HARBIR ANTIL: RESEARCH HIGHLIGTS

The research of Prof. Harbir Antil is currently supported by National Science Foundation(NSF), Air Force Office of Scientific Research (AFOSR), and Department of Energy (DOE).

Some of the latest research highlights are:

(1) Gravitational waves (GWs). During the recent years, GWs discovery has been one ofthe most prominent. In fact, the LIGO gravitational wave discoverers won the 2017 Nobelprize in Physics. The three key difficulties in detecting GWs are the weak magnitude ofthe GW signals (cf. Figure 1, left), signals are noisy (cf. Figure 1, right), and it takesfrom weeks-to-months to analyze the data on supercomputers. In 2013 we introduced

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Figure 1. Gravitational wave (GW). Source: [3].

the notion of Reduced Order Quadrature (ROQ) in [3] which reduced the data analysistime by several orders of magnitude. ROQ then played a central role in the first GWsdetection and is currently used as in the LIGO’s low-latency parameter inference analysis.See also our recent article and code here [1].

(2) Fractional Helmholtz in Geophysical Electromagnetics. Magnetotellurics (MT)is a widely used technique in geophysics to infer the earth’s subsurface electrical conduc-tivity. Typically, the standard Helmholtz partial differential equation (PDE) is used tomodel this. Recently, in [9], starting from electromagnetic equations we have derived afractional derivative based Helmholtz PDE model. The left panel in the Figure 2 showsour numerical simulations for various fractional exponents s, the exponent s = 1 corre-sponds to the well-known classical case. In the middle panel we have displayed the realdata from USArray MT station for KSP34 located NW of Kansas City, KS, USA fromthe US Array. The rightmost panel shows the location map for USArray MT stationKSP34. Notice the significant difference between the curves (s = 1, the classical model)in the left and middle panels. On the other hand, for smaller values of the fractional

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fractional Helmholtz MT sounding curves

Figure 2. Magnetotelluric (MT). Source: [9]

exponent s, our model, qualitatively matches well with the real data. This is the firstinstance that directly shows the benefits of using a fractional model.

(3) Variable order fractional Laplacian with application to imaging science. Re-cently, in [7], we have introduced a completely new model in imaging science. In Figure 3,the left panel indicates a noisy image, the middle panel shows the result using a total-variation based model, and the right panel shows results using our new model. Our new

Figure 3. Image denoising using fractional variable order models. Source: [7]

model is based on the fractional derivatives but where the fractional exponent is spatiallydependent. Notice that, we can obtain almost perfect reconstructions using our model.

(4) Exterior optimal control. Optimal control and inverse problems are ubiquitous inscience and engineering. The classical models for diffusion processes only allow con-trol/source placement either inside the domain or on its boundary. Recently in [4, 8] wehave introduced a new notion of optimal control where the control can be placed in theexterior, i.e., away from the domain and its boundary. We carry out this using fractionaldiffusion operators. The left panel in Figure 4 shows a spatial discretization. Our goal isto control the diffusion in the letter M and the control is placed in the small structure onthe top right corner of the letter M. The middle and the right panels show the optimalstate and control, respectively.

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Figure 4. Exterior optimal control. Source: [4]

(5) Magnetic drug targeting (MDT). MDT has recently received a significant amountof attention, including several successful human trials. The principle of MDT is: a drugis injected in the body along with the ferrofluid (“fluid with nano-size magnets”) andthis mixture is steered to a desired location by an external magnetic field. Motivated byMDT, in [5, 6] we have introduced models that allows generating desired magnetic forcein a subdomain, moving along a prescribed trajectory. Figure 5 shows such an examplewhere with the help of 3 external magnets, we have created a targeted magnetic force ina subdomain inside the large circle. The control variables in this case are the location ofmagnets and magnetic field strength.

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Figure 5. Magnetic drug targeting (MDT). Source: [5]

We have also shown that such a magnetic force can effectively steer the drug con-centration governed by a drift-diffusion partial differential equation, from an initial to adesired location with limited spreading.

(6) Fractional Laplacian regularization and deep learning. In imaging science, oftenthe reconstruction or recovery of solution is modeled as an optimization problem. Thesemodels are improved by adding some form of regularization. There are two key challengesassociated with solving such inverse problems, namely the choice of regularization, anddetermination of regularization strength. In [2], we propose the use of fractional Lapla-cian as a regularizer which enforces a reduced smoothness and an improved solutionquality. Furthermore, to determine the strength of regularization we use machine learn-ing. We develop a Bilevel Optimization Neural Network (BONNet) which requires a

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training and a testing dataset. The training dataset is used to learn the optimal regu-larization parameters, and the testing dataset is used to check their effectiveness. Weshow the effectiveness of our algorithm on a tomographic reconstruction problem wherewe learn the regularization parameters using BONNet and compare fractional Laplacianregularization with total variation regularization (cf. Figure 6).

Figure 6. Tomographic reconstruction of a sample (panel 3 ) via total varia-tion regularization (panel 1 ) and fractional Laplacian regularization (panel 2 ).Mean squared errors of the reconstructions for various amounts of tomographicmeasurement data clearly shows that fractional Laplacian regularization out-performs total variation regularization (panel 4 ). Optimal regularization pa-rameters labeled in panel 4 are learned via BONNet. Source: [2]

References

[1] H. Antil, D. Chen, and S. Field. A note on qr-based model reduction: Algorithm, software, and gravita-tional wave applications. Computing in Science & Engineering, 20(4):10, 2018.

[2] H. Antil, Z. Di, and R. Khatri. Bilevel optimization, deep learning and fractional laplacian regularizationwith applications in tomography. arXiv preprint arXiv:1907.09605, 2019.

[3] H. Antil, S.E. Field, F. Herrmann, R.H. Nochetto, and M. Tiglio. Two-step greedy algorithm for reducedorder quadratures. J. Sci. Comput., 57(3):604–637, 2013.

[4] H. Antil, R. Khatri, and M. Warma. External optimal control of nonlocal pdes. Inverse Problems, 35(8),2019.

[5] H. Antil, R.H. Nochetto, and P. Venegas. Controlling the Kelvin force: basic strategies and applicationsto magnetic drug targeting. Optim. Eng., 19(3):559–589, 2018.

[6] H. Antil, R.H. Nochetto, and P. Venegas. Optimizing the Kelvin force in a moving target subdomain.Math. Models Methods Appl. Sci., 28(1):95–130, 2018.

[7] H. Antil and C.N. Rautenberg. Sobolev spaces with non-Muckenhoupt weights, fractional elliptic opera-tors, and applications. SIAM J. Math. Anal., 51(3):2479–2503, 2019.

[8] H. Antil, D. Verma, and M. Warma. External optimal control of fractional parabolic pdes. arXiv preprintarXiv:1904.07123, 2019.

[9] C. Weiss, B. van Bloemen Waanders, and H. Antil. Fractional operators applied to geophysical electro-magnetics. arXiv preprint arXiv:1902.05096, 2019.