51,910 research outputs found
A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces
The closest point method (Ruuth and Merriman, J. Comput. Phys.
227(3):1943-1961, [2008]) is an embedding method developed to solve a variety
of partial differential equations (PDEs) on smooth surfaces, using a closest
point representation of the surface and standard Cartesian grid methods in the
embedding space. Recently, a closest point method with explicit time-stepping
was proposed that uses finite differences derived from radial basis functions
(RBF-FD). Here, we propose a least-squares implicit formulation of the closest
point method to impose the constant-along-normal extension of the solution on
the surface into the embedding space. Our proposed method is particularly
flexible with respect to the choice of the computational grid in the embedding
space. In particular, we may compute over a computational tube that contains
problematic nodes. This fact enables us to combine the proposed method with the
grid based particle method (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024,
[2009]) to obtain a numerical method for approximating PDEs on moving surfaces.
We present a number of examples to illustrate the numerical convergence
properties of our proposed method. Experiments for advection-diffusion
equations and Cahn-Hilliard equations that are strongly coupled to the velocity
of the surface are also presented
Efficient option pricing with transaction costs
A fast numerical algorithm is developed to price European options with proportional transaction costs using the utility-maximization framework of Davis (1997). This approach allows option prices to be computed by solving the investor’s basic portfolio selection problem without insertion of the option payoff into the terminal value function. The properties of the value function can then be used to drastically reduce the number of operations needed to locate the boundaries of the no-transaction region, which leads to very efficient option valuation. The optimization problem is solved numerically for the case of exponential utility, and comparisons with approximately replicating strategies reveal tight bounds for option prices even as transaction costs become large. The computational technique involves a discrete-time Markov chain approximation to a continuous-time singular stochastic optimal control problem. A general definition of an option hedging strategy in this framework is developed. This involves calculating the perturbation to the optimal portfolio strategy when an option trade is executed
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
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