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) |
|---|---|
| Pages (from-to) | 1-28 |
| Number of pages | 28 |
| Journal | Journal of Scientific Computing |
| DOIs | |
| State | Accepted/In press - Aug 11 2017 |
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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 journal › Article
}
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 -