581 research outputs found
A Note on the Importance of Weak Convergence Rates for SPDE Approximations in Multilevel Monte Carlo Schemes
It is a well-known rule of thumb that approximations of stochastic partial
differential equations have essentially twice the order of weak convergence
compared to the corresponding order of strong convergence. This is already
known for many approximations of stochastic (ordinary) differential equations
while it is recent research for stochastic partial differential equations. In
this note it is shown how the availability of weak convergence results
influences the number of samples in multilevel Monte Carlo schemes and
therefore reduces the computational complexity of these schemes for a given
accuracy of the approximations.Comment: 16 pages, 3 figures, updated to version published in the Proceedings
of MCQMC1
Regression-based variance reduction approach for strong approximation schemes
In this paper we present a novel approach towards variance reduction for
discretised diffusion processes. The proposed approach involves specially
constructed control variates and allows for a significant reduction in the
variance for the terminal functionals. In this way the complexity order of the
standard Monte Carlo algorithm () can be reduced down to
in case of the Euler
scheme with being the precision to be achieved. These theoretical
results are illustrated by several numerical examples.Comment: arXiv admin note: text overlap with arXiv:1510.0314
Discrete adjoint approximations with shocks
This paper is concerned with the formulation and discretisation of adjoint equations when there are shocks in the underlying solution to the original nonlinear hyperbolic p.d.e. For the model problem of a scalar unsteady one-dimensional p.d.e. with a convex flux function, it is shown that the analytic formulation of the adjoint equations requires the imposition of an interior boundary condition along any shock. A 'discrete adjoint' discretisation is defined by requiring the adjoint equations to give the same value for the linearised functional as a linearisation of the original nonlinear discretisation. It is demonstrated that convergence requires increasing numerical smoothing of any shocks. Without this, any consistent discretisation of the adjoint equations without the inclusion of the shock boundary condition may yield incorrect values for the adjoint solution
Blocked All-Pairs Shortest Paths Algorithm on Intel Xeon Phi KNL Processor: A Case Study
Manycores are consolidating in HPC community as a way of improving
performance while keeping power efficiency. Knights Landing is the recently
released second generation of Intel Xeon Phi architecture. While optimizing
applications on CPUs, GPUs and first Xeon Phi's has been largely studied in the
last years, the new features in Knights Landing processors require the revision
of programming and optimization techniques for these devices. In this work, we
selected the Floyd-Warshall algorithm as a representative case study of graph
and memory-bound applications. Starting from the default serial version, we
show how data, thread and compiler level optimizations help the parallel
implementation to reach 338 GFLOPS.Comment: Computer Science - CACIC 2017. Springer Communications in Computer
and Information Science, vol 79
A Dimension-Adaptive Multi-Index Monte Carlo Method Applied to a Model of a Heat Exchanger
We present an adaptive version of the Multi-Index Monte Carlo method,
introduced by Haji-Ali, Nobile and Tempone (2016), for simulating PDEs with
coefficients that are random fields. A classical technique for sampling from
these random fields is the Karhunen-Lo\`eve expansion. Our adaptive algorithm
is based on the adaptive algorithm used in sparse grid cubature as introduced
by Gerstner and Griebel (2003), and automatically chooses the number of terms
needed in this expansion, as well as the required spatial discretizations of
the PDE model. We apply the method to a simplified model of a heat exchanger
with random insulator material, where the stochastic characteristics are
modeled as a lognormal random field, and we show consistent computational
savings
Eigen-AD: Algorithmic Differentiation of the Eigen Library
In this work we present useful techniques and possible enhancements when
applying an Algorithmic Differentiation (AD) tool to the linear algebra library
Eigen using our in-house AD by overloading (AD-O) tool dco/c++ as a case study.
After outlining performance and feasibility issues when calculating derivatives
for the official Eigen release, we propose Eigen-AD, which enables different
optimization options for an AD-O tool by providing add-on modules for Eigen.
The range of features includes a better handling of expression templates for
general performance improvements, as well as implementations of symbolically
derived expressions for calculating derivatives of certain core operations. The
software design allows an AD-O tool to provide specializations to automatically
include symbolic operations and thereby keep the look and feel of plain AD by
overloading. As a showcase, dco/c++ is provided with such a module and its
significant performance improvements are validated by benchmarks.Comment: Updated with accepted version for ICCS 2020 conference proceedings.
The final authenticated publication is available online at
https://doi.org/10.1007/978-3-030-50371-0_51. See v1 for the original,
extended preprint. 14 pages, 7 figure
Construction of a Mean Square Error Adaptive Euler--Maruyama Method with Applications in Multilevel Monte Carlo
A formal mean square error expansion (MSE) is derived for Euler--Maruyama
numerical solutions of stochastic differential equations (SDE). The error
expansion is used to construct a pathwise a posteriori adaptive time stepping
Euler--Maruyama method for numerical solutions of SDE, and the resulting method
is incorporated into a multilevel Monte Carlo (MLMC) method for weak
approximations of SDE. This gives an efficient MSE adaptive MLMC method for
handling a number of low-regularity approximation problems. In low-regularity
numerical example problems, the developed adaptive MLMC method is shown to
outperform the uniform time stepping MLMC method by orders of magnitude,
producing output whose error with high probability is bounded by TOL>0 at the
near-optimal MLMC cost rate O(TOL^{-2}log(TOL)^4).Comment: 43 pages, 12 figure
Rapid covariance-based sampling of linear SPDE approximations in the multilevel Monte Carlo method
The efficient simulation of the mean value of a non-linear functional of the
solution to a linear stochastic partial differential equation (SPDE) with
additive Gaussian noise is considered. A Galerkin finite element method is
employed along with an implicit Euler scheme to arrive at a fully discrete
approximation of the mild solution to the equation. A scheme is presented to
compute the covariance of this approximation, which allows for rapid sampling
in a Monte Carlo method. This is then extended to a multilevel Monte Carlo
method, for which a scheme to compute the cross-covariance between the
approximations at different levels is presented. In contrast to traditional
path-based methods it is not assumed that the Galerkin subspaces at these
levels are nested. The computational complexities of the presented schemes are
compared to traditional methods and simulations confirm that, under suitable
assumptions, the costs of the new schemes are significantly lower.Comment: 18 pages, 5 figures; numerical simulations revised, implementation
section added; To appear in Monte Carlo and Quasi-Monte Carlo Methods -
MCQMC, Rennes, France, July 201
The VOLNA-OP2 tsunami code (version 1.5)
In this paper, we present the VOLNA-OP2
tsunami model and implementation; a finite-volume nonlinear
shallow-water equation (NSWE) solver built on
the OP2 domain-specific language (DSL) for unstructured
mesh computations. VOLNA-OP2 is unique among tsunami
solvers in its support for several high-performance computing
platforms: central processing units (CPUs), the Intel
Xeon Phi, and graphics processing units (GPUs). This is
achieved in a way that the scientific code is kept separate
from various parallel implementations, enabling easy maintainability.
It has already been used in production for several
years; here we discuss how it can be integrated into various
workflows, such as a statistical emulator. The scalability of
the code is demonstrated on three supercomputers, built with
classical Xeon CPUs, the Intel Xeon Phi, and NVIDIA P100
GPUs. VOLNA-OP2 shows an ability to deliver productivity
as well as performance and portability to its users across a
number of platforms
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