Optimal Refinement Strategy Research Articles

Enabling four-dimensional conformal hybrid meshing with cubic pyramids

The main purpose of this article is to develop a novel refinement strategy for four-dimensional hybrid meshes based on cubic pyramids. This optimal refinement strategy subdivides a given cubic pyramid into a conforming set of congruent cubic pyramids and invariant bipentatopes. The theoretical properties of the refinement strategy are rigorously analyzed and evaluated. In addition, a new class of fully symmetric quadrature rules with positive weights are generated for the cubic pyramid. These rules are capable of exactly integrating polynomials with degrees up to 12. Their effectiveness is successfully demonstrated on polynomial and transcendental functions. Broadly speaking, the refinement strategy and quadrature rules in this paper open new avenues for four-dimensional hybrid meshing, and space-time finite element methods.

Optimal refinancing strategy for mortgage rate with regime switching

AbstractWe investigate an optimal refinancing problem for an interest‐only mortgage, where the market mortgage rate is defined as the classical Black‐Scholes geometric Brownian motion with regime switching. We verify that the optimal refinancing strategy only depends on the quotient between the market mortgage rate and the borrowing rate. We show that the optimal refinancing strategy is of threshold type. By reducing the two‐dimensional optimal refinancing problem to a one‐dimensional problem, we obtain the system of equations that the function with respect to quotient satisfies. Analytic solutions are obtained in one‐ and two‐regime cases. Finally, we present some numerical results to illustrate the influence of the model parameters on the optimal refinancing strategy.

Recurrent neural networks as optimal mesh refinement strategies

We show that optimal mesh refinement algorithms for a large class of PDEs can be learned by a recurrent neural network with a fixed number of trainable parameters independent of the desired accuracy and the input size, i.e., number of elements of the mesh. This includes problems for which no optimal adaptive strategy is known yet. The proposed algorithm is problem independent in the sense that it only requires the current numerical approximation in order to optimally refine the mesh. Thus, the method is a provably optimal black-box mesh refinement tool for a wide variety of PDE problems.

A penteract partition by means of the optimal subdivision of cells

Freudental's algorithm obtained way back in early forties have been traditionally used for simplicial triangulating of the hypercube. The main advantage of this algorithm is that it only generates one congruence class. Unfortunately, Freudental's algorithm is not optimal with respect to the measure of degeneracy. The multigrid methods require the degeneracy measure to be as small as possible. The minimal subdivision in the 3-dimensional case and the uniform tesseract corner subdivision in the 4-dimensional case are optimal in regards the measure of degeneracy and multigrid applications. The question about the optimal refinement strategy in more dimensional cases is still an open problem. This paper deals with a penteract subdivision with degeneracy measure much better than one obtained by the Freudental algorithm.

Adaptive refinement strategies for the simulation of gas flow in networks using a model hierarchy

A model hierarchy that is based on the one-dimensional isothermal Euler equations of fluid dynamics is used for the simulation and optimisation of natural gas flow through a pipeline network. Adaptive refinement strategies have the aim of bringing the simulation error below a prescribed tolerance while keeping the computational costs low. While spatial and temporal stepsize adaptivity is well studied in the literature, model adaptivity is a new field of research. The problem of finding an optimal refinement strategy that combines these three types of adaptivity is a generalisation of the unbounded knapsack problem. A refinement strategy that is currently used in gas flow simulation software is compared to two novel greedy-like strategies. Both a theoretical experiment and a realistic gas flow simulation show that the novel strategies significantly outperform the current refinement strategy with respect to the computational cost incurred.

Simulation Solution to a Two-Dimensional Mortgage Refinancing Problem

This work studies a mortgage borrower’s optimal refinancing strategy, which is formulated as the solution to a stochastic minimization problem with contingent conditions. The problem is framed in a business economic environment where the underlying discounting factor and mortgage interest rate are assumed to follow a two-dimensional stochastic process of Vasicek type. A complete Monte Carlo algorithm is developed and implemented. This algorithm generates the optimal refinancing surface as a function of time and the risk-free rate. Numerical examples with financial implications are provided.

