About grid generation in constructions bounded by the surfaces of revolution

For constructions bounded by the surfaces of revolution, structured grid generation technique is presented. Its technology has been elaborated within the variational approach for constructing optimal grids satisfying optimality criteria: closeness of grids to uniform ones, closeness of grids to orthogonal ones and adaptation to a given function. Grid generation has been designed for numerical solution of the differential equations modeling the vortex processes of multi-component hydrodynamics. In the technology, the three-dimensional construction in which it is required to construct a grid is represented in the form of the curvilinear hexahedron defining its configuration. The specific feature of the required configurations is that some faces of a curvilinear hexahedron lie in one plane and along edges of adjoining faces grid cells degenerate into prisms. Grid generation in the considered constructions has started to be developed by the elaboration of algorithms for the volume of revolution which has become the basic construction. The basic construction is obtained by the rotation through 180? around the axis of a generatrix consisting of straight line segments, arcs of circles and ellipses. Then the deformed volumes of revolutions are considered along with the generalizations of the volume of revolution which represent constructions obtained by the surfaces of revolution with parallel axis of rotation. The aim of the further development of the technology is to consider more and more complicated constructions and elaborate the technology for them. In the presentation, the current state of the development of the technology is given. Examples of generated grids are supplied.

A backward Monte Carlo approach to exotic option pricing

We propose a novel algorithm which allows to sample paths from an underlying price process in a local volatility model and to achieve a substantial variance reduction when pricing exotic options. The new algorithm relies on the construction of a discrete multinomial tree. The crucial feature of our approach is that – in a similar spirit to the Brownian Bridge – each random path runs backward from a terminal fixed point to the initial spot price. We characterize the tree in two alternative ways: (i) in terms of the optimal grids originating from the Recursive Marginal Quantization algorithm, (ii) following an approach inspired by the finite difference approximation of the diffusion's infinitesimal generator. We assess the reliability of the new methodology comparing the performance of both approaches and benchmarking them with competitor Monte Carlo methods.

Frequency-domain finite-element simulations of 2D sonic wireline borehole measurements acquired in fractured and thinly bedded formations

We simulated wireline borehole sonic waveforms to appraise modal frequency dispersions across fractured and thinly bedded formations. Simulations included monopole and dipole sources of excitation and explicitly took into account the borehole, the mandrel tool, and casing whenever present. Calculations were performed in the frequency domain with a highly accurate finite-element method that automatically generates optimal grids for each problem/frequency combination. The method guarantees solutions at significantly reduced computational time with relative energy errors below 0.5% (even in the presence of singularities in the solution that can originate from complex geometries, high material contrasts, and simultaneous presence of large and fine structures). Such a high accuracy in the simulation of sonic waveforms is necessary to accurately quantify the effects of fractures and thin beds on acoustic logs. Simulations indicate that fractures mainly influence propagation modes related to the formation shear velocity. In slow thinly bedded formations, effective properties are similar to those of average layer properties, as predicted by the Reuss lower bound. However, presence of intralayer interfaces gives rise to multiple reflections that can deleteriously affect the estimation of elastic properties with dispersion processing. Casing effectively functions as a low-pass frequency filter of sonic waveforms, significantly distorting the original borehole modes, hence the estimation of elastic properties.

Reverse flood routing with the inverted Muskingum storage routing scheme

This work treats reverse flood routing aiming at signal identification: inflows are inferred from observed outflows by orienting the Muskingum scheme against the wave propagation direction. Routing against the wave propagation is an ill-posed, inverse problem (small errors amplify, leading to large spurious responses); therefore, the reverse solution must be smoothness-constrained towards stability and uniqueness (regularised). Theoretical constrains on the coefficients of the reverse routing scheme assist in error control, but optimal grids are derived by numerical experimentation. Exact solutions of the convection-diffusion equation, for a single and a composite wave, are reverse-routed and in both instances the wave is backtracked well for a range of grid parameters. In the arduous test of a square pulse, the result is comparable to those of more complex methods. Seeding outflow data with random errors enhances instability; to cope with the spurious oscillations, the reversed solution is conditioned by smoothing via low-pass filtering or optimisation. Good-quality inflow hydrographs are recovered with either smoothing treatment, yet the computationally demanding optimisation is superior. Finally, the reverse Muskingum routing method is compared to a reverse-solution method of the St. Venant equations of flood wave motion and is found to perform equally well, at a fraction of the computing effort. This study leads us to conclude that the efficiently attained good inflow identification rests on the simplicity of the Muskingum reverse routing scheme that endows it with numerical robustness.

AN OPTIMAL MARKOVIAN QUANTIZATION ALGORITHM FOR MULTI-DIMENSIONAL STOCHASTIC CONTROL PROBLEMS

We propose a probabilistic numerical method based on optimal quantization to solve some multi-dimensional stochastic control problems that arise, for example, in mathematical finance for portfolio optimization. We then consider some controlled diffusions with most components control free. The Euler scheme of the uncontrolled diffusion part is approximated by a discrete time process obtained by a nearest neighbor projection on some grids optimally fitted to its dynamics. The resulting process is also designed to preserve the Markov property with respect to the filtration of the Euler scheme. This Markovian quantization approach leads to an approximate control problem that can be solved numerically by the dynamic programming formula. This approach seems promising in higher dimension. A prioriLp-error bounds are stated and we show that the spatial discretization error term is minimal at some specific grids. A simple recursive algorithm is devised to compute these optimal grids by induction based on a Monte Carlo simulation. Some numerical illustrations are processed for solving a mean-variance hedging problem.