High dimensional finite elements for multiscale wave equations

For locally periodic multiscale wave equations in $\mathbb{R}^d$ that depend on a macroscopic scale and n microscopic separated scales, we solve the high dimensional limiting multiscale homogenized problem that is posed in $(n+1)d$ dimensions and is obtained by multiscale convergence. We consider the full and sparse tensor product finite element methods, and analyze both the spatial semidiscrete and the fully (both temporal and spatial) discrete approximating problems. With sufficient regularity, the sparse tensor product approximation achieves a convergence rate essentially equal to that for the full tensor product approximation, but requires only an essentially equal number of degrees of freedom as for solving an equation in $\mathbb{R}^d$ for the same level of accuracy. For the initial condition $u(0,x)=0$, we construct a numerical corrector from the finite element solution. In the case of two scales, we derive an explicit homogenization error which, together with the finite element error, produces an explicit rate of convergence for the numerical corrector. Numerical examples for two- and three-scale problems in one or two dimensions confirm our analysis.Published versio

Best N-term GPC approximations for a class of stochastic linear elasticity equations

We consider a class of stochastic linear elasticity problems whose elastic moduli depend linearly on a countable set of random variables. The stochastic equation is studied via a deterministic parametric problem on an infinite-dimensional parameter space. We first study the best N-term approximation of the generalized polynomial chaos (gpc) expansion of the solution to the displacement formula by considering a Galerkin projection onto the space obtained by truncating the gpc expansion. We provide sufficient conditions on the coefficients of the elastic moduli’s expansion so that a rate of convergence for this approximation holds. We then consider two classes of stochastic and parametric mixed elasticity problems. The first one is the Hellinger–Reissner formula for approximating directly the gpc expansion of the stress. For isotropic problems, the multiplying constant of the best N-term convergence rate for the displacement formula grows with the ratio of the Lame constants. We thus consider stochastic and parametric mixed problems for nearly incompressible isotropic materials whose best N-term approximation rate is uniform with respect to the ratio of the Lame constants

Alternative targets in optimal monetary policy rules in an open economy

This report examines the optimal monetary policy rules in a two-country DSGE model with real and nominal rigidities. The model is solved by simulation in discrete time and the optimal rule is selected as the one giving highest utility. We find that (1) targeting wage inflation always delivers welfare improvements; (2) targeting exchange rate brings non-decreasing welfare; (3) with interest rate smoothing, monetary authority should give equal weight to inflation targeting and wage inflation targeting. Also, the two-country DSGE model shows a good forecast power after technological shocks and transaction cost shocks.Bachelor of Art

Polynomial approximations of a class of stochastic multiscale elasticity problems

We consider a class of elasticity equations in Rd whose elastic moduli depend on n separated microscopic scales. The moduli are random and expressed as a linear expansion of a countable sequence of random variables which are independently and identically uniformly distributed in a compact interval. The multiscale Hellinger–Reissner mixed problem that allows for computing the stress directly and the multiscale mixed problem with a penalty term for nearly incompressible isotropic materials are considered. The stochastic problems are studied via deterministic problems that depend on a countable number of real parameters which represent the probabilistic law of the stochastic equations. We study the multiscale homogenized problems that contain all the macroscopic and microscopic information. The solutions of these multiscale homogenized problems are written as generalized polynomial chaos (gpc) expansions. We approximate these solutions by semidiscrete Galerkin approximating problems that project into the spaces of functions with only a finite number of N gpc modes. Assuming summability properties for the coefficients of the elastic moduli’s expansion, we deduce bounds and summability properties for the solutions’ gpc expansion coefficients. These bounds imply explicit rates of convergence in terms of N when the gpc modes used for the Galerkin approximation are chosen to correspond to the best N terms in the gpc expansion. For the mixed problem with a penalty term for nearly incompressible materials, we show that the rate of convergence for the best N term approximation is independent of the Lamé constants’ ratio when it goes to ∞ . Correctors for the homogenization problem are deduced. From these we establish correctors for the solutions of the parametric multiscale problems in terms of the semidiscrete Galerkin approximations. For two-scale problems, an explicit homogenization error which is uniform with respect to the parameters is deduced. Together with the best N term approximation error, it provides an explicit convergence rate for the correctors of the parametric multiscale problems. For nearly incompressible materials, we obtain a homogenization error that is independent of the ratio of the Lamé constants, so that the error for the corrector is also independent of this ratio.ASTAR (Agency for Sci., Tech. and Research, S’pore)MOE (Min. of Education, S’pore