The accurate numerical solution of partial differential equations and optimization problems, including the use of linear algebra techniques for solving the resulting matrix systems, presents vast challenges for researchers in applied mathematics and engineering. In this workshop, we considered the fast and efficient discretization and solution of PDEs arising from a variety of scientific applications, including optimization problems where PDEs act as constraints. This workshop consisted of presentations from a leading researchers to encourage an exchange of ideas to further advance the state of the art in the numerical solution of PDEs, applied linear algebra, and computational optimization.