Numerical Simulation of a Multi-Frequency Resistivity Logging-While-Drilling Tool Using a Highly Accurate and Adaptive Higher-Order Finite Element Method

A novel, highly efficient and accurate adaptive higher-order finite ele- ment method (hp-FEM) is used to simulate a multi-frequency resistivity logging- while-drilling (LWD) tool response in a borehole environment. Presented in this study are the vector expression of Maxwell's equations, three kinds of boundary conditions, stability weak formulation of Maxwell's equations, and automatic hp- adaptivity strategy. The new hp-FEM can select optimal refinement and calculation strategies based on the practical formation model and error estimation. Numeri- cal experiments show that the new hp-FEM has an exponential convergence rate in terms of relative error in a user-prescribed quantity of interest against the degrees of freedom, which provides more accurate results than those obtained using the adaptive h-FEM. The numerical results illustrate the high efficiency and accuracy of the method at a given LWD tool structure and parameters in different physical models, which further confirm the accuracy of the results using the Hermes library (http://hpfem.org/hermes) with a multi-frequency resistivity LWD tool response in a borehole environment.

An isogeometric analysis for elliptic homogenization problems

A novel and efficient approach which is based on the framework of isogeometric analysis for elliptic homogenization problems is proposed. These problems possess highly oscillating coefficients leading to extremely high computational expenses while using traditional finite element methods. The isogeometric analysis heterogeneous multiscale method (IGA-HMM) investigated in this paper is regarded as an alternative approach to the standard finite element heterogeneous multiscale method (FE-HMM) which is currently an effective framework to solve these problems. The method utilizes non-uniform rationalB-splines (NURBS) in both macro and micro levels instead of standard Lagrange basis. Besides the ability to describe exactly the geometry, it tremendously facilitates high-order macroscopic/microscopic discretizations thanks to the flexibility of refinement and degree elevation with an arbitrary continuity level provided by NURBS basis functions. Furthermore, the nearly optimal quadrature rule for IGA (Auricchio et al., 2012) introduced recently is utilized to reduce significantly the number of micro problems, which is the main factor contributing into the computational cost in heterogeneous multiscale method. A priori error estimates of the discretization error coming from macro and micro meshes and optimal micro refinement strategies for macro/micro NURBS basis functions of arbitrary orders are derived. An efficient coupling between degrees of macro and micro basis functions is introduced. Numerical results show the excellent performance of the proposed method.

Numerical Simulation of a Multi-Frequency Resistivity Logging-While-Drilling Tool Using a Highly Accurate and Adaptive Higher-Order Finite Element Method

AbstractA novel, highly efficient and accurate adaptive higher-order finite element method (hp-FEM) is used to simulate a multi-frequency resistivity logging-while-drilling (LWD) tool response in a borehole environment. Presented in this study are the vector expression of Maxwell’s equations, three kinds of boundary conditions, stability weak formulation of Maxwell’s equations, and automatichp-adaptivity strategy. The newhp-FEM can select optimal refinement and calculation strategies based on the practical formation model and error estimation. Numerical experiments show that the newhp-FEM has an exponential convergence rate in terms of relative error in a user-prescribed quantity of interest against the degrees of freedom, which provides more accurate results than those obtained using the adaptiveh-FEM. The numerical results illustrate the high efficiency and accuracy of the method at a given LWD tool structure and parameters in different physical models, which further confirm the accuracy of the results using the Hermes library (http://hpfem.org/hermes) with a multi-frequency resistivity LWD tool response in a borehole environment.

Adaptive <i>hp</i>-Finite Element Method for Electromagnetic Field Logging Problems

A novel, highly efficient and accurate adaptive higher-order finite element method (hp-FEM) is proposed for electromagnetic field problems. Presented in this paper are the vector expression of Maxwell's equations, three kinds of boundary conditions, stability weak formulation of Maxwell's equations, and automatic hp-adaptivity strategy. This method can select optimal refinement and calculation strategies based on the practical formation model and error estimation. Numerical experiments show that the new hp-FEM has an exponential convergence rate in terms of relative error in a user-prescribed quantity of interest against the degrees of freedom, which provides more accurate results than those obtained using the adaptive h-FEM. The methodology is freely available online in the form of a general public licensed C++ library Hermes (http://hpfem.org/hermes).

