Adaptive Mesh Model Research Articles

Oil Spill Visualization Based on the Numeric Simulation of Tidal Current

This paper proposes an approach to implement the 3D visualization of oil spill based on tidal hydrodynamic model. It simulates tidal current of M2 component tide in Jiaozhou Bay. The simulation results conform to the tidal theory and probably conform to the flow measurement report of crude oil pier Phase III at Qingdao Harbor. Based on tidal current and eye-point related adaptive ocean surface mesh model, by analyzing the drift and diffusion mathematical models of oil spill on the sea, the dynamic visualization of drift and diffusion course of oil on the sea were implemented, the visualization result is satisfactory.

A POD reduced‐order 4D‐Var adaptive mesh ocean modelling approach

AbstractA novel proper orthogonal decomposition (POD) inverse model, developed for an adaptive mesh ocean model (the Imperial College Ocean Model, ICOM), is presented here. The new POD model is validated using the Munk gyre flow test case, where it inverts for initial conditions. The optimized velocity fields exhibit overall good agreement with those generated by the full model. The correlation between the inverted and the true velocity is 80–98% over the majority of the domain. Error estimation was used to judge the quality of reduced‐order adaptive mesh models. The cost function is reduced by 20% of its original value, and further by 70% after the POD bases are updated.In this study, the reduced adjoint model is derived directly from the discretized reduced forward model. The whole optimization procedure is undertaken completely in reduced space. The computational cost for the four‐dimensional variational (4D‐Var) data assimilation is significantly reduced (here a decrease of 70% in the test case) by decreasing the dimensional size of the control space, in both the forward and adjoint models. Computational efficiency is further enhanced since both the reduced forward and adjoint models are constructed by a series of time‐independent sub‐matrices. The reduced forward and adjoint models can be run repeatedly with negligible computational costs.An adaptive POD 4D‐Var is employed to update the POD bases as minimization advances and loses control, thus adaptive updating of the POD bases is necessary. Previously developed numerical approaches Fang et al. (Int. J. Numer. Meth. Fluids 2008) are employed to accurately represent the geostrophic balance and improve the efficiency of the POD simulation. Copyright © 2008 John Wiley & Sons, Ltd.

Adaptive placement method on pricing arithmetic average options

Since there is no analytic solution for arithmetic average options until present, developing an efficient numerical algorithm becomes a promising alternative. One of the most famous numerical algorithms is introduced by Hull and White (J Deriv 1:21–31, 1993). Motivated by the common idea of reducing the nonlinearity error in the adaptive mesh model in Figlewski and Gao (J Financ Econ 53:313–351, 1999) and the adaptive quadrature method, we propose an adaptive placement method to replace the logarithmically equally-spaced placement rule in the Hull and White’s model by placing more representative average prices in the highly nonlinear area of the option value as the function of the arithmetic average stock price. The basic idea of this method is to design a recursive algorithm to limit the error of the linear interpolation between each pair of adjacent representative average prices. Numerical experiments verify the superior performance of this method for reducing the interpolation error and hence improving the convergence rate. To show that the adaptive placement method can improve any numerical algorithm with the techniques of augmented state variables and the piece-wise linear interpolation approximation, we also demonstrate how to integrate the adaptive placement method into the GARCH option pricing algorithm in Ritchken and Trevor (J Finance 54:377–402, 1999). Similarly great improvement of the convergence rate suggests the potential applications of this novel method to a broad class of numerical pricing algorithms for exotic options and complex underlying processes.

Conditional entropy maximization for PET image reconstruction using adaptive mesh model

Iterative image reconstruction algorithms have been widely used in the field of positron emission tomography (PET). However, such algorithms are sensitive to noise artifacts so that the reconstruction begins to degrade when the number of iterations is high. In this paper, we propose a new algorithm to reconstruct an image from the PET emission projection data by using the conditional entropy maximization and the adaptive mesh model. In a traditional tomography reconstruction method, the reconstructed image is directly computed in the pixel domain. Unlike this kind of methods, the proposed approach is performed by estimating the nodal values from the observed projection data in a mesh domain. In our method, the initial Delaunay triangulation mesh is generated from a set of randomly selected pixel points, and it is then modified according to the pixel intensity value of the estimated image at each iteration step in which the conditional entropy maximization is used. The advantage of using the adaptive mesh model for image reconstruction is that it provides a natural spatially adaptive smoothness mechanism. In experiments using the synthetic and clinical data, it is found that the proposed algorithm is more robust to noise compared to the common pixel-based MLEM algorithm and mesh-based MLEM with a fixed mesh structure.

