Computing Optimal Partition Problems via Lagrange Multiplier approach
Copyright policy )Computing Optimal Partition Problems via Lagrange Multiplier approach
In this paper, we consider numerical approximations for the optimal partition problem using Lagrange multipliers. By rewriting it into constrained gradient flows, three and four steps numerical schemes based on the Lagrange multiplier approach \cite{ChSh22,ChSh_II22} are proposed to solve the constrained gradient system. Numerical schemes proposed for the constrained gradient flows satisfy the nice properties of orthogonality-preserving, norm-preserving, positivity-preserving and energy dissipating. The proposed schemes are very efficient in which only linear Poisson equations are solved at each time step. Extensive numerical results in 2D and 3D for optimal partition problem are presented to validate the effectiveness and accuracy of the proposed numerical schemes.
25 pages, 15 figures
- Shenzhen Research Institute of Big Data China (People's Republic of)
- Tongji University China (People's Republic of)
- THE CHINESE UNIVERSITY OF HONG KONG China (People's Republic of)
- Chinese University of Hong Kong China (People's Republic of)
Numerical optimization and variational techniques, Eulerian and Hamiltonian graphs, Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation, Numerical computation of solutions to systems of equations, Positive solutions to PDEs, PDE constrained optimization (numerical aspects), Lagrange multiplier approach, optimal partition problem, Integro-partial differential equations, gradient flow, Optimization of shapes other than minimal surfaces, Optimization and Control (math.OC), Finite difference methods for initial value and initial-boundary value problems involving PDEs, FOS: Mathematics, Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs, Mathematics - Optimization and Control
Numerical optimization and variational techniques, Eulerian and Hamiltonian graphs, Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation, Numerical computation of solutions to systems of equations, Positive solutions to PDEs, PDE constrained optimization (numerical aspects), Lagrange multiplier approach, optimal partition problem, Integro-partial differential equations, gradient flow, Optimization of shapes other than minimal surfaces, Optimization and Control (math.OC), Finite difference methods for initial value and initial-boundary value problems involving PDEs, FOS: Mathematics, Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs, Mathematics - Optimization and Control
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