The Second Order Adjoint Analysis Team of Stochastic Department Of Civil Engineering Chuo University Kawahara Lab.

What is Hessian? Performance Function Taylor Expansion : The model solution : Observation Taylor Expansion

The First Order

The Second Order Hessian

Hessian Action Calculation DFP method High computational burden EFGS method SOA ( Second Order Adjoint )

Basic Equation Thermal Conduction Equation Initial Condition Specific heat Density Thermal Conductivity Temperature Initial Condition

Boundary Condition Heat Flux Error

Uncertain Estimation via Hessian Calculation Performance Function Exact Solution Noisy Solution

Lagrange Multiplier Method First Order Adjoint Variation

First Order Adjoint (FOA) The First Order Adjoint Equation Boundary Conditions Final Condition

Tangent Linear Tangent Liner Problem Boundary Conditions Initial Condition

Second Order Adjoint (SOA) The Second Order Adjoint Equation Boundary Conditions Final Condition

and The Hessian Action

Newton’s Method Gradient Hessian

Numerical Example

Performance Function

Heat Flux

Temperature

Conclusion The calculation of the Hessian can be performed using the solution of the SOA problem. The optimal control of thermal conduction can be analyzed using Hessian.