191,040 research outputs found
The Differential Scheme and Quantum Computation
It is well-known that standard models of computation are representable as simple dynamical systems that evolve in discrete time, and that systems that evolve in continuous time are often representable by dynamical systems governed by ordinary differential equations. In many applications, e.g., molecular networks and hybrid Fermi-Pasta-Ulam systems, one must work with dynamical systems comprising both discrete and continuous components.
Reasoning about and verifying the properties of the evolving state of such systems is currently a piecemeal affair that depends on the nature of major components of a system: e.g., discrete vs. continuous components of state, discrete vs. continuous time, local vs. distributed clocks, classical vs. quantum states and state evolution.
We present the Differential Scheme as a unifying framework for reasoning about and verifying the properties of the evolving state of a system, whether the system in question evolves in discrete time, as for standard models of computation, or continuous time, or a combination of both. We show how instances of the differential scheme can accommodate classical computation.
We also generalize a relatively new model of quantum computation, the quantum cellular automaton, with an eye towards extending the differential scheme to accommodate quantum computation and hybrid classical/quantum computation.
All the components of a specific instance of the differential scheme are Convergence Spaces. Convergence spaces generalize notions of continuity and convergence. The category of convergence spaces, Conv, subsumes both simple discrete structures (e.g., digraphs), and complex continuous structures (e.g., topological spaces, domains, and the standard fields of analysis: R and C). We present novel uses for convergence spaces, and extend their theory by defining differential calculi on Conv. It is to the use of convergence spaces that the differential scheme owes its generality and flexibility
Towards higher order lattice Boltzmann schemes
In this contribution we extend the Taylor expansion method proposed
previously by one of us and establish equivalent partial differential equations
of DDH lattice Boltzmann scheme at an arbitrary order of accuracy. We derive
formally the associated dynamical equations for classical thermal and linear
fluid models in one to three space dimensions. We use this approach to adjust
relaxation parameters in order to enforce fourth order accuracy for thermal
model and diffusive relaxation modes of the Stokes problem. We apply the
resulting scheme for numerical computation of associated eigenmodes and compare
our results with analytical references
Fast and Efficient Numerical Methods for an Extended Black-Scholes Model
An efficient linear solver plays an important role while solving partial
differential equations (PDEs) and partial integro-differential equations
(PIDEs) type mathematical models. In most cases, the efficiency depends on the
stability and accuracy of the numerical scheme considered. In this article we
consider a PIDE that arises in option pricing theory (financial problems) as
well as in various scientific modeling and deal with two different topics. In
the first part of the article, we study several iterative techniques
(preconditioned) for the PIDE model. A wavelet basis and a Fourier sine basis
have been used to design various preconditioners to improve the convergence
criteria of iterative solvers. We implement a multigrid (MG) iterative method.
In fact, we approximate the problem using a finite difference scheme, then
implement a few preconditioned Krylov subspace methods as well as a MG method
to speed up the computation. Then, in the second part in this study, we analyze
the stability and the accuracy of two different one step schemes to approximate
the model.Comment: 29 pages; 10 figure
Development and application of the GIM code for the Cyber 203 computer
The GIM computer code for fluid dynamics research was developed. Enhancement of the computer code, implicit algorithm development, turbulence model implementation, chemistry model development, interactive input module coding and wing/body flowfield computation are described. The GIM quasi-parabolic code development was completed, and the code used to compute a number of example cases. Turbulence models, algebraic and differential equations, were added to the basic viscous code. An equilibrium reacting chemistry model and implicit finite difference scheme were also added. Development was completed on the interactive module for generating the input data for GIM. Solutions for inviscid hypersonic flow over a wing/body configuration are also presented
Pricing exotic options using strong convergence properties?
