Gyrokinetic Simulations of Alfvénic Turbulence
journal, July 2015

Told, D.; Jenko, F.; TenBarge, J. M.
Physical Review Letters, Vol. 115, Issue 2
DOI: 10.1103/PhysRevLett.115.025003

Electron temperature gradient driven turbulence
journal, May 2000

Jenko, F.; Dorland, W.; Kotschenreuther, M.
Physics of Plasmas, Vol. 7, Issue 5
DOI: 10.1063/1.874014

Two-dimensional time dependent Riemann solvers for neutron transport
journal, November 2005

Brunner, Thomas A.; Holloway, James Paul
Journal of Computational Physics, Vol. 210, Issue 1
DOI: 10.1016/j.jcp.2005.04.011

A Collision-Based Hybrid Method for Time-Dependent, Linear, Kinetic Transport Equations
journal, January 2013

Hauck, Cory D.; McClarren, Ryan G.
Multiscale Modeling & Simulation, Vol. 11, Issue 4
DOI: 10.1137/110846610

Robust and accurate filtered spherical harmonics expansions for radiative transfer
journal, August 2010

McClarren, Ryan G.; Hauck, Cory D.
Journal of Computational Physics, Vol. 229, Issue 16
DOI: 10.1016/j.jcp.2010.03.043

Simulating radiative transfer with filtered spherical harmonics
journal, May 2010

McClarren, Ryan G.; Hauck, Cory D.
Physics Letters A, Vol. 374, Issue 22
DOI: 10.1016/j.physleta.2010.02.041

Diffusive Corrections to $P_N$ Approximations
journal, January 2011

Schäfer, Matthias; Frank, Martin; Levermore, C. David
Multiscale Modeling & Simulation, Vol. 9, Issue 1
DOI: 10.1137/090764542

An arbitrary-order, fully implicit, hybrid kinetic solver for linear radiative transport using integral deferred correction
journal, October 2017

Crockatt, Michael M.; Christlieb, Andrew J.; Garrett, C. Kristopher
Journal of Computational Physics, Vol. 346
DOI: 10.1016/j.jcp.2017.06.017

Small-scale structure of two-dimensional magnetohydrodynamic turbulence
journal, January 1979

Orszag, Steven A.; Tang, Cha-Mei
Journal of Fluid Mechanics, Vol. 90, Issue 1
DOI: 10.1017/S002211207900210X

Similar Records in DOE PAGES and OSTI.GOV collections:

Graph interpolating activation improves both natural and robust accuracies in data-efficient deep learning

Journal Article WANG, BAO ; OSHER, STAN J. - European Journal of Applied Mathematics

Improving the accuracy and robustness of deep neural nets (DNNs) and adapting them to small training data are primary tasks in deep learning (DL) research. In this paper, we replace the output activation function of DNNs, typically the data-agnostic softmax function, with a graph Laplacian-based high-dimensional interpolating function which, in the continuum limit, converges to the solution of a Laplace–Beltrami equation on a high-dimensional manifold. Furthermore, we propose end-to-end training and testing algorithms for this new architecture. The proposed DNN with graph interpolating activation integrates the advantages of both deep learning and manifold learning. Compared to the conventional DNNs withmore »« less
https://doi.org/10.1017/s0956792520000406
Error-Bounded Graph Construction for Semi-supervised Manifold Learning

Conference Symons, Christopher T.

Graphs are commonly used in semi-supervised learning to represent a manifold on which the data reside in a high-dimensional ambient space. The graph can then be utilized in different ways, typically via the Laplacian of the graph, in order to leverage associations among the unlabeled data to improve learning. One common way to leverage the graph Laplacian is as a regularization term, where models that would disagree with the graph are penalized. More often the spectrum of the graph Laplacian is used to find a lower dimensional embedding in which neighboring relations encoded via the graph are preserved. Most manifold-basedmore »« less
Full Text Available
Time-series forecasting using manifold learning, radial basis function interpolation, and geometric harmonics

Journal Article Papaioannou, Panagiotis G. ; Talmon, Ronen ; Kevrekidis, Ioannis G. ; ... - Chaos: An Interdisciplinary Journal of Nonlinear Science

We address a three-tier numerical framework based on nonlinear manifold learning for the forecasting of high-dimensional time series, relaxing the “curse of dimensionality” related to the training phase of surrogate/machine learning models. At the first step, we embed the high-dimensional time series into a reduced low-dimensional space using nonlinear manifold learning (local linear embedding and parsimonious diffusion maps). Then, we construct reduced-order surrogate models on the manifold (here, for our illustrations, we used multivariate autoregressive and Gaussian process regression models) to forecast the embedded dynamics. Finally, we solve the pre-image problem, thus lifting the embedded time series back to themore »« less
https://doi.org/10.1063/5.0094887

Full Text Available
An exactly solvable deformation of the Coulomb problem associated with the Taub–NUT metric

Journal Article Ballesteros, Ángel ; Enciso, Alberto ; Herranz, Francisco J., E-mail: fjherranz@ubu.es ; ... - Annals of Physics (New York)

In this paper we quantize the N-dimensional classical Hamiltonian system H=(|q|)/(2(η+|q|)) p{sup 2}−k/(η+|q|) , that can be regarded as a deformation of the Coulomb problem with coupling constant k, that it is smoothly recovered in the limit η→0. Moreover, the kinetic energy term in H is just the one corresponding to an N-dimensional Taub–NUT space, a fact that makes this system relevant from a geometric viewpoint. Since the Hamiltonian H is known to be maximally superintegrable, we propose a quantization prescription that preserves such superintegrability in the quantum mechanical setting. We show that, to this end, one must choose asmore »« less
https://doi.org/10.1016/J.AOP.2014.09.013
The concept of spin ice graphs and a field theory for their charges

