214,548 research outputs found
Equivalence of operator-splitting schemes for the integration of the Langevin equation
We investigate the equivalence of different operator-splitting schemes for
the integration of the Langevin equation. We consider a specific problem, so
called the directed percolation process, which can be extended to a wider class
of problems. We first give a compact mathematical description of the
operator-splitting method and introduce two typical splitting schemes that will
be useful in numerical studies. We show that the two schemes are essentially
equivalent through the map that turns out to be an automorphism. An associated
equivalent class of operator-splitting integrations is also defined by
generalizing the specified equivalence.Comment: 4 page
Testing weighted splitting schemes on a one-column transport-chemistry model
In many transport-chemistry models, a huge system of ODE’s of the advection-diffusion-reaction type has to be integrated in time. Typically, this is done with the help of operator splitting. Rosenbrock schemes combined with approximate matrix factorization (ROS-AMF) are an alternative to operator splitting which does not suffer from splitting errors. However, implementation of ROS-AMF schemes often requires serious changes in the code. In this paper we test another classical second order splitting introduced by Strang in 1963, which, unlike the popular Strang splitting, seemed to be forgotten and rediscovered recently (partially due to its intrinsic parallellism). This splitting, called symmetrically weighted sequential (SWS) splitting, is simple and straightforward to apply, independent of the order of the operators and has an operator-level parallelism. In the experiments, the SWS scheme compares favorably to the Strang splitting, but is less accurate than ROS-AMF
OSQP: An Operator Splitting Solver for Quadratic Programs
We present a general-purpose solver for convex quadratic programs based on
the alternating direction method of multipliers, employing a novel operator
splitting technique that requires the solution of a quasi-definite linear
system with the same coefficient matrix at almost every iteration. Our
algorithm is very robust, placing no requirements on the problem data such as
positive definiteness of the objective function or linear independence of the
constraint functions. It can be configured to be division-free once an initial
matrix factorization is carried out, making it suitable for real-time
applications in embedded systems. In addition, our technique is the first
operator splitting method for quadratic programs able to reliably detect primal
and dual infeasible problems from the algorithm iterates. The method also
supports factorization caching and warm starting, making it particularly
efficient when solving parametrized problems arising in finance, control, and
machine learning. Our open-source C implementation OSQP has a small footprint,
is library-free, and has been extensively tested on many problem instances from
a wide variety of application areas. It is typically ten times faster than
competing interior-point methods, and sometimes much more when factorization
caching or warm start is used. OSQP has already shown a large impact with tens
of thousands of users both in academia and in large corporations
Asymptotic wave-splitting in anisotropic linear acoustics
Linear acoustic wave-splitting is an often used tool in describing sound-wave
propagation through earth's subsurface. Earth's subsurface is in general
anisotropic due to the presence of water-filled porous rocks. Due to the
complexity and the implicitness of the wave-splitting solutions in anisotropic
media, wave-splitting in seismic experiments is often modeled as isotropic.
With the present paper, we have derived a simple wave-splitting procedure for
an instantaneously reacting anisotropic media that includes spatial variation
in depth, yielding both a traditional (approximate) and a `true amplitude'
wave-field decomposition. One of the main advantages of the method presented
here is that it gives an explicit asymptotic representation of the linear
acoustic-admittance operator to all orders of smoothness for the smooth,
positive definite anisotropic material parameters considered here. Once the
admittance operator is known we obtain an explicit asymptotic wave-splitting
solution.Comment: 20 page
Application of Operator Splitting Methods in Finance
Financial derivatives pricing aims to find the fair value of a financial
contract on an underlying asset. Here we consider option pricing in the partial
differential equations framework. The contemporary models lead to
one-dimensional or multidimensional parabolic problems of the
convection-diffusion type and generalizations thereof. An overview of various
operator splitting methods is presented for the efficient numerical solution of
these problems.
Splitting schemes of the Alternating Direction Implicit (ADI) type are
discussed for multidimensional problems, e.g. given by stochastic volatility
(SV) models. For jump models Implicit-Explicit (IMEX) methods are considered
which efficiently treat the nonlocal jump operator. For American options an
easy-to-implement operator splitting method is described for the resulting
linear complementarity problems.
Numerical experiments are presented to illustrate the actual stability and
convergence of the splitting schemes. Here European and American put options
are considered under four asset price models: the classical Black-Scholes
model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV
model with jumps
Wave splitting of Maxwell's equations with anisotropic heterogeneous constitutive relations
The equations for the electromagnetic field in an anisotropic media are
written in a form containing only the transverse field components relative to a
half plane boundary. The operator corresponding to this formulation is the
electromagnetic system's matrix. A constructive proof of the existence of
directional wave-field decomposition with respect to the normal of the boundary
is presented.
In the process of defining the wave-field decomposition (wave-splitting), the
resolvent set of the time-Laplace representation of the system's matrix is
analyzed. This set is shown to contain a strip around the imaginary axis. We
construct a splitting matrix as a Dunford-Taylor type integral over the
resolvent of the unbounded operator defined by the electromagnetic system's
matrix. The splitting matrix commutes with the system's matrix and the
decomposition is obtained via a generalized eigenvalue-eigenvector procedure.
The decomposition is expressed in terms of components of the splitting matrix.
The constructive solution to the question on the existence of a decomposition
also generates an impedance mapping solution to an algebraic Riccati operator
equation. This solution is the electromagnetic generalization in an anisotropic
media of a Dirichlet-to-Neumann map.Comment: 45 pages, 2 figure
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