An adaptive mesh model for 3D reconstruction from unorganized data points

Reconstructing exact topology mesh from data points is one of the most important tasks in the fields of industrial CAD/CAE/CAM, computational vision and reverse engineering. In this paper, a deflation algorithm that integrates an adaptive mesh and physical constraint model is presented for the 3D reconstruction of geometric-closed shape (genus 0) from unorganized data points. First, an initial mesh is formed using the Delaunay algorithm. Second, an asymptotic deforming performance is accomplished to deflate initial mesh towards the local concave boundary step-by-step. In this phase, a physical constraint model of coupled particle systems based on particle dynamics and Newtonian law of motion is constructed, and the model dynamically controls mesh deformation as a behavior constraint. To guarantee that the resultant mesh is homeomorphous to the original surface of data points, a continuously deforming mechanism, visibility cone and collision-detecting criterions are designed. At last, experimental results in reverse engineering which supports the usefulness of this method for reconstruction .

Adaptively merging large-scale range data with reflectance properties

In this paper, we tackle the problem of geometric and photometric modeling of large intricately shaped objects. Typical target objects we consider are cultural heritage objects. When constructing models of such objects, we are faced with several important issues that have not been addressed in the past-issues that mainly arise due to the large amount of data that has to be handled. We propose two novel approaches to efficiently handle such large amounts of data: A highly adaptive algorithm for merging range images and an adaptive nearest-neighbor search to be used with the algorithm. We construct an integrated mesh model of the target object in adaptive resolution, taking into account the geometric and/or photometric attributes associated with the range images. We use surface curvature for the geometric attributes and (laser) reflectance values for the photometric attributes. This adaptive merging framework leads to a significant reduction in the necessary amount of computational resources. Furthermore, the resulting adaptive mesh models can be of great use for applications such as texture mapping, as we will briefly demonstrate. Additionally, we propose an additional test for the k-d tree nearest-neighbor search algorithm. Our approach successfully omits back-tracking, which is controlled adaptively depending on the distance to the nearest neighbor. Since the main consumption of computational cost lies in the nearest-neighbor search, the proposed algorithm leads to a significant speed-up of the whole merging process. In this paper, we present the theories and algorithms of our approaches with pseudo code and apply them to several real objects, including large-scale cultural assets.

The adaptive mesh model: a new approach to efficient option pricing

Most derivative securities must be priced by numerical techniques. These models contain “distribution error” and “nonlinearity error”. The Adaptive Mesh Model (AMM) sharply reduces nonlinearity error by grafting one or more small sections of fine high-resolution lattice onto a tree with coarser time and price steps. Three different AMM structures are presented, one for pricing ordinary options, one for barrier options, and one for computing delta and gamma efficiently. The AMM approach can be adapted to a wide variety of contingent claims. For some common problems, accuracy increases by several orders of magnitude with no increase in execution time.

Pricing Discrete Barrier Options with an Adaptive Mesh Model

The binomial option pricing model and the trinomial model, its more versatile relative, are invaluable tools for pricing complex derivatives, especially those with American exercise. But while these models converge to the correct option values as the time and price step sizes go to zero, for certain kinds of problems getting close enough may require a very large amount of calculation. The cause is often non-linearity or discontinuity in the option payoff that occurs only in a small region. One example is a barrier option with a barrier that is only monitored at discrete intervals. This article describes how to solve the discrete barrier problem with an adaptive mesh model, a general approach to lattice building that constructs smell sections of fine high-resolution mesh in the critical areas and then grafts them onto a base lattice with coarser time and price steps elsewhere. The technique achieves an increase in accuracy that is quite remarkable.

Pricing Discrete Barrier Options with an Adaptive Mesh Model

Many exotic derivatives do not have closed-form valuation equations, and must be priced using approximation methods. Where they can be applied, standard lattice techniques based on binomial and trinomial trees will achieve correct valuations asymptotically. They can also generally handle American exercise. But for many problems, including pricing barrier options, convergence may be slow and erratic, producing large errors even with thousands of time steps and millions of node calculations. Options with price barriers that are only monitored at discrete points in time present additional difficulty for lattice models. Standard tree methods increase accuracy by shrinking the time and price step size throughout the lattice, but this increases the number of calculations sharply and much of the additional computation is in regions of the tree where it makes little difference to accuracy. A previous paper, Figlewski and Gao [1999], introduced the Adaptive Mesh Model (AMM), a very flexible approach that greatly increases efficiency in trinomial lattices. Coarse time and price steps are used in most of the tree, but small sections of finer mesh are constructed to improve resolution in specific critical areas. This paper presents an especially effective AMM structure for pricing options with discrete barriers. In a basic example, an AMM with 60 time steps is ten times more accurate than a 5000-step trinomial, but runs more than 1000 times faster.