33 research outputs found
Multi-Index Monte Carlo: When Sparsity Meets Sampling
We propose and analyze a novel Multi-Index Monte Carlo (MIMC) method for weak
approximation of stochastic models that are described in terms of differential
equations either driven by random measures or with random coefficients. The
MIMC method is both a stochastic version of the combination technique
introduced by Zenger, Griebel and collaborators and an extension of the
Multilevel Monte Carlo (MLMC) method first described by Heinrich and Giles.
Inspired by Giles's seminal work, we use in MIMC high-order mixed differences
instead of using first-order differences as in MLMC to reduce the variance of
the hierarchical differences dramatically. This in turn yields new and improved
complexity results, which are natural generalizations of Giles's MLMC analysis
and which increase the domain of the problem parameters for which we achieve
the optimal convergence, Moreover, in MIMC, the
rate of increase of required memory with respect to is independent
of the number of directions up to a logarithmic term which allows far more
accurate solutions to be calculated for higher dimensions than what is possible
when using MLMC.
We motivate the setting of MIMC by first focusing on a simple full tensor
index set. We then propose a systematic construction of optimal sets of indices
for MIMC based on properly defined profits that in turn depend on the average
cost per sample and the corresponding weak error and variance. Under standard
assumptions on the convergence rates of the weak error, variance and work per
sample, the optimal index set turns out to be the total degree (TD) type. In
some cases, using optimal index sets, MIMC achieves a better rate for the
computational complexity than the corresponding rate when using full tensor
index sets..
Nested Multilevel Monte Carlo with Biased and Antithetic Sampling
We consider the problem of estimating a nested structure of two expectations
taking the form , where .
Terms of this form arise in financial risk estimation and option pricing. When
requires approximation, but exact samples of and are
available, an antithetic multilevel Monte Carlo (MLMC) approach has been
well-studied in the literature. Under general conditions, the antithetic MLMC
estimator obtains a root mean squared error with order
cost. If, additionally, and require approximate
sampling, careful balancing of the various aspects of approximation is required
to avoid a significant computational burden. Under strong convergence criteria
on approximations to and , randomised multilevel Monte Carlo techniques
can be used to construct unbiased Monte Carlo estimates of , which can be
paired with an antithetic MLMC estimate of to recover order
computational cost. In this work, we instead consider biased
multilevel approximations of , which require less strict assumptions on
the approximate samples of . Extensions to the method consider an
approximate and antithetic sampling of . Analysis shows the resulting
estimator has order asymptotic cost under the conditions
required by randomised MLMC and order
cost under more general assumptions.Comment: 28 pages, 2 figure
Multi-index Stochastic Collocation convergence rates for random PDEs with parametric regularity
We analyze the recent Multi-index Stochastic Collocation (MISC) method for
computing statistics of the solution of a partial differential equation (PDEs)
with random data, where the random coefficient is parametrized by means of a
countable sequence of terms in a suitable expansion. MISC is a combination
technique based on mixed differences of spatial approximations and quadratures
over the space of random data and, naturally, the error analysis uses the joint
regularity of the solution with respect to both the variables in the physical
domain and parametric variables. In MISC, the number of problem solutions
performed at each discretization level is not determined by balancing the
spatial and stochastic components of the error, but rather by suitably
extending the knapsack-problem approach employed in the construction of the
quasi-optimal sparse-grids and Multi-index Monte Carlo methods. We use a greedy
optimization procedure to select the most effective mixed differences to
include in the MISC estimator. We apply our theoretical estimates to a linear
elliptic PDEs in which the log-diffusion coefficient is modeled as a random
field, with a covariance similar to a Mat\'ern model, whose realizations have
spatial regularity determined by a scalar parameter. We conduct a complexity
analysis based on a summability argument showing algebraic rates of convergence
with respect to the overall computational work. The rate of convergence depends
on the smoothness parameter, the physical dimensionality and the efficiency of
the linear solver. Numerical experiments show the effectiveness of MISC in this
infinite-dimensional setting compared with the Multi-index Monte Carlo method
and compare the convergence rate against the rates predicted in our theoretical
analysis
Sub-sampling and other considerations for efficient risk estimation in large portfolios
Computing risk measures of a financial portfolio comprising thousands of
