Hilbert class library: A library of abstract C++ classes for optimization and inversion

AbstractAccording to the Object-Oriented Programming paradigm, a computer program should be organized around the fundamental objects it manipulates. In the C++ programming languages, these objects are embodied in classes. The Hilbert Class Library (HCL) is a collection of C++ classes designed for implementing numerical optimization algorithms in the context of Hilbert spaces. HCL includes base classes for defining vectors, linear operators, nonlinear operators and functionals, and related mathematical objects. Using these base classes, algorithms can be coded in a natural style that does not refer to application-specific details; nonetheless, the code can be applied to arbitrarily complex applications. Thus, HCL is intended to provide a way to bridge the often large gap between sophisticated numerical optimization routines and complicated simulation-based applications

Proximal methods for the elastography inverse problem of tumor identification using an equation error approach

In this chapter, we study a nonlinear inverse problem in linear elasticity relating to tumor identification by an equation error formulation. This approach leads to a variational inequality as a necessary and sufficient optimality condition. We give complete convergence analysis for the proposed equation error method. Since the considered problem is highly ill-posed, we develop a stable computational framework by employing a variety of proximal point methods and compare their performance with the more commonly used Tikhonov regularization