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Much of uncertainty quantification to date has focused on determining the effect of variables modeled probabilistically, and with a known distribution, on some physical or engineering system. We develop methods to obtain information on the system when the distributions of some variables are known exactly, others are known only approximately, and perhaps others are not modeled as random variables at all.The main tool used is the duality between risk-sensitive integrals and relative entropy, and we obtain explicit bounds on standard performance measures (variances, exceedance probabilities) over families of distributions whose distance from a nominal distribution is measured by relative entropy. The evaluation of the risk-sensitive expectations is based on polynomial chaos expansions, which help keep the computational aspects tractable.
We consider the Euler equation for an incompressible fluid on a three dimensional torus,
and the construction of its solution as a power series in time. We point out some general
facts on this subject, from convergence issues for the power series to the role of
symmetries of the initial datum. We then turn the attention to a paper by Behr, Nečas and
Wu, ESAIM: M2AN 35 (2001) 229–238; here, the authors chose a
very simple Fourier polynomial as an initial datum for the Euler equation and analyzed the
power series in time for the solution, determining the first 35 terms by computer algebra.
Their calculations suggested for the series a finite convergence radius
τ3 in the H3 Sobolev space, with
0.32 < τ3 < 0.35; they regarded this as an indication
that the solution of the Euler equation blows up. We have repeated the calculations of E.
Behr, J. Nečas and H. Wu, ESAIM: M2AN 35 (2001) 229–238,
using again computer algebra; the order has been increased from 35 to 52, using the
symmetries of the initial datum to speed up computations. As for
τ3, our results agree with the original computations of E.
Behr, J. Nečas and H. Wu, ESAIM: M2AN 35 (2001) 229–238
(yielding in fact to conjecture that 0.32 < τ3 < 0.33).
Moreover, our analysis supports the following conclusions: (a) The finiteness of
τ3 is not at all an indication of a possible blow-up. (b)
There is a strong indication that the solution of the Euler equation does not blow up at a
time close to τ3. In fact, the solution is likely to exist, at
least, up to a time θ3 > 0.47. (c) There is a weak
indication, based on Padé analysis, that the solution might blow up at a later time.
We introduce a piecewise P2-nonconforming quadrilateral
finite element. First, we decompose a convex quadrilateral into the union of four
triangles divided by its diagonals. Then the finite element space is defined by the set of
all piecewise P2-polynomials that are quadratic in each
triangle and continuously differentiable on the quadrilateral. The degrees of freedom
(DOFs) are defined by the eight values at the two Gauss points on each of the four edges
plus the value at the intersection of the diagonals. Due to the existence of one linear
relation among the above DOFs, it turns out the DOFs are eight. Global basis functions are
defined in three types: vertex-wise, edge-wise, and element-wise types. The corresponding
dimensions are counted for both Dirichlet and Neumann types of elliptic problems. For
second-order elliptic problems and the Stokes problem, the local and global interpolation
operators are defined. Also error estimates of optimal order are given in both broken
energy and L2(Ω) norms. The proposed element
is also suitable to solve Stokes equations. The element is applied to approximate each
component of velocity fields while the discontinuous
P1-nonconforming quadrilateral element is adopted to
approximate the pressure. An optimal error estimate in energy norm is derived. Numerical
results are shown to confirm the optimality of the presented piecewise
P2-nonconforming element on quadrilaterals.
The parareal in time algorithm allows for efficient parallel numerical simulations of
time-dependent problems. It is based on a decomposition of the time interval into
subintervals, and on a predictor-corrector strategy, where the propagations over each
subinterval for the corrector stage are concurrently performed on the different processors
that are available. In this article, we are concerned with the long time integration of
Hamiltonian systems. Geometric, structure-preserving integrators are preferably employed
for such systems because they show interesting numerical properties, in particular
excellent preservation of the total energy of the system. Using a symmetrization procedure
and/or a (possibly also symmetric) projection step, we introduce here several variants of
the original plain parareal in time algorithm [L. Baffico, et al. Phys. Rev. E
66 (2002) 057701; G. Bal and Y. Maday, A parareal time
discretization for nonlinear PDE’s with application to the pricing of an American put, in
Recent developments in domain decomposition methods, Lect.
Notes Comput. Sci. Eng. 23 (2002) 189–202; J.-L. Lions, Y. Maday
and G. Turinici, C. R. Acad. Sci. Paris, Série I 332 (2001)
661–668.] that are better adapted to the Hamiltonian context. These variants are
compatible with the geometric structure of the exact dynamics, and are easy to implement.
Numerical tests on several model systems illustrate the remarkable properties of the
proposed parareal integrators over long integration times. Some formal elements of
understanding are also provided.
