# Using ℓ1 Regularization to Improve Numerical Partial Differential Equation Solvers

Theresa Scarnati, Anne Gelb, Rodrigo Platte

Research output: Contribution to journalArticle

2 Citations (Scopus)

### 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 (Formula presented.) 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.

Original language English (US) 1-28 28 Journal of Scientific Computing https://doi.org/10.1007/s10915-017-0530-8 Accepted/In press - Aug 11 2017

### Fingerprint

Partial differential equations
Shock
Regularization
Partial differential equation
Method of multipliers
Alternating Direction Method
Hyperbolic Partial Differential Equations
Prior Information
Spectral Methods
Sparsity
Difference Method
Imaging techniques
Resolve
Finite Difference
Imaging
Transform
Higher Order

### Keywords

• $$\ell _1$$ℓ1regularization
• Alternating direction method of multipliers
• Image denoising
• Numerical conservation laws

### ASJC Scopus subject areas

• Theoretical Computer Science
• Software
• Engineering(all)
• Computational Theory and Mathematics

### Cite this

Using ℓ1 Regularization to Improve Numerical Partial Differential Equation Solvers. / Scarnati, Theresa; Gelb, Anne; Platte, Rodrigo.

In: Journal of Scientific Computing, 11.08.2017, p. 1-28.

Research output: Contribution to journalArticle

@article{b20cfab6c892471d99ffc47e448d514a,
title = "Using ℓ1 Regularization to Improve Numerical Partial Differential Equation Solvers",
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 (Formula presented.) 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.",
keywords = "$$\ell _1$$ℓ1regularization, Alternating direction method of multipliers, Image denoising, Numerical conservation laws",
author = "Theresa Scarnati and Anne Gelb and Rodrigo Platte",
year = "2017",
month = "8",
day = "11",
doi = "10.1007/s10915-017-0530-8",
language = "English (US)",
pages = "1--28",
journal = "Journal of Scientific Computing",
issn = "0885-7474",
publisher = "Springer New York",

}

TY - JOUR

T1 - Using ℓ1 Regularization to Improve Numerical Partial Differential Equation Solvers

AU - Scarnati, Theresa

AU - Gelb, Anne

AU - Platte, Rodrigo

PY - 2017/8/11

Y1 - 2017/8/11

N2 - 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 (Formula presented.) 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.

AB - 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 (Formula presented.) 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.

KW - $$\ell _1$$ℓ1regularization

KW - Alternating direction method of multipliers

KW - Image denoising

KW - Numerical conservation laws

UR - http://www.scopus.com/inward/record.url?scp=85027332108&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85027332108&partnerID=8YFLogxK

U2 - 10.1007/s10915-017-0530-8

DO - 10.1007/s10915-017-0530-8

M3 - Article

AN - SCOPUS:85027332108

SP - 1

EP - 28

JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

SN - 0885-7474

ER -