In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems involving (a) nonaffine dependence on the parameter, and (b) nonlinear dependence on the field variable. The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational decomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateral reduced-basis approximation space, and (ii) a stable and inexpensive interpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in each instance, we discuss the reduced-basis approximation and the associated offline-online computational procedures. Numerical results are presented to assess our approach.

Lagrangian interpolation is a classical way to approximate general functions by finite sums of well chosen, pre-defined, linearly independent interpolating functions; it is much simpler to implement than determining the best fits with respect to some Banach (or even Hilbert) norms. In addition, only partial knowledge is required (here values on some set of points). The problem of defining the best sample of points is nevertheless rather complex and is in general open. In this paper we propose a way to derive such sets of points. We do not claim that the points resulting from the construction explained here are optimal in any sense. Nevertheless, the resulting interpolation method is proven to work under certain hypothesis, the process is very general and simple to implement, and compared to situations where the best behavior is known, it is relatively competitive.

The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the elements that constitute the “reduced basis”. The purpose of this paper is to analyze the a priori convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we prove that three greedy algorithms converge; the last algorithm, based on the use of an a posteriori estimator, is the approach actually employed in the calculations.

We report here on the recent application of a now classical general reduction technique, the Reduced-Basis (RB) approach initiated by C. Prud’homme et al. in J. Fluids Eng. 124(1), 70–80, 2002, to the specific context of differential equations with random coefficients. After an elementary presentation of the approach, we review two contributions of the authors: in Comput. Methods Appl. Mech. Eng. 198(41–44), 3187–3206, 2009, which presents the application of the RB approach for the discretization of a simple second order elliptic equation supplied with a random boundary condition, and in Commun. Math. Sci., 2009, which uses a RB type approach to reduce the variance in the Monte-Carlo simulation of a stochastic differential equation. We conclude the review with some general comments and also discuss possible tracks for further research in the direction.

ADAPT-VQE is a robust algorithm for hybrid quantum-classical simulations of quantum chemical systems on near-term quantum computers. While its iterative process systematically reaches the ground state energy, practical implementations of ADAPT-VQE are sensitive to local energy minima, leading to over-parameterized ansätze. We introduce the Overlap-ADAPT-VQE to grow wave-functions by maximizing their overlap with any intermediate target wave-function that already captures some electronic correlation. By avoiding building the ansatz in the energy landscape strewn with local minima, the Overlap-ADAPT-VQE produces ultra-compact ansätze suitable for high-accuracy initialization of a new ADAPT procedure. Significant advantages over ADAPT-VQE are observed for strongly correlated systems including substantial savings in circuit depth. Since this compression strategy can also be initialized with accurate Selected-Configuration Interaction (SCI) classical target wave-functions, it paves the way for chemically accurate simulations of larger systems, and strengthens the promise of decisively surpassing classical quantum chemistry through the power of quantum computing.