74,709 research outputs found
Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations
We construct quantum algorithms to compute the solution and/or physical
observables of nonlinear ordinary differential equations (ODEs) and nonlinear
Hamilton-Jacobi equations (HJE) via linear representations or exact mappings
between nonlinear ODEs/HJE and linear partial differential equations (the
Liouville equation and the Koopman-von Neumann equation). The connection
between the linear representations and the original nonlinear system is
established through the Dirac delta function or the level set mechanism. We
compare the quantum linear systems algorithms based methods and the quantum
simulation methods arising from different numerical approximations, including
the finite difference discretisations and the Fourier spectral discretisations
for the two different linear representations, with the result showing that the
quantum simulation methods usually give the best performance in time
complexity. We also propose the Schr\"odinger framework to solve the Liouville
equation for the HJE, since it can be recast as the semiclassical limit of the
Wigner transform of the Schr\"odinger equation. Comparsion between the
Schr\"odinger and the Liouville framework will also be made.Comment: quantum algorithms,linear representations,noninea
Parallel algorithm with spectral convergence for nonlinear integro-differential equations
We discuss a numerical algorithm for solving nonlinear integro-differential
equations, and illustrate our findings for the particular case of Volterra type
equations. The algorithm combines a perturbation approach meant to render a
linearized version of the problem and a spectral method where unknown functions
are expanded in terms of Chebyshev polynomials (El-gendi's method). This
approach is shown to be suitable for the calculation of two-point Green
functions required in next to leading order studies of time-dependent quantum
field theory.Comment: 15 pages, 9 figure
XMDS2: Fast, scalable simulation of coupled stochastic partial differential equations
XMDS2 is a cross-platform, GPL-licensed, open source package for numerically
integrating initial value problems that range from a single ordinary
differential equation up to systems of coupled stochastic partial differential
equations. The equations are described in a high-level XML-based script, and
the package generates low-level optionally parallelised C++ code for the
efficient solution of those equations. It combines the advantages of high-level
simulations, namely fast and low-error development, with the speed, portability
and scalability of hand-written code. XMDS2 is a complete redesign of the XMDS
package, and features support for a much wider problem space while also
producing faster code.Comment: 9 pages, 5 figure
Numerical Approximations Using Chebyshev Polynomial Expansions
We present numerical solutions for differential equations by expanding the
unknown function in terms of Chebyshev polynomials and solving a system of
linear equations directly for the values of the function at the extrema (or
zeros) of the Chebyshev polynomial of order N (El-gendi's method). The
solutions are exact at these points, apart from round-off computer errors and
the convergence of other numerical methods used in connection to solving the
linear system of equations. Applications to initial value problems in
time-dependent quantum field theory, and second order boundary value problems
in fluid dynamics are presented.Comment: minor wording changes, some typos have been eliminate
Exact solutions to quantum field theories and integrable equations
The exact solutions to quantum string and gauge field theories are discussed
and their formulation in the framework of integrable systems is presented. In
particular I consider in detail several examples of appearence of solutions to
the first-order integrable equations of hydrodynamical type and stress that all
known examples can be treated as partial solutions to the same problem in the
theory of integrable systems.Comment: revised version, some details and formulations are changed, few
references added; LaTeX, 12 p
Using Spectral Method as an Approximation for Solving Hyperbolic PDEs
We demonstrate an application of the spectral method as a numerical
approximation for solving Hyperbolic PDEs. In this method a finite basis is
used for approximating the solutions. In particular, we demonstrate a set of
such solutions for cases which would be otherwise almost impossible to solve by
the more routine methods such as the Finite Difference Method. Eigenvalue
problems are included in the class of PDEs that are solvable by this method.
Although any complete orthonormal basis can be used, we discuss two
particularly interesting bases: the Fourier basis and the quantum oscillator
eigenfunction basis. We compare and discuss the relative advantages of each of
these two bases.Comment: 19 pages, 14 figures. to appear in Computer Physics Communicatio
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