Numerical solution to the inverse problem of the calculus of variations
MetadataShow full item record
Variational formalisms provide a framework that allows for invariant functional representations of operators. Calculus of variations is oftentimes concerned with the study of stationarity and the conditions that make a functional representation stationary, these are notions discussed in Gelfand and Fomin classic monograph. The inverse problem of calculus of variations studies the same problem in reverse: given an operator which is dependent on some state input, can one formulate a functional representation of this operator such that the functional is stationary with respect to the given operator once the input state is varied. One of the benefits of these functional representations is that they are particularly amenable to discretization and allow for compact representation of a numerical method that approximates the solution to the associated operator. Although there are many benefits to representing an operator using a functional formalism, not all operators can be represented within the "classical'' inner product framework. Additionally, in order to use the inner product formulation, some assumptions about the operators, which may not be true, must be made. To resolve both the aforementioned issues, kernel (or transform) based variational formalisms, which can be extended to almost any operator, have been developed. Research on this topic has previously been carried out by Tonti, but no numerical methods that make use of this framework have been developed. Here, a numerical approach is developed that discretizes the state input to the operator, as well as the kernel inherent to this formulation. This new approach approximates the kernel and the state. Previous attempts at numerical solutions would approximate the state only, while seeking an analytical solution to the kernel. The new approach presented here allows for the development of variational formalism-based numerical approaches for almost any operator, a capability that has thus far been limited to specific types of operators. The ability to find these numerical solutions to arbitrary operators, which have these invariant and stable properties, is extremely useful for simulating these systems and can lead to more accurate simulations of physical phenomena.