Error analysis of large-eddy simulation of the turbulent non-premixed sydney bluff-body flame

A computational error analysis is applied to the large-eddy simulation of the turbulent non-premixed Sydney bluff-body flame, where the error is defined with respect to experimental data. The error-landscape approach is extended to heterogeneous compressible turbulence, which is coupled to combustion as described by a flamelet model. The Smagorinsky model formulation is used to model the unknown turbulent stresses. We introduce several measures to quantify the total simulation error and observe a striking ‘valley-structure’ in the error that arises as function of the spatial resolution and the Smagorinsky length parameter. The optimal refinement strategy that can be extracted from this error-landscape is reminiscent of that for non-reacting turbulent flow.

Analysis of errors occurring in large eddy simulation

We analyse the effect of second- and fourth-order accurate central finite-volume discretizations on the outcome of large eddy simulations of homogeneous, isotropic, decaying turbulence at an initial Taylor-Reynolds number Re(lambda)=100. We determine the implicit filter that is induced by the spatial discretization and show that a higher order discretization also induces a higher order filter, i.e. a low-pass filter that keeps a wider range of flow scales virtually unchanged. The effectiveness of the implicit filtering is correlated with the optimal refinement strategy as observed in an error-landscape analysis based on Smagorinsky's subfilter model. As a point of reference, a finite-volume method that is second-order accurate for both the convective and the viscous fluxes in the Navier-Stokes equations is used. We observe that changing to a fourth-order accurate convective discretization leads to a higher value of the Smagorinsky coefficient C(S) required to achieve minimal total error at given resolution. Conversely, changing only the viscous flux discretization to fourth-order accuracy implies that optimal simulation results are obtained at lower values of C(S). Finally, a fully fourth-order discretization yields an optimal C(S) that is slightly lower than the reference fully second-order method.

A fully automatic hp-adaptivity for elliptic PDEs in three dimensions

We present an algorithm that determines an optimal refinement strategy for a given finite element mesh. The algorithm automatically selects optimal (possibly anisotropic) h-refinements and distribution of (also anisotropic) order of approximation p. The local selection between h-refinement and p-enrichment is made to decrease the projection-based interpolation error of a reference (fine grid) solution with maximum rate with respect to the number of local degrees of freedom (dof) added. A reconciliation step maintains the 1-irregularity of the mesh. When applied iteratively, the algorithm delivers exponential convergence in the energy norm for model elliptic problems with singular or nearly singular solutions.

Interacting errors in large-eddy simulation: a review of recent developments

The accuracy of large-eddy simulations is limited, among others, by the quality of the subgrid parameterisation and the numerical contamination of the smaller retained flow structures. We review the effects of discretisation and modelling errors from two different perspectives. We first show that spatial discretisation induces its own filter and compare the dynamic importance of this numerical filter to the basic large-eddy filter. The spatial discretisation modifies the large-eddy closure problem as is expressed by the difference between the discrete ‘numerical stress tensor’ and the continuous ‘turbulent stress tensor’. This difference consists of a high-pass contribution associated with the specific numerical filter. Several central differencing methods are analysed and the importance of the subgrid resolution is established. Second, we review a database approach to assess the total simulation error and its numerical and modelling contributions. The interaction between the different sources of error is shown to lead to their partial cancellation. From this analysis one may identify an ‘optimal refinement strategy’ for a given subgrid model, discretisation method and flow conditions, leading to minimal total simulation error at a given computational cost. We provide full detail for homogeneous decaying turbulence in a ‘Smagorinsky fluid’. The optimal refinement strategy is compared with the error reduction that arises from grid refinement of the dynamic eddy-viscosity model. The main trends of the optimal refinement strategy as a function of resolution and Reynolds number are found to be adequately followed by the dynamic model. This yields significant error reduction upon grid refinement although at coarse resolutions significant error levels remain. To address this deficiency, a new successive inverse polynomial interpolation procedure is proposed with which the optimal Smagorinsky constant may be efficiently approximated at a given resolution. The computational overhead of this optimisation procedure is shown to be well justified in view of the achieved reduction of the error level relative to the ‘no-model’ and dynamic model predictions.

Borrower Credit and the Valuation of Mortgage-Backed Securities

We study the valuation of mortgage-backed securities when borrowers may have to refinance at premium rates because of their credit. The optimal refinancing strategy often results in prepayment being delayed significantly relative to traditional models. Furthermore, mortgage values can exceed par by much more than the cost of refinancing. Applying the model to an extensive sample of mortgage-backed security prices, we find that the implied credit spreads that match these prices closely parallel borrowers' actual spreads at the origination of the mortgage. These results suggest that models that incorporate borrower credit into the analysis may provide a promising alternative to the reduced-form prepayment models widely used in practice.

