Abstract
This paper presents a numerical method for PDE-constrained optimization problems. These problems arise in many fields of science and engineering including those dealing with real applications. The physical problem is modeled by partial differential equations (PDEs) and involve optimization of some quantity. The PDEs are in most cases nonlinear and solved using numerical methods. Since such numerical solutions are being used routinely, the recent trend has been to develop numerical methods and algorithms so that the optimization problems can be solved numerically as well using the same PDE-solver. We present here one such numerical method which is based on simultaneous pseudo-time stepping. The efficiency of the method is increased with the help of a multigrid strategy. Application example is included for an aerodynamic shape optimization problem.
Highlights
Numerical methods for partial differential equations (PDEs) are being used routinely for analysis of problems in scientific research and industrial applications as they are cost effective alternative to experiments
This paper presents a numerical method for PDE-constrained optimization problems
The physical problem is modeled by partial differential equations (PDEs) and involve optimization of some quantity
Summary
Numerical methods for PDEs ( known as simulation techniques) are being used routinely for analysis of problems in scientific research and industrial applications as they are cost effective alternative to experiments. Adjoint based gradient methods are advantageous as long as the number of design variables are large in comparison to the number of state constraints present in the problem. The number of optimization iterations is comparatively large since we update the design parameters after each time-step of the state and costate runs To overcome this problem, we use a multigrid strategy to accelerate the optimization convergence. We use simultaneous pseudo-time stepping for solving above problem (4) In this method, to determine the solution of (4), we look for the steady state solutions of the following pseudo-time embedded evolution equations dw dt c. L w, q, This formulation is advantageous, for the problem class in which the steady-state forward (as well as adjoint) solution is obtained by integrating the pseudo-unsteady state equations (e.g. in our application example). 4) Solve the quadratic subproblem (9) to get q . 5) March in time one step for the design equation. 6) Use the correction step for the new qk + 1. 7) Compute wk + 1 marching one step in time for the state equations. 8) Set k := k 1 ; go to 1) until convergence
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