Journal Article Nisoli, Cristiano - AIP Advances

Originally detected in rare earth pyrochlores, spin ice physics is now being artificially extended to a variety of geometries that control collective behavior and exotic properties, making graph theory their proper framework. We relate spin ice notions, such as ice rule, ice manifold, Coulomb phases, charges, and monopoles, to graph-theoretical notions, such as balance, in/out-degrees, and Euler paths. We then propose a field-theoretical treatment in which topological charges and monopoles are the degrees of freedom, while the binary spins are subsumed in an entropic interaction among charges. We show that for a spin ice on a graph in a Gaussianmore »« less
https://doi.org/10.1063/5.0010079

Full Text Available

Similar Records

Title: Scientific data interpolation with low dimensional manifold model

Abstract

Citation Formats

Exemplar-based interpolation of sparsely sampled images dataset, January 2013

Beyond alias hierarchical scale curvelet interpolation of regularly and irregularly sampled seismic data journal, November 2010

Wave‐equation trace interpolation journal, July 1987

2-D continuation operators and their applications journal, November 1996

Seismic data mapping and reconstruction journal, May 2002

Seismic reflection data interpolation with differential offset and shot continuation journal, March 2003

Nonlinear total variation based noise removal algorithms journal, November 1992

Nontexture Inpainting by Curvature-Driven Diffusions journal, December 2001

Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA) journal, November 2005

Inpainting and Zooming Using Sparse Representations journal, February 2008

A Review of Image Denoising Algorithms, with a New One journal, January 2005

Nonlocal Operators with Applications to Image Processing journal, January 2009

Sparse Representation for Color Image Restoration journal, January 2008

Solving Inverse Problems With Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity journal, May 2012

Nonparametric Bayesian Dictionary Learning for Analysis of Noisy and Incomplete Images journal, January 2012

Diffuse Interface Models on Graphs for Classification of High Dimensional Data journal, January 2012

An Algorithm for Finding Best Matches in Logarithmic Expected Time journal, September 1977

Unsupervised Classification in Hyperspectral Imagery With Nonlocal Total Variation and Primal-Dual Hybrid Gradient Algorithm journal, May 2017

Algorithm 862: MATLAB tensor classes for fast algorithm prototyping journal, December 2006

Multiscale Nature of the Dissipation Range in Gyrokinetic Simulations of Alfvénic Turbulence journal, July 2015

Electron temperature gradient driven turbulence journal, May 2000

Two-dimensional time dependent Riemann solvers for neutron transport journal, November 2005

A Collision-Based Hybrid Method for Time-Dependent, Linear, Kinetic Transport Equations journal, January 2013

Robust and accurate filtered spherical harmonics expansions for radiative transfer journal, August 2010

Simulating radiative transfer with filtered spherical harmonics journal, May 2010

Diffusive Corrections to $P_N$ Approximations journal, January 2011

An arbitrary-order, fully implicit, hybrid kinetic solver for linear radiative transport using integral deferred correction journal, October 2017

Small-scale structure of two-dimensional magnetohydrodynamic turbulence journal, January 1979

Exemplar-based interpolation of sparsely sampled images
dataset, January 2013

Beyond alias hierarchical scale curvelet interpolation of regularly and irregularly sampled seismic data
journal, November 2010

Wave‐equation trace interpolation
journal, July 1987

2-D continuation operators and their applications
journal, November 1996

Seismic data mapping and reconstruction
journal, May 2002

Seismic reflection data interpolation with differential offset and shot continuation
journal, March 2003

Nonlinear total variation based noise removal algorithms
journal, November 1992

Nontexture Inpainting by Curvature-Driven Diffusions
journal, December 2001

Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA)
journal, November 2005

Inpainting and Zooming Using Sparse Representations
journal, February 2008

A Review of Image Denoising Algorithms, with a New One
journal, January 2005

Nonlocal Operators with Applications to Image Processing
journal, January 2009

Sparse Representation for Color Image Restoration
journal, January 2008

Solving Inverse Problems With Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity
journal, May 2012

Nonparametric Bayesian Dictionary Learning for Analysis of Noisy and Incomplete Images
journal, January 2012

Diffuse Interface Models on Graphs for Classification of High Dimensional Data
journal, January 2012

An Algorithm for Finding Best Matches in Logarithmic Expected Time
journal, September 1977

Unsupervised Classification in Hyperspectral Imagery With Nonlocal Total Variation and Primal-Dual Hybrid Gradient Algorithm
journal, May 2017

Algorithm 862: MATLAB tensor classes for fast algorithm prototyping
journal, December 2006

Multiscale Nature of the Dissipation Range in Gyrokinetic Simulations of Alfvénic Turbulence
journal, July 2015

Electron temperature gradient driven turbulence
journal, May 2000

Two-dimensional time dependent Riemann solvers for neutron transport
journal, November 2005

A Collision-Based Hybrid Method for Time-Dependent, Linear, Kinetic Transport Equations
journal, January 2013

Robust and accurate filtered spherical harmonics expansions for radiative transfer
journal, August 2010

Simulating radiative transfer with filtered spherical harmonics
journal, May 2010

Diffusive Corrections to $P_N$ Approximations
journal, January 2011

An arbitrary-order, fully implicit, hybrid kinetic solver for linear radiative transport using integral deferred correction
journal, October 2017

Small-scale structure of two-dimensional magnetohydrodynamic turbulence
journal, January 1979