of the well-known Douglas−Rachford and Peaceman−Rachford numerical algorithms for computation of solutions to parabolic partial differential equations. The Lions−Mercier algorithms and their proof of convergence have been particularly influential in the literature on operator theory and its applications to numerical analysis. A similar method was studied at the same time by Gregory Passty. Calculus of variations The mathematical study of the steady-state Schrödinger–Newton equation, also called the Choquard equation, was initiated in a seminal article of Elliott Lieb. It is inspired by plasma physics via a standard approximation technique in quantum chemistry. Lions showed that one could apply standard methods such as the mountain pass theorem, together with some technical work of Walter Strauss, in order to show that a generalized steady-state Schrödinger–Newton equation with a radially symmetric generalization of the gravitational potential is necessarily solvable by a radially symmetric function. The partial differential equation has received a great deal of attention in the mathematical literature. Lions' extensive work on this equation is concerned with the existence of rotationally symmetric solutions as well as estimates and existence for boundary value problems of various type. In the interest of studying solutions on all of Euclidean space, where standard compactness theory does not apply, Lions established a number of compactness results for functions with symmetry. With Henri Berestycki and Lambertus Peletier, Lions used standard ODE shooting methods to directly study the existence of rotationally symmetric solutions. However, sharper results were obtained two years later by Berestycki and Lions by variational methods. They considered the solutions of the equation as rescalings of minima of a constrained optimization problem, based upon a modified Dirichlet energy. Making use of the Schwarz symmetrization, there exists a minimizing sequence for the infimization problem which consists of positive and rotationally symmetric functions. So they were able to show that there is a minimum which is also rotationally symmetric and nonnegative. By adapting the critical point methods of Felix Browder, Paul Rabinowitz, and others, Berestycki and Lions also demonstrated the existence of infinitely many (not always positive) radially symmetric solutions to the PDE. Maria Esteban and Lions investigated the nonexistence of solutions in a number of unbounded domains with Dirichlet boundary data. Their basic tool is a Pohozaev-type identity, as previously reworked by Berestycki and Lions. They showed that such identities can be effectively used with Nachman Aronszajn's unique continuation theorem to obtain the triviality of solutions under some