2,559 research outputs found
Adjoint recovery of superconvergent functionals from PDE approximations
Motivated by applications in computational fluid dynamics, a method is presented for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed flow solution on which they are based. This is achieved through error analysis that uses an adjoint PDE to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest. Numerical evaluation of the local residual error together with an approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy. Numerical results are presented for the Poisson equation in one and two dimensions and for the nonlinear quasi-one-dimensional Euler equations. The theory is equally applicable to nonlinear equations in complex multi-dimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations
Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without L\'{e}vy area simulation
In this paper we introduce a new multilevel Monte Carlo (MLMC) estimator for
multi-dimensional SDEs driven by Brownian motions. Giles has previously shown
that if we combine a numerical approximation with strong order of convergence
with MLMC we can reduce the computational complexity to estimate
expected values of functionals of SDE solutions with a root-mean-square error
of from to . However, in
general, to obtain a rate of strong convergence higher than
requires simulation, or approximation, of L\'{e}vy areas. In this paper,
through the construction of a suitable antithetic multilevel correction
estimator, we are able to avoid the simulation of L\'{e}vy areas and still
achieve an multilevel correction variance for smooth payoffs,
and almost an variance for piecewise smooth payoffs, even
though there is only strong convergence. This results in an
complexity for estimating the value of European and Asian
put and call options.Comment: Published in at http://dx.doi.org/10.1214/13-AAP957 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Decision-making under uncertainty: using MLMC for efficient estimation of EVPPI
In this paper we develop a very efficient approach to the Monte Carlo
estimation of the expected value of partial perfect information (EVPPI) that
measures the average benefit of knowing the value of a subset of uncertain
parameters involved in a decision model. The calculation of EVPPI is inherently
a nested expectation problem, with an outer expectation with respect to one
random variable and an inner conditional expectation with respect to the
other random variable . We tackle this problem by using a Multilevel Monte
Carlo (MLMC) method (Giles 2008) in which the number of inner samples for
increases geometrically with level, so that the accuracy of estimating the
inner conditional expectation improves and the cost also increases with level.
We construct an antithetic MLMC estimator and provide sufficient assumptions on
a decision model under which the antithetic property of the estimator is well
exploited, and consequently a root-mean-square accuracy of can be
achieved at a cost of . Numerical results confirm the
considerable computational savings compared to the standard, nested Monte Carlo
method for some simple testcases and a more realistic medical application
Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance
In this article, we propose a Milstein finite difference scheme for a
stochastic partial differential equation (SPDE) describing a large particle
system. We show, by means of Fourier analysis, that the discretisation on an
unbounded domain is convergent of first order in the timestep and second order
in the spatial grid size, and that the discretisation is stable with respect to
boundary data. Numerical experiments clearly indicate that the same convergence
order also holds for boundary-value problems. Multilevel path simulation,
previously used for SDEs, is shown to give substantial complexity gains
compared to a standard discretisation of the SPDE or direct simulation of the
particle system. We derive complexity bounds and illustrate the results by an
application to basket credit derivatives
Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation
The multilevel Monte Carlo path simulation method introduced by Giles ({\it
Operations Research}, 56(3):607-617, 2008) exploits strong convergence
properties to improve the computational complexity by combining simulations
with different levels of resolution. In this paper we analyse its efficiency
when using the Milstein discretisation; this has an improved order of strong
convergence compared to the standard Euler-Maruyama method, and it is proved
that this leads to an improved order of convergence of the variance of the
multilevel estimator. Numerical results are also given for basket options to
illustrate the relevance of the analysis.Comment: 33 pages, 4 figures, to appear in Discrete and Continuous Dynamical
Systems - Series
Generation and use of unstructured grids for turbomachinery calculations
A wavefront mesh generator for two dimensional triangular meshes as well as a brief description of the solution method used with these meshes are presented. The interest is in creating meshes for solving the equations of fluid mechanics in complex turbomachinery problems, although the mesh generator and flow solver may be used for a larger variety of applications. The focus is on the flexibility and power of the mesh generation method for triangulating extremely complex geometries and in changing the geometry to create a new mesh. Two turbomachinery applications are presented which take advantage of this method: the analysis of pylon/strut and pylon/OGV interaction in the bypass of a turbofan
Random Bit Multilevel Algorithms for Stochastic Differential Equations
We study the approximation of expectations \E(f(X)) for solutions of
SDEs and functionals by means of restricted
Monte Carlo algorithms that may only use random bits instead of random numbers.
We consider the worst case setting for functionals from the Lipschitz class
w.r.t.\ the supremum norm. We construct a random bit multilevel Euler algorithm
and establish upper bounds for its error and cost. Furthermore, we derive
matching lower bounds, up to a logarithmic factor, that are valid for all
random bit Monte Carlo algorithms, and we show that, for the given quadrature
problem, random bit Monte Carlo algorithms are at least almost as powerful as
general randomized algorithms
Random Bit Quadrature and Approximation of Distributions on Hilbert Spaces
We study the approximation of expectations \E(f(X)) for Gaussian random
elements with values in a separable Hilbert space and Lipschitz
continuous functionals . We consider restricted Monte Carlo
algorithms, which may only use random bits instead of random numbers. We
determine the asymptotics (in some cases sharp up to multiplicative constants,
in the other cases sharp up to logarithmic factors) of the corresponding -th
minimal error in terms of the decay of the eigenvalues of the covariance
operator of . It turns out that, within the margins from above, restricted
Monte Carlo algorithms are not inferior to arbitrary Monte Carlo algorithms,
and suitable random bit multilevel algorithms are optimal. The analysis of this
problem leads to a variant of the quantization problem, namely, the optimal
approximation of probability measures on by uniform distributions supported
by a given, finite number of points. We determine the asymptotics (up to
multiplicative constants) of the error of the best approximation for the
one-dimensional standard normal distribution, for Gaussian measures as above,
and for scalar autonomous SDEs
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