In finance, the strong convergence properties of discretisations of stochastic differential equations (SDEs) are very important for the hedging and valuation of exotic options. In this paper we show how the use of the Milstein scheme can improve the convergence of the multi-level Monte Carlo method, so that the computational cost to achieve an accuracy of O(e) is reduced to O() for a Lipschitz payoff. The Milstein scheme gives first order strong convergence for all 1âdimensional systems (one Wiener process). However, for processes with two or more Wiener processes, such as correlated portfolios and stochastic volatility models, there is no exact solution for the iterated integrals of second order (LĂ©vy area) and the Milstein scheme neglecting the LĂ©vy area gives the same order of convergence as the Euler-Maruyama scheme. The purpose of this paper is to show that if certain conditions are satisfied, we can avoid the calculation of the LĂ©vy area and obtain first convergence order by applying an orthogonal transformation. We demonstrate when the conditions of the 2âDimensional problem permit this and give an exact solution for the orthogonal transformation. We present examples of pricing exotic options to demonstrate that the use of both the orthogonal Milstein scheme and the Multi-level Monte Carlo give a substantial reduction in the computation cost
Quantum kinetic perturbation theory for near-integrable spin chains with weak long-range interactions
For a transverse-field Ising chain with weak long-range interactions we
develop a perturbative scheme, based on quantum kinetic equations, around the
integrable nearest-neighbour model. We introduce, discuss, and benchmark
several truncations of the time evolution equations up to eighth order in the
Jordan-Wigner fermionic operators. The resulting set of differential equations
can be solved for lattices with sites and facilitates the computation
of spin expectation values and correlation functions to high accuracy, at least
for moderate timescales. We use this scheme to study the relaxation dynamics of
the model, involving prethermalisation and thermalisation. The techniques
developed here can be generalised to other spin models with weak
integrability-breaking terms.Comment: 31 pages, 6 figure
A Numerical Scheme for Invariant Distributions of Constrained Diffusions
Reflected diffusions in polyhedral domains are commonly used as approximate
models for stochastic processing networks in heavy traffic. Stationary
distributions of such models give useful information on the steady state
performance of the corresponding stochastic networks and thus it is important
to develop reliable and efficient algorithms for numerical computation of such
distributions. In this work we propose and analyze a Monte-Carlo scheme based
on an Euler type discretization of the reflected stochastic differential
equation using a single sequence of time discretization steps which decrease to
zero as time approaches infinity. Appropriately weighted empirical measures
constructed from the simulated discretized reflected diffusion are proposed as
approximations for the invariant probability measure of the true diffusion
model. Almost sure consistency results are established that in particular show
that weighted averages of polynomially growing continuous functionals evaluated
on the discretized simulated system converge a.s. to the corresponding
integrals with respect to the invariant measure. Proofs rely on constructing
suitable Lyapunov functions for tightness and uniform integrability and
characterizing almost sure limit points through an extension of Echeverria's
criteria for reflected diffusions. Regularity properties of the underlying
Skorohod problems play a key role in the proofs. Rates of convergence for
suitable families of test functions are also obtained. A key advantage of
Monte-Carlo methods is the ease of implementation, particularly for high
dimensional problems. A numerical example of a eight dimensional Skorohod
problem is presented to illustrate the applicability of the approach
General tooth boundary conditions for equation free modelling
We are developing a framework for multiscale computation which enables models
at a ``microscopic'' level of description, for example Lattice Boltzmann, Monte
Carlo or Molecular Dynamics simulators, to perform modelling tasks at
``macroscopic'' length scales of interest. The plan is to use the microscopic
rules restricted to small "patches" of the domain, the "teeth'', using
interpolation to bridge the "gaps". Here we explore general boundary conditions
coupling the widely separated ``teeth'' of the microscopic simulation that
achieve high order accuracy over the macroscale. We present the simplest case
when the microscopic simulator is the quintessential example of a partial
differential equation. We argue that classic high-order interpolation of the
macroscopic field provides the correct forcing in whatever boundary condition
is required by the microsimulator. Such interpolation leads to Tooth Boundary
Conditions which achieve arbitrarily high-order consistency. The high-order
consistency is demonstrated on a class of linear partial differential equations
in two ways: firstly through the eigenvalues of the scheme for selected
numerical problems; and secondly using the dynamical systems approach of
holistic discretisation on a general class of linear \textsc{pde}s. Analytic
modelling shows that, for a wide class of microscopic systems, the subgrid
fields and the effective macroscopic model are largely independent of the tooth
size and the particular tooth boundary conditions. When applied to patches of
microscopic simulations these tooth boundary conditions promise efficient
macroscale simulation. We expect the same approach will also accurately couple
patch simulations in higher spatial dimensions.Comment: 22 page
- âŠ