Numerical recipes: the art of scientific computing
Numerical recipes: the art of scientific computing
Numerical computation of internal & external flows: fundamentals of numerical discretization
Numerical computation of internal & external flows: fundamentals of numerical discretization
A multiresolution method for distributed parameter estimation
SIAM Journal on Scientific Computing
Iterative solution methods
Multiscale Algorithm for Atmospheric Data Assimilation
SIAM Journal on Scientific Computing
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
An interface optimization and application for the numerical solution of optimal control problems
ACM Transactions on Mathematical Software (TOMS)
Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition
Journal of Optimization Theory and Applications
Nonlinearity, Scale, and Sensitivity for Parameter Estimation Problems
SIAM Journal on Scientific Computing
Basis Norm Rescaling for Nonlinear Parameter Estimation
SIAM Journal on Scientific Computing
Computational Methods for Inverse Problems
Computational Methods for Inverse Problems
On the smoothness constraints for four-dimensional data assimilation
Journal of Computational Physics
Perspectives in Flow Control and Optimization
Perspectives in Flow Control and Optimization
Energy minimization using Sobolev gradients: application to phase separation and ordering
Journal of Computational Physics
Adjoint-based optimization of PDEs in moving domains
Journal of Computational Physics
Adjoint-based optimization of PDE systems with alternative gradients
Journal of Computational Physics
An inverse model for a free-boundary problem with a contact line: Steady case
Journal of Computational Physics
Level-set, penalization and cartesian meshes: A paradigm for inverse problems and optimal design
Journal of Computational Physics
Adjoint algorithms for the Navier-Stokes equations in the low Mach number limit
Journal of Computational Physics
Optimal reconstruction of material properties in complex multiphysics phenomena
Journal of Computational Physics
Journal of Computational Physics
Hi-index | 31.48 |
This paper examines the regularization opportunities available in the adjoint analysis and optimization of multiscale PDE systems. Regularization may be introduced into such optimization problems by modifying the form of the evolution equation and the forms of the norms and inner products used to frame the adjoint analysis. Typically, L2 brackets are used in the definition of the cost functional, the adjoint operator, and the cost functional gradient. If instead we adopt the more general Sobolev brackets, the various fields involved in the adjoint analysis may be made smoother and therefore easier to resolve numerically. The present paper identifies several relationships which illustrate how the different regularization options fit together to form a general framework. The regularization strategies proposed are exemplified using a ID Kuramoto-Sivashinsky forecasting problem, and computational examples are provided which exhibit their utility. A multiscale preconditioning algorithm is also proposed that noticeably accelerates convergence of the optimization procedure. Application of the proposed regularization strategies to more complex optimization problems of physical and engineering relevance is also discussed.