# A primal-dual approach for solving conservation laws with implicit in time approximations

@article{Liu2022APA, title={A primal-dual approach for solving conservation laws with implicit in time approximations}, author={Siting Liu and Stanley J. Osher and Wuchen Li and Chi-Wang Shu}, journal={J. Comput. Phys.}, year={2022}, volume={472}, pages={111654} }

. In this work, we propose a novel framework for the numerical solution of time-dependent conservation laws with implicit schemes via primal-dual hybrid gradient methods. We solve an initial value problem (IVP) for the partial diﬀerential equation (PDE) by casting it as a saddle point of a min-max problem and using iterative optimization methods to ﬁnd the saddle point. Our approach is ﬂexible with the choice of both time and spatial discretization schemes. It beneﬁts from the implicit…

## One Citation

### High order computation of optimal transport, mean field planning, and mean field games

- Computer ScienceArXiv
- 2023

This work explores applying general high-order numerical schemes with finite element methods in the space-time domain for computing the optimal transport, mean-field planning, and MFG problems.

## References

SHOWING 1-10 OF 16 REFERENCES

### Primal Dual Methods for Wasserstein Gradient Flows

- Computer Science, MathematicsFoundations of Computational Mathematics
- 2021

It is proved that minimizers of the fully discrete problem converge to minimizer of the spatially continuous, discrete time problem as the spatial discretization is refined, including higher-order convergence the novel Crank–Nicolson-type method, when compared to the classical JKO method.

### Controlling conservation laws II: compressible Navier-Stokes equations

- Mathematics, Computer ScienceJ. Comput. Phys.
- 2022

### A new Lagrange multiplier approach for constructing structure preserving schemes, I. Positivity preserving

- Mathematics, Computer ScienceComputer Methods in Applied Mechanics and Engineering
- 2022

### A primal–dual hybrid gradient method for nonlinear operators with applications to MRI

- Mathematics
- 2013

We study the solution of minimax problems min xmax yG(x) + 〈K(x), y〉 − F*(y) in finite-dimensional Hilbert spaces. The functionals G and F* we assume to be convex, but the operator K we allow to be…

### Primal-Dual Extragradient Methods for Nonlinear Nonsmooth PDE-Constrained Optimization

- MathematicsSIAM J. Optim.
- 2017

The applicability of the accelerated algorithm to examples of inverse problems with $L^1$- and $L$-fitting terms as well as of state-constrained optimal control problems, where convergence can be guaranteed after introducing an (arbitrary small, still nonsmooth) Moreau--Yosida regularization is shown.

### Computational mean-field information dynamics associated with reaction-diffusion equations

- Mathematics, Computer ScienceJ. Comput. Phys.
- 2022

### A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging

- Mathematics, Computer ScienceJournal of Mathematical Imaging and Vision
- 2010

A first-order primal-dual algorithm for non-smooth convex optimization problems with known saddle-point structure can achieve O(1/N2) convergence on problems, where the primal or the dual objective is uniformly convex, and it can show linear convergence, i.e. O(ωN) for some ω∈(0,1), on smooth problems.

### Weak Adversarial Networks for High-dimensional Partial Differential Equations

- Computer Science, MathematicsJ. Comput. Phys.
- 2020

### Weighted essentially non-oscillatory schemes

- Computer Science
- 1994

A new version of ENO (essentially non-oscillatory) shock-capturing schemes which is called weighted ENO, where, instead of choosing the "smoothest" stencil to pick one interpolating polynomial for the ENO reconstruction, a convex combination of all candidates is used.

### Controlling conservation laws I: entropy-entropy flux

- Mathematics, Computer ScienceJ. Comput. Phys.
- 2023

This work introduces a modified optimal transport space based on conservation laws with diffusion and constructs variational problems for these flows, for which dual PDE systems for regularized conservation laws are derived.