Time Stepping Classes for Optimization

This report introduces the “Time Stepping Package for Optimization”, or TSOpt, which is an interface for time-stepping simulation written in C++. It packages a simulator together with its derivatives (“sensitivities”) and adjoint derivatives with respect to simulation parameters in a single object called a Jet, which can be used in conjunction with an optimization algorithm to solve a simulation-driven optimization problem. Further, TSOpt interfaces with the Rice Vector Library (RVL), allowing Jet objects to define a Operator subclass. INTRODUCTION We are interested in solving optimization problems of the form min c J(c) = G(u(c, ·)) (1) where the state u(t) and the control c(t) solve the state equation H̄ ( du dt , u, c ) = 0 . (2) Though many types of control and inverse problems fit the mathematical framework above, we note that the procedure of solving such problems numerically oft involve the same procedures (e.g., formation of derivatives). To exploit these commonalities, hence, we created TSOpt, or the “Time-Stepping for Optimization” package. TSOpt is a “middleware” package written in C++, designed to act as an “interface for time-stepping simulation”, providing a way for simulation software to inter-operate with optimization software. TSOpt is capable of encapsulating the reference, linearized and adjoint simulators in a single object, and properly arrange their execution. TSOpt also aids in providing necessary data structures for the optimization algorithm (e.g., the gradient, formed via the adjointstate method). This report is organized as follows: the first section introduces RVL and section two discusses the Alg framework developed by Tony Padula. RVL and the Alg framework provides the foundation for TSOpt. The most notable features of TSOpt include its modular code structure, due to use of the Alg framework from the Rice Vector Library (RVL), and also accommodation of a generic data structure type through templating. The specifics of the structure of TSOpt and its features will be discussed in more detail in section three.