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Using \(\ell _1\) Regularization to Improve Numerical Partial Differential Equation Solvers

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Abstract

Sparse regularization plays a central role in many recent developments in imaging and other related fields. However, it is still of limited use in numerical solvers for partial differential equations (PDEs). In this paper we investigate the use of \(\ell _1\) regularization to promote sparsity in the shock locations of hyperbolic PDEs. We develop an algorithm that uses a high order sparsifying transform which enables us to effectively resolve shocks while still maintaining stability. Our method does not require a shock tracking procedure nor any prior information about the number of shock locations. It is efficiently implemented using the alternating direction method of multipliers. We present our results on one and two dimensional examples using both finite difference and spectral methods as underlying PDE solvers.

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Notes

  1. Even orders were not used in [2] because the post-processing techniques used for pinpointing the edges assumed that maximum (minimum) values occurred at the edge, which is true only for odd orders.

  2. We also applied \(\ell _1\) enhancement to the spectral viscosity method for the Fourier and Chebyshev cases. Both resulted in improved accuracy that essentially mirrored the approximations displayed in Figs. 4 and 10. Hence they are not reported here.

  3. With a small decrease in accuracy, the mapped Chebyshev method allows the time step to increase to \(\mathcal {O}(\frac{1}{N})\), [19].

  4. The explicit matrix entries for (30) for (24) can be found in [5, 16]. In our examples, we use the mapped Chebyshev points, [19], so the derivative matrix depends on the chosen grid points \(x_j\).

  5. MATLAB code is available at [32].

  6. We did not separately analyze our method for Euler’s equations, (47).

  7. More information about these images can be found at [14, 29], and [34].

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Authors and Affiliations

  1. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287, USA

    Theresa Scarnati & Rodrigo B. Platte

  2. Department of Mathematics, Dartmouth College, 27 N. Main St., Hanover, NH, 03755, USA

    Anne Gelb

Authors
  1. Theresa Scarnati
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  2. Anne Gelb
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  3. Rodrigo B. Platte
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Correspondence to Anne Gelb.

Additional information

This work is supported in part by the Grants NSF-DMS 1502640, NSF-DMS 1522639 and AFOSR FA9550-15-1-0152.

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Scarnati, T., Gelb, A. & Platte, R.B. Using \(\ell _1\) Regularization to Improve Numerical Partial Differential Equation Solvers. J Sci Comput 75, 225–252 (2018). https://doi.org/10.1007/s10915-017-0530-8

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