For a two phase incompressible flow we consider a diffuse interface model aimed at addressing the movement of three-phase (fluid-fluid-solid) contact lines. The model consists of the Cahn Hilliard Navier Stokes system with a variant of the Navier slip boundary conditions. We show that this model possesses a natural energy law. For this system, a new numerical technique based on operator splitting and fractional time-stepping is proposed. The method is shown to be unconditionally stable. We present several numerical illustrations of this scheme.
In the article an optimal control problem subject to a stationary variational inequality
is investigated. The optimal control problem is complemented with pointwise control
constraints. The convergence of a smoothing scheme is analyzed. There, the variational
inequality is replaced by a semilinear elliptic equation. It is shown that solutions of
the regularized optimal control problem converge to solutions of the original one. Passing
to the limit in the optimality system of the regularized problem allows to prove
C-stationarity of local solutions of the original problem. Moreover, convergence rates
with respect to the regularization parameter for the error in the control are obtained,
which turn out to be sharp. These rates coincide with rates obtained by numerical
experiments, which are included in the paper.
A mixed finite element method for the Navier–Stokes equations is introduced in which the
stress is a primary variable. The variational formulation retains the mathematical
structure of the Navier–Stokes equations and the classical theory extends naturally to
this setting. Finite element spaces satisfying the associated inf–sup conditions are
We analyze the regularity of random entropy solutions to scalar hyperbolic conservation laws with random initial data. We prove regularity theorems for statistics of random entropy solutions like expectation, variance, space-time correlation functions and polynomial moments such as gPC coefficients. We show how regularity of such moments (statistical and polynomial chaos) of random entropy solutions depends on the regularity of the distribution law of the random shock location of the initial data. Sufficient conditions on the law of the initial data for moments of the random entropy solution to be piece-wise smooth functions of space and time are identified, even in cases where path-wise random entropy solutions are discontinuous almost surely. We extrapolate the results to hyperbolic systems of conservation laws in one space dimension. We then exhibit a class of stochastic Galerkin discretizations which allows to derive closed deterministic systems of hyperbolic conservation laws for the coefficients in truncated polynomial chaos expansions of the random entropy solution. Based on the regularity theory developed here, we show that depending on the smoothness of the law of the initial data, arbitrarily high convergence rates are possible for the computation of coefficients in gPC approximations of random entropy solutions for Riemann problems with random shock location by combined Stochastic Galerkin Finite Volume schemes.
We discuss a numerical formulation for the cell problem related to a homogenization approach for the study of wetting on micro rough surfaces. Regularity properties of the solution are described in details and it is shown that the problem is a convex one. Stability of the solution with respect to small changes of the cell bottom surface allows for an estimate of the numerical error, at least in two dimensions. Several benchmark experiments are presented and the reliability of the numerical solution is assessed, whenever possible, by comparison with analytical one. Realistic three dimensional simulations confirm several interesting features of the solution, improving the classical models of study of wetting on roughness.
Iterative approximation algorithms are successfully applied in parametric approximation tasks. In particular, reduced basis methods make use of the so-called Greedy algorithm for approximating solution sets of parametrized partial differential equations. Recently, a priori convergence rate statements for this algorithm have been given (Buffa et al. 2009, Binev et al. 2010). The goal of the current study is the extension to time-dependent problems, which are typically approximated using the POD–Greedy algorithm (Haasdonk and Ohlberger 2008). In this algorithm, each greedy step is invoking a temporal compression step by performing a proper orthogonal decomposition (POD). Using a suitable coefficient representation of the POD–Greedy algorithm, we show that the existing convergence rate results of the Greedy algorithm can be extended. In particular, exponential or algebraic convergence rates of the Kolmogorov n-widths are maintained by the POD–Greedy algorithm.
The aim of this paper is to analyze a formulation of the eddy current problem in terms of a time-primitive of the electric field in a bounded domain with input current intensities or voltage drops as source data. To this end, we introduce a Lagrange multiplier to impose the divergence-free condition in the dielectric domain. Thus, we obtain a time-dependent weak mixed formulation leading to a degenerate parabolic problem which we prove is well-posed. We propose a finite element method for space discretization based on Nédélec edge elements for the main variable and standard finite elements for the Lagrange multiplier, for which we obtain error estimates. Then, we introduce a backward Euler scheme for time discretization and prove error estimates for the fully discrete problem, too. Finally, the method is applied to solve a couple of test problems.
We consider here the Interior Penalty Discontinuous Galerkin (IPDG) discretization of the wave equation. We show how to derive the optimal penalization parameter involved in this method in the case of regular meshes. Moreover, we provide necessary stability conditions of the global scheme when IPDG is coupled with the classical Leap–Frog scheme for the time discretization. Numerical experiments illustrate the fact that these conditions are also sufficient.