Various properties of PDE solutions can be inferred by constructing functionals that satisfy suitable inequalities. A familiar example is the use of Lyapunov functionals to prove nonlinear stability. Similar methods exist for estimating time averages, extreme values, absorbing sets, and other properties. Often the simplest useful functionals are quadratic. In the case of the Navier--Stokes equations, using quadratic functionals to prove nonlinear stability corresponds to the "energy method", while using quadratic functionals to bounds time averages or absorbing sets corresponds to the "background method". In this talk I will describe ways to generalize beyond the quadratic case, thereby generalizing the energy and background methods. The desired functionals are constructed with computer assistance using methods of polynomial optimization and semidefinite programming -- a standard type of convex optimization problem. I will illustrate some applications, including estimating mean energy in the Kuramoto--Sivashinsky equation, and verifying nonlinear stability of solutions to the Navier--Stokes equations.