Hi Josh, Luis Caffarelli has some writings from his Fermi lecture on the Obstacle problem that may or may not be helpful. He primarily looks at it in the context of variational inequalities.
http://user.math.uzh.ch/ros-oton/Caffarelli-The-obstacle-problem-SNS-1998.pdf I don’t have an academic reference for the solution of this as an optimization problem, but I have implemented it before as a TAO/MFEM test case for the ATPESC 2018 Hands-On Lessons. https://xsdk-project.github.io/ATPESC2018HandsOnLessons/lessons/obstacle_tao/ Hope that helps. —— Alp Dener Argonne National Laboratory http://www.mcs.anl.gov/person/alp-dener On Sep 12, 2018, at 11:01 AM, Josh L <[email protected]<mailto:[email protected]>> wrote: Hi Alp, Yes, my minimal energy problem reduces to the solution of (1). Is there any reference on the problem you mention in the last paragraph? Thanks, Josh 2018-09-12 9:26 GMT-05:00 Dener, Alp <[email protected]<mailto:[email protected]>>: Hi Josh, In the FormFunctionAndGradient() function, the gradient you compute and provide TAO must be the gradient of the objective function with respect to the optimization variables. If your optimization variables are u, and your objective function is J(u), then FormFunctionGradient would need to compute J(u) and dJ/du. I’m not entirely sure if I understand your problem correctly, but it seems that your minimal energy problem reduces to the solution of equation (1). Is this correct? TAO is not the right package to solve time-dependent PDEs like (1) directly. You would want to use TS for that. However, TAO can be used to solve the original energy minimization problem. A good example of this is the canonical obstacle problem that aims to minimize the Dirichlet energy function subject to a constraint that represents the obstacle. That problem reduces to the solution of the Laplace equation with the appropriate boundary conditions matching the obstacle, which would be discretized and solved as a stand-alone PDE. However, the same problem can also be solved as an optimization problem where the objective function is the Dirichlet energy function, and the associated gradient is related to the stiffness matrix associated with the Laplace equation. The two formulations are mathematically equivalent. If I’m interpreting your problem correctly, you should be able to take a similar approach, but you would need to derive the gradient of your energy function and see how you can construct it in relation to equation (1). —— Alp Dener Argonne National Laboratory http://www.mcs.anl.gov/person/alp-dener On Sep 11, 2018, at 3:51 PM, Josh L <[email protected]<mailto:[email protected]>> wrote: Hi, I am using PETSc TAO to solve the following equation obtained from minimizing certain energy and constitutive model: u_xx - u - u_t =0 (1) For simplicity, the coefficients are neglected. In the routine to form function value and gradient, I use (1) to form my gradient vector, and my energy definition to calculate function value.Is it correct? I only found one tao example that solves time dependent problem, but it is using TS which I am not using. Thanks, Josh
