Fast iterative solvers for an optimal transport problem

@article{Herzog2019FastIS,
  title={Fast iterative solvers for an optimal transport problem},
  author={Roland Herzog and John W. Pearson and M. Stoll},
  journal={Advances in Computational Mathematics},
  year={2019},
  volume={45},
  pages={495-517}
}
Optimal transport problems pose many challenges when considering their numerical treatment. We investigate the solution of a PDE-constrained optimisation problem subject to a particular transport equation arising from the modelling of image metamorphosis. We present the nonlinear optimisation problem, and discuss the discretisation and treatment of the nonlinearity via a Gauss–Newton scheme. We then derive preconditioners that can be used to solve the linear systems at the heart of the (Gauss… 

Interior‐point methods and preconditioning for PDE‐constrained optimization problems involving sparsity terms

TLDR
This work proposes the use of an interior‐point scheme applied to a smoothed reformulation of the discretized problem and illustrates that such a scheme exhibits robust performance with respect to parameter changes.

On the numerical solution of state- and control-constrained optimal control problems

We consider the iterative solution of algebraic systems, arising in optimal control problems, constrained by a partial differential equation, with additional box constraints on the state and the

Fast solver of optimal control problems constrained by Ohta-Kawasaki equations

TLDR
This paper proves that the eigenvalues of the corresponding preconditioned system are all real, which leads to a fast Krylov subspace solver, that is robust with respect to mesh sizes, model parameters, and regularization parameters.

Numerical optimal control of a size-structured PDE model for metastatic cancer treatment.

A Parallel-In-Time Block-Circulant Preconditioner for Optimal Control of Wave Equations

TLDR
A new efficient preconditioner for iteratively solving the large-scale indefinite saddle-point sparse linear system, which arises from discretizing the optimality system i.

PDE-constrained optimization in medical image analysis

TLDR
The ultimate goal of the work is the design of inversion methods that integrate complementary data, and rigorously follow mathematical and physical principles, in an attempt to support clinical decision making, which requires reliable, high-fidelity algorithms with a short time-to-solution.

CLAIRE: A distributed-memory solver for constrained large deformation diffeomorphic image registration

TLDR
This work releases CLAIRE, a distributed-memory implementation of an effective solver for constrained large deformation diifeomorphic image registration problems in three dimensions, and introduces an improved preconditioner for the reduced space Hessian to speed up the convergence of the solver.

Interior Point Methods for PDE-Constrained Optimization with Sparsity Constraints.

TLDR
This work proposes the use of an Interior Point scheme applied to a smoothed reformulation of the discretized problem, and illustrates that such a scheme exhibits robust performance with respect to parameter changes.

References

SHOWING 1-10 OF 43 REFERENCES

A preconditioning technique for a class of PDE-constrained optimization problems

We investigate the use of a preconditioning technique for solving linear systems of saddle point type arising from the application of an inexact Gauss–Newton scheme to PDE-constrained optimization

Fast Iterative Solution of Reaction-Diffusion Control Problems Arising from Chemical Processes

TLDR
Important aspects of the solvers are saddle point theory, mass matrix representation, and effective Schur complement approximation, as well as the incorporation of control constraints and application of the outer iteration to take into account the nonlinearity of the underlying PDEs.

Regularization-Robust Preconditioners for Time-Dependent PDE-Constrained Optimization Problems

In this article, we motivate, derive, and test effective preconditioners to be used with the Minres algorithm for solving a number of saddle point systems which arise in PDE-constrained optimization

FAST ITERATIVE SOLVERS FOR CONVECTION-DIFFUSION CONTROL PROBLEMS ∗

TLDR
Effective solvers for the optimal control of stabilized convectiondiffusion control problems are described and numerical results show that these preconditioners result in convergence in a small number of iterations, which is robust with respect to the step-size h and the regularization parameter β for a range of problems.

Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems

The governing dynamics of fluid flow is stated as a system of partial differential equations referred to as the Navier-Stokes system. In industrial and scientific applications, fluid flow control

Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems

TLDR
This paper considers distributed optimal control for various types of scalar stationary partial differential equations and compares the efficiency of several numerical solution methods to find one that is favourably compared with other published methods.

Low-rank solvers for unsteady Stokes–Brinkman optimal control problem with random data

A new approximation of the Schur complement in preconditioners for PDE‐constrained optimization

TLDR
This short manuscript presents a new Schur complement approximation for PDE‐constrained optimization, an important class of these problems with constraints, and designs such preconditioners for which this optimality property holds independently of both the mesh size and the Tikhonov regularization parameter β that is used.

Efficient iterative solvers for elliptic finite element problems on nonmatching grids

TLDR
A block diagonal preconditioner is proposed which is spectrally equivalent to the original saddle-point matrix and has the optimal order of arithmetical complexity for the algebraic systems that occur in using the mortar finite element method.

On optimization techniques for solving nonlinear inverse problems

This paper considers optimization techniques for the solution of nonlinear inverse problems where the forward problems, like those encountered in electromagnetics, are modelled by differential