Application of the Double Descent Color Intermittent Diffusion Method to a Variational Data Assimilation Problem Open Access

Gonzalez Hernandez, Rodrigo Jose (Spring 2022)

Permanent URL: https://etd.library.emory.edu/concern/etds/2227mq900?locale=en
Published

Abstract

The Double Descent Color Intermittent Diffusion Method (DD-CID) is a global optimization algorithm that identifies the global minimum of a smooth function through the combination of the Double Descent method and a basin-escape technique with stochastic features. Data Assimilation (DA) is the branch of applied mathematics that involves the optimal blend of mathematical models and observed data. In this work, two data assimilation problems in the form of constrained optimization problems are proposed. Specifically, the goal is to identify some unknown parameters in two boundary-value problems. The DD-CID method is tested against these and numerical results are presented.

Overview ...................................................................................................................................... 1

1 Double Descent and Color Intermittent Diffusion for Landscape Exploration .................................. 3

1.1 Global Optimization ................................................................................................................. 3

1.1.1 Global Optimization: an introduction through a simple working example .................................. 4

1.1.2 Numerical solution to the simple working example ................................................................. 12

1.1.3 Some difficulties of the global optimization problem .............................................................. 14

1.2 DDCID .................................................................................................................................... 20

1.3 Local search ............................................................................................................................ 20

1.3.1 Local search: saddle point ..................................................................................................... 20

1.3.2 Local search: local minimum.................................................................................................. 21

1.4 Landscape exploration ............................................................................................................. 23

1.4.1 Local minimum to saddle point .............................................................................................. 23

1.4.2 Saddle point to local minimum ............................................................................................... 24

1.5 DD-CID at a glance ................................................................................................................... 25

1.6 DD-CID visualized.................................................................................................................... 26

1.6.1 Contour plot and Hessian regions ........................................................................................... 26

1.6.2 Finding an initial minimum through Double Descent ............................................................... 27

1.6.3 Escaping the basin of attraction of the minimum through color noise diffusion .......................... 28

1.6.4 Finding a saddle point through Damped Newton ...................................................................... 29

1.6.5 Escaping the basin of attraction of the saddle through Diffused Double Descent ......................... 30

1.6.6 Finding a local minimum point through Double Descent ........................................................... 31

2 Data Assimilation and Global Constrained Optimization ................................................................ 33

2.1 Data Assimilation: the big picture .............................................................................................. 33

2.2 Parameter estimation ............................................................................................................... 35

2.2.1 The state equation ................................................................................................................. 37

2.2.2 The adjoint problem ............................................................................................................... 37

2.2.3 The optimality condition......................................................................................................... 38

2.2.4 Two test problems .................................................................................................................. 38

2.3 One-dimensional problem ......................................................................................................... 39

2.3.1 Boundary-value problem ......................................................................................................... 39

2.3.2 Function to be optimized ........................................................................................................ 40

2.3.3 Derivation of the gradient ....................................................................................................... 40

2.3.4 Summary ................................................................................................................................43

2.4 Multi-dimensional problem.........................................................................................................44

2.4.1 Boundary-value problem .........................................................................................................44

2.4.2 Function to be optimized......................................................................................................... 45

2.4.3 Derivation of the gradient ....................................................................................................... 45

2.4.4 Summary ................................................................................................................................48

2.5 Numerical Hessian .....................................................................................................................49

3 Results and Conclusions ............................................................................................................... 51

3.1 The coding environment: FEniCS................................................................................................ 51

3.2 From MATLAB to Python ........................................................................................................... 54

3.3 Visualization of the functionals ................................................................................................. 57

3.4 Results...................................................................................................................................... 58

3.5 Conclusions .............................................................................................................................. 59

Bibliography ...................................................................................................................................61