A Posteriori Finite-Element Output Bounds for the Incompressible Navier–Stokes Equations: Application to a Natural Convection Problem

We present a new Neumann subproblem a posteriori finite-element procedure for the efficient calculation of rigorous, constant-free, sharp lower and upper bounds for linear and nonlinear functional outputs of the incompressible Navier–Stokes equations. We first formulate the bound procedure; we derive and discuss a bound error expression; and we then demonstrate the capabilities of the method with numerical results obtained for natural convection problems. We also implement an optimal adaptive refinement strategy based on a local elemental decomposition of the bound gap.

Declining Mortgage Prepayment Penalties and the Value of Delay

Mortgages are complicated instruments, and the pace of financial innovation complicates them further. Unlike standard single family mortgages, the terms on commercial loans are heterogeneous, and the motivations of property owners are not well understood. Single family lenders are increasingly adopting the prepayment penalties found in the commercial market. This paper fills in a gap in the study of mortgage finance by determining the optimal refinancing strategy under common forms of prepayment penalties. Our paper uses a bivariate binomial options pricing technique to value call options within various mortgages with different types of prepayment penalties. Using simulations, we calculate the spread between contract and prevailing interest rates needed to trigger an optimal refinance, and find that it is often substantially larger for mortgages with declining penalties than for mortgages with static penalties. The larger required spreads caused by the enhanced value of delay explain much of the slow prepayment behavior discovered in the multifamily market by Follain, Ondrich, and Sinha (1997). Therefore, borrower decisions which appear slow (or irrational) in a static setting may be quite rational in a dynamic setting. Our results have important implications for both the interpretation of regression results based on historical data, and for the pricing of newly originated mortgages. Declining prepayment penalties were used extensively for multifamily mortgages in the 1970's and 1980's. While their use has declined in the 1990's in favor of yield maintainance agreements, empirical studies which use 1970's and 1980's data need to incorporate the effects of these penalty structures or risk specification bias. At the same time that their use is declining in the multifamily market it is growing in the single-family market. Understanding the behavior of these mortgages will be crucial to their pricing.

Progressive refinement using local variance estimators

When transmitting an image over a slow communications link, it is desired to provide the best display possible in early stages of the transmission. Progressive refinement attempts to continuously minimize the deviation between the image and its displayed approximation. It is shown that this minimization is accomplished by transmitting average values of regions of the image at increasingly fine levels of resolution. The optimal refinement strategy is to subdivide the region which maximizes the product of area and pixel variance. Typically, the receiving process cannot determine the pixel variance. Estimators of the variance can be computed from information already received are developed. They are applied to refinement over the spatial distribution of values and simultaneously to refinement over their gray-scale quantization. The estimators are shown to apply equally well to monochrome and full-color image. These techniques are implemented with no transmission overhead and are shown to generate qualitatively and quantitatively superior displays. >

On an Adaptive Grid Refining Technique for Finite Element Approximations

We consider a family of finite element spaces and minimize an energy functional over each space. The space which allows the lowest energy is considered “optimal.” Such a family is constructed by starting with an initial “triangulation” and refining one or more “triangles” at a time. We estimate the profit in energy gained by refining a triangle and set up a discrete optimization problem which determines the optimal refinement strategy according to a prescribed bound of the costs. This enables us to construct the final grid by using as few as possible intermediate grids. Instead of solving the original optimization problem we set up a partially dualized form of it which produces a nearly optimal solution and can be solved very efficiently.

Propagation of phase errors during phase refinement and extension

A formalism is developed for estimating phase errors during refinement using non-crystallographic symmetry, solvent flattening or density modification. This formalism, based on a separation of all structure factors into known (best estimate) and unknown (random variable) parts, leads directly to simple expressions for the propagation of phase errors during a refinement process. Phase extension and treatment of unmeasured reflections are readily incorporated into this scheme. The formulation provides a direct method for evaluating the success of a refinement process. This may be useful in cases where examination of the resulting electron density map does not provide a quantitative evaluation of the calculations, such as at low to moderate resolution or when phase extension has been used. It may also provide a basis for designing optimal refinement strategies.