A modified balanced POD model reduction algorithm for parabolic PDEs with unbounded inputs

We develop an algorithm to approximate the balanced reduced order model for linear parabolic PDE systems with an unbounded input operator and a bounded output operator. We use balanced POD, a widely used simulation-based algorithm for balanced truncation, however we modify the algorithm to avoid unusual and challenging simulations due to the unbounded input operator. We develop the modified algorithm for two classes of unbounded input operators; the first class includes Neumann boundary input operators, and the second class includes Dirichlet boundary input operators. We present numerical results for a convection diffusion equation.