options is a challenging problem because (a) it involves a nested expectation
requiring multiple evaluations of the loss of the financial portfolio for
different risk scenarios and (b) evaluating the loss of the portfolio is
expensive and the cost increases with its size. In this work, we look at
applying Multilevel Monte Carlo (MLMC) with adaptive inner sampling to this
problem and discuss several practical considerations. In particular, we discuss
a sub-sampling strategy that results in a method whose computational complexity
does not increase with the size of the portfolio. We also discuss several
control variates that significantly improve the efficiency of MLMC in our
setting
Multilevel Path Branching for Digital Options
We propose a new Monte Carlo-based estimator for digital options with assets
modelled by a stochastic differential equation (SDE). The new estimator is
based on repeated path splitting and relies on the correlation of approximate
paths of the underlying SDE that share parts of a Brownian path. Combining this
new estimator with Multilevel Monte Carlo (MLMC) leads to an estimator with a
complexity that is similar to the complexity of a MLMC estimator when applied
to options with Lipschitz payoffs
Multilevel nested simulation for efficient risk estimation
We investigate the problem of computing a nested expectation of the form where is the Heaviside function. This nested expectation appears, for example, when estimating the probability of a large loss from a financial portfolio. We present a method that combines the idea of using Multilevel Monte Carlo (MLMC) for nested expectations with the idea of adaptively selecting the number of samples in the approximation of the inner expectation, as proposed by [M. Broadie, Y. Du, and C. C. Moallemi, Manag. Sci., 57 (2011), pp. 1172--1194]. We propose and analyze an algorithm that adaptively selects the number of inner samples on each MLMC level and prove that the resulting MLMC method with adaptive sampling has an complexity to achieve a root mean-squared error . The theoretical analysis is verified by numerical experiments on a simple model problem. We also present a stochastic root-finding algorithm that, combined with our adaptive methods, can be used to compute other risk measures such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR), the latter being achieved with complexity.
Read More: https://epubs.siam.org/doi/10.1137/18M117318
Optimization of mesh hierarchies in Multilevel Monte Carlo samplers
We perform a general optimization of the parameters in the Multilevel Monte
Carlo (MLMC) discretization hierarchy based on uniform discretization methods
with general approximation orders and computational costs. We optimize
hierarchies with geometric and non-geometric sequences of mesh sizes and show
that geometric hierarchies, when optimized, are nearly optimal and have the
same asymptotic computational complexity as non-geometric optimal hierarchies.
We discuss how enforcing constraints on parameters of MLMC hierarchies affects
the optimality of these hierarchies. These constraints include an upper and a
lower bound on the mesh size or enforcing that the number of samples and the
number of discretization elements are integers. We also discuss the optimal
tolerance splitting between the bias and the statistical error contributions
and its asymptotic behavior. To provide numerical grounds for our theoretical
results, we apply these optimized hierarchies together with the Continuation
MLMC Algorithm. The first example considers a three-dimensional elliptic
partial differential equation with random inputs. Its space discretization is
based on continuous piecewise trilinear finite elements and the corresponding
linear system is solved by either a direct or an iterative solver. The second
example considers a one-dimensional It\^o stochastic differential equation
discretized by a Milstein scheme
A Continuation Multilevel Monte Carlo algorithm
We propose a novel Continuation Multi Level Monte Carlo (CMLMC) algorithm for
weak approximation of stochastic models. The CMLMC algorithm solves the given
approximation problem for a sequence of decreasing tolerances, ending when the
required error tolerance is satisfied. CMLMC assumes discretization hierarchies
that are defined a priori for each level and are geometrically refined across
levels. The actual choice of computational work across levels is based on
parametric models for the average cost per sample and the corresponding weak
and strong errors. These parameters are calibrated using Bayesian estimation,
taking particular notice of the deepest levels of the discretization hierarchy,
where only few realizations are available to produce the estimates. The
resulting CMLMC estimator exhibits a non-trivial splitting between bias and
statistical contributions. We also show the asymptotic normality of the
statistical error in the MLMC estimator and justify in this way our error
estimate that allows prescribing both required accuracy and confidence in the
final result. Numerical results substantiate the above results and illustrate
the corresponding computational savings in examples that are described in terms
of differential equations either driven by random measures or with random
coefficients