Generalized Nash equilibrium problems with partial differential operators: Theory, algorithms, and risk aversion

PDE-constrained (generalized) Nash equilibrium problems (GNEPs) are considered in a deterministic setting as well as under uncertainty. This includes a study of deterministic GNEPs with nonlinear and/or multivalued operator equations as forward problems and PDE-constrained GNEPs with uncertain data. The deterministic nonlinear problems are analyzed using the theory of generalized convexity for set-valued operators, and a variational approximation approach is proposed. The stochastic setting includes a detailed overview of the recently developed theory and algorithms for risk-averse PDE-constrained optimization problems. These new results open the way to a rigorous study of stochastic PDE-constrained GNEPs

Generalized Nash equilibrium problems with partial differential operators: Theory, algorithms, and risk aversion

PDE-constrained (generalized) Nash equilibrium problems (GNEPs) are considered in a deterministic setting as well as under uncertainty. This includes a study of deterministic GNEPs with nonlinear and/or multivalued operator equations as forward problems and PDE-constrained GNEPs with uncertain data. The deterministic nonlinear problems are analyzed using the theory of generalized convexity for set-valued operators, and a variational approximation approach is proposed. The stochastic setting includes a detailed overview of the recently developed theory and algorithms for risk-averse PDE-constrained optimization problems. These new results open the way to a rigorous study of stochastic PDE-constrained GNEPs

Approximate Models and Robust Decisions

Decisions based partly or solely on predictions from probabilistic models may be sensitive to model misspecification. Statisticians are taught from an early stage that "all models are wrong", but little formal guidance exists on how to assess the impact of model approximation on decision making, or how to proceed when optimal actions appear sensitive to model fidelity. This article presents an overview of recent developments across different disciplines to address this. We review diagnostic techniques, including graphical approaches and summary statistics, to help highlight decisions made through minimised expected loss that are sensitive to model misspecification. We then consider formal methods for decision making under model misspecification by quantifying stability of optimal actions to perturbations to the model within a neighbourhood of model space. This neighbourhood is defined in either one of two ways. Firstly, in a strong sense via an information (Kullback-Leibler) divergence around the approximating model. Or using a nonparametric model extension, again centred at the approximating model, in order to `average out' over possible misspecifications. This is presented in the context of recent work in the robust control, macroeconomics and financial mathematics literature. We adopt a Bayesian approach throughout although the methods are agnostic to this position

Phonon-mediated superconductivity in strongly correlated electron systems: a Luttinger-Ward functional approach

We use a Luttinger-Ward functional approach to study the problem of phonon-mediated superconductivity in electron systems with strong electron-electron interactions (EEIs). Our derivation does not rely on an expansion in skeleton diagrams for the EEI and the resulting theory is therefore nonperturbative in the strength of the latter. We show that one of the building blocks of the theory is the irreducible six-leg vertex related to EEIs. Diagrammatically, this implies five contributions (one of the Fock and four of the Hartree type) to the electronic self-energy, which, to the best of our knowledge, have never been discussed in the literature. Our approach is applicable to (and in fact designed to tackle superconductivity in) strongly correlated electron systems described by generic lattice models, as long as the glue for electron pairing is provided by phonons.Comment: To be published in the special issue of Annals of Physics "Eliashberg-90" dedicated to Gerasim (Sima) Eliashber

Relativistic nuclear energy density functional constrained by low-energy QCD

A relativistic nuclear energy density functional is developed, guided by two important features that establish connections with chiral dynamics and the symmetry breaking pattern of low-energy QCD: a) strong scalar and vector fields related to in-medium changes of QCD vacuum condensates; b) the long- and intermediate-range interactions generated by one-and two-pion exchange, derived from in-medium chiral perturbation theory, with explicit inclusion of $\Delta(1232)$ excitations. Applications are presented for binding energies, radii of proton and neutron distributions and other observables over a wide range of spherical and deformed nuclei from $^{16}O$ to $^{210}Po$. Isotopic chains of $Sn$ and $Pb$ nuclei are studied as test cases for the isospin dependence of the underlying interactions. The results are at the same level of quantitative comparison with data as the best phenomenological relativistic mean-field models.Comment: 48 pages, 12 figures, elsart.cls class file. Revised version, accepted for publication in Nucl. Phys.

Quantum information approach to electronic equilibria : molecular fragments and non-equilibrium thermodynamic description

The quantum-generalized Information Theory is applied to explore mole- cular equilibrium states by using the resultant information content of electronic states, determind by the classical (probability based) measures and their non -classical (phase/current related) complements, in the extremum entropy/information princi- ples.The“ vertical ”(probability-constrained)entropicrulesareinvestigatedwithinthe familiar Levy and Harriman–Zumbach–Maschke constructions of Density Functional Theory. A close parallelism between the vertical maximum-entropy and minimum- energy principles in quantum mechanics and their thermodynamic analogs is empha- sized and a relation between the probability and phase distributions in the “ horizontal ” (probability-unconstrained) phase -equilibria is examined. These solutions are shown to involve the spatial phase contribution related to the system electron density.The complete specification of the equilibrium states of molecular/promolecular fragments, including the subsystem density and the equilibrium phase of the system as a whole, is advocatedandillustratedforbondedhydrogensinH 2 .Elementsofthe non -equilibrium thermodynamic description of molecular systems are formulated. They recognize the independent probability and phase state parameters, the associated currents, and their contributions to the quantum entropy density and its current. The phase and entropy continuity equations are explored and the local sources of these quantities are identi- fied