Approximation of Fractional Harmonic Maps

The article of record as published may be located at https://doi.org/10.48550/arXiv.2104.10049Funded by Naval Postgraduate SchoolThis paper addresses the approximation of fractional harmonic maps. Besides a unit-length constraint, one has to tackle the difficulty of nonlocality. We establish weak compactness results for critical points of the fractional Dirichlet energy on unit-length vector fields. We devise and analyze numerical methods for the approximation of various partial differential equations related to fractional harmonic maps. The compactness results imply the convergence of numerical approximations. Numerical examples on spin chain dynamics and point defects are presented to demonstrate the effectiveness of the proposed methods.HA is partially supported by NSF grants DMS-1818772 and DMS-1913004, the Air Force Office of Scientific Research under Award NO: FA9550-19-1-0036, and the Department of the Navy, Naval Postgraduate School under Award NO: N00244-20-1-0005. SB acknowledges support by the DFG via the Research Unit FOR 3013 Vector- and tensor-valued surface PDEs. AS is supported by NSF Career DMS-2044898 and Simons foundation grant no 579261.HA is partially supported by NSF grants DMS-1818772 and DMS-1913004, the Air Force Office of Scientific Research under Award NO: FA9550-19-1-0036, and the Department of the Navy, Naval Postgraduate School under Award NO: N00244-20-1-0005. SB acknowledges support by the DFG via the Research Unit FOR 3013 Vector- and tensor-valued surface PDEs. AS is supported by NSF Career DMS-2044898 and Simons foundation grant no 579261

Bilevel optimization, deep learning and fractional Laplacian regularization with applications in tomography

The article of record as published may be located at https://doi.org/10.1088/1361-6420/ab80d7Funded by Naval Postgraduate SchoolIn this work we consider a generalized bilevel optimization framework for solv- ing inverse problems. We introduce fractional Laplacian as a regularizer to improve the reconstruction quality, and compare it with the total variation regularization. We emphasize that the key advantage of using fractional Laplacian as a regularizer is that it leads to a linear operator, as opposed to the total varia- tion regularization which results in a nonlinear degenerate operator. Inspired by residual neural networks, to learn the optimal strength of regularization and the exponent of fractional Laplacian, we develop a dedicated bilevel opti- mization neural network with a variable depth for a general regularized inverse problem. We illustrate how to incorporate various regularizer choices into our proposed network. As an example, we consider tomographic reconstruction as a model problem and show an improvement in reconstruction quality, especially for limited data, via fractional Laplacian regularization. We successfully learn the regularization strength and the fractional exponent via our proposed bilevel optimization neural network. We observe that the fractional Laplacian regular- ization outperforms total variation regularization. This is specially encouraging, and important, in the case of limited and noisy data.The first and third authors are partially supported by NSF grants DMS-1818772, DMS-1913004, the Air Force Office of Scientific Research under Award No.: FA9550-19-1-0036, and the Department of Navy, Naval PostGraduate School under Award No.: N00244-20-1-0005. The third author is also partially supported by a Provost award at George Mason University under the Industrial Immersion Program. The second author is partially supported by DOE Office of Science under Contract No. DE-AC02-06CH11357.The first and third authors are partially supported by NSF grants DMS-1818772, DMS-1913004, the Air Force Office of Scientific Research under Award No.: FA9550-19-1-0036, and the Department of Navy, Naval PostGraduate School under Award No.: N00244-20-1-0005. The third author is also partially supported by a Provost award at George Mason University under the Industrial Immersion Program. The second author is partially supported by DOE Office of Science under Contract No. DE-AC02-06CH11357

Autonomous Electron Tomography Reconstruction with Machine Learning

Modern electron tomography has progressed to higher resolution at lower doses by leveraging compressed sensing methods that minimize total variation (TV). However, these sparsity-emphasized reconstruction algorithms introduce tunable parameters that greatly influence the reconstruction quality. Here, Pareto front analysis shows that high-quality tomograms are reproducibly achieved when TV minimization is heavily weighted. However, in excess, compressed sensing tomography creates overly smoothed 3D reconstructions. Adding momentum into the gradient descent during reconstruction reduces the risk of over-smoothing and better ensures that compressed sensing is well behaved. For simulated data, the tedious process of tomography parameter selection is efficiently solved using Bayesian optimization with Gaussian processes. In combination, Bayesian optimization with momentum-based compressed sensing greatly reduces the required compute time$-$an 80% reduction was observed for the 3D reconstruction of SrTiO$_3$ nanocubes. Automated parameter selection is necessary for large scale tomographic simulations that enable the 3D characterization of a wider range of inorganic and biological materials.Comment: 8 pages, 4 figure

Imaging 3D Chemistry at 1 nm Resolution with Fused Multi-Modal Electron Tomography

Measuring the three-dimensional (3D) distribution of chemistry in nanoscale matter is a longstanding challenge for metrological science. The inelastic scattering events required for 3D chemical imaging are too rare, requiring high beam exposure that destroys the specimen before an experiment completes. Even larger doses are required to achieve high resolution. Thus, chemical mapping in 3D has been unachievable except at lower resolution with the most radiation-hard materials. Here, high-resolution 3D chemical imaging is achieved near or below one nanometer resolution in a Au-Fe$_3$O$_4$ metamaterial, Co$_3$O$_4$ - Mn$_3$O$_4$ core-shell nanocrystals, and ZnS-Cu$_{0.64}$S$_{0.36}$ nanomaterial using fused multi-modal electron tomography. Multi-modal data fusion enables high-resolution chemical tomography often with 99\% less dose by linking information encoded within both elastic (HAADF) and inelastic (EDX / EELS) signals. Now sub-nanometer 3D resolution of chemistry is measurable for a broad class of geometrically and compositionally complex materials