Search

In this paper we enhance the well-known fifth order WENO shock-capturing scheme by using deep learning techniques. This fine-tuning of an existing algorithm is implemented by training a rather small neural network to modify the smoothness indicators of the WENO scheme in order to improve the numerical results especially at discontinuities. In our approach no further post-processing is needed to ensure the consistency of the method, which simplifies the method and increases the effect of the neural network. Moreover, the convergence of the resulting scheme can be theoretically proven. We demonstrate our findings with the inviscid Burgers' equation, the Buckley-Leverett equation and the 1-D Euler equations of gas dynamics. Hereby we investigate the classical Sod problem and the Lax problem and show that our novel method outperforms the classical fifth order WENO schemes in simulations where the numerical solution is too diffusive or tends to overshoot at shocks., Comment: 31 pages 12 figures, 6 tables

In this paper we introduce high dimensional tensor product interpolation for the combination technique. In order to compute the sparse grid solution, the discrete numerical subsolutions have to be extended by interpolation. If unsuitable interpolation techniques are used, the rate of convergence is deteriorated. We derive the necessary framework to preserve the error structure of high order finite difference solutions of elliptic partial differential equations within the combination technique framework. This strategy enables us to obtain high order sparse grid solutions on the full grid. As exemplifications for the case of order four we illustrate our theoretical results by two test examples with up to four dimensions.

The concept of the iterative parabolic approximation based on the multiscale technique is discussed. This approach is compared with the traditional ways to derive the wide-angle parabolic equation. While the latter fail in the nonlinear case, the multiscale derivation technique leading to iterative parabolic equations can be easily adapted to handle it. The nonlinear iterative parabolic approximations for the wave propagation in Kerr media are presented. An example demonstrating the capability of iterative parabolic equations to take nonparaxial propagation effects in Kerr media into account is considered.

We consider the reflectionless transport of solitons in networks. The system is modeled in terms of the nonlinear Schrödinger equation on metric graphs, for which transparent boundary conditions at the branching points are imposed. This approach allows to derive simple constraints, which link the equivalent usual Kirchhoff-type vertex conditions to the transparent ones. Our method is applied to a metric star graph. An extension to more complicated graph topologies is straightforward.

A wide-angle mode parabolic equation is obtained from the horizontal refraction equation by using the rational-linear approximation of the square-root operator. A finite-difference scheme for the numerical solution of the derived equation is developed. The scheme is based on the standard Crank–Nicolson method and fully-discrete transparent boundary conditions which allow for an accurate simulation of sound propagation on an unbounded domain.

In this paper, we introduce a novel, explicit, wide-stencil, two-dimensional (2D) tree–grid method for solving stochastic control problems (SCPs) with two space dimensions and one time dimension, or, equivalently, the corresponding Hamilton– Jacobi–Bellman equation. This new method can be seen as a generalization of the tree–grid method for SCPs with one space dimension that was recently developed by the authors. The method is unconditionally stable and no 2D interpolation is needed in the stencil construction. We prove the convergence of the method and exemplify it in our application to a two-factor uncertain volatility model.

Specifying time-dependent correlation matrices is a problem that occurs in several important areas of finance and risk management. The goal of this work is to tackle this problem by applying techniques of geometric integration in financial mathematics, i.e., to combine two fields of numerical mathematics that have not been studied yet jointly. Based on isospectral flows we create valid time-dependent correlation matrices, so called correlation flows, by solving a stochastic differential equation (SDE) that evolves in the special orthogonal group. Since the geometric structure of the special orthogonal group needs to be preserved we use stochastic Lie group integrators to solve this SDE. An application example is presented to illustrate this novel methodology.

Recently a new approach to the construction of high-order parabolic approximations for the Helmholtz equation was developed. These approximations have the form of the system of iterative parabolic equations, where the solution of the n-th equation is used as an input term for the ( n + 1 ) -th equation. In this study the transparent boundary conditions for such systems of coupled parabolic equations are derived. The existence and uniqueness of the solution of the initial boundary value problem for the system of iterative parabolic equations with the derived boundary conditions are proved. The well-posedness of this problem is also established and an unconditionally stable finite difference scheme for its solution is proposed.

In this paper we consider two numerical scheme based on trapezoidal rule in time for the linearized KdV equation in one space dimension. The goal is to derive some suitable artificial boundary conditions for these two full discretization using Z-transformation. We give some numerical benchmark examples from the literature to illustrate our findings.

In this article we combine the ideas of high-order (HO) and alternating direction implicit (ADI) schemes on sparse grids for diffusion equations with mixed derivatives. With the help of HO and ADI schemes solutions can be computed, which are fourth-order accurate in space and second-order accurate in time. For each implicit step of the ADI scheme we use a high-order-compact (HOC) discretisation such that the computational effort consists of only solving tridiagonal systems. In order to reduce the number of grid points, we use the combination technique to construct a solution defined on the sparse grid. This approach allows to further reduce the computational effort and memory consumption.

In this work we present a result on the non-existence of monotone, consistent linear discrete approximation of order higher than 2. This is an essential ingredient, if we want to solve numerically nonlinear and particularly Hamilton–Jacobi–Bellman (HJB) equations.

To confine a spatial domain to a smaller computational domain, one needs artificial boundaries. This work considers the lattice Boltzmann method and deals with boundary conditions for these open boundaries. Ideally, such a condition does not interact with the fluid at all. We present novel two-dimensional discrete artificial boundary conditions to pursue that goal and we discuss four different versions. This type of condition is formulated on the discrete lattice Boltzmann level and does not require a PDE formulation of the fluid. We set a special focus on the D2Q9 model. Our numerical results compare the novel discrete artificial boundary conditions to simulations using the existing non-reflecting characteristic boundary condition and an exit boundary condition.

The modelling of sound propagation in the ocean by the solution of mode parabolic equations is discussed. Mode parabolic equations can be obtained as the one-way approximation to horizontal refraction equations for modal amplitudes. Their wide-angle capabilities depend on the order of the Pade approximation of the involved pseudo-differential operators. Various aspects of numerical solution methods for wide-angle mode parabolic equations are considered in detail, including artificial domain truncation and Cauchy initial data for the point source field approximation. The capabilities of the discussed numerical approaches are demonstrated in several important test cases, including the problems of sound propagation in a penetrable wedge and in a sea with an underwater canyon.

Recently a new approach to the modeling of one-way wave propagation in Kerr media was proposed [1]. Within this approach the solution of the nonlinear Helmholtz equation is approximated by a series of solutions of iterative parabolic equations (IPEs). It was also shown that IPEs take the nonparaxial propagation effects into account. In this study we develop an efficient pseudospectral numerical method for solving the system of IPEs. The method is a generalization of an exponential time differencing (ETD) method for the nonlinear Schrodinger equation [2]. The ETD technique is well-suited for the system of IPEs, as it allows to reduce the order of the derivative in the input term.

Stochastic correlation models have become increasingly important in financial markets. In order to be able to price vanilla options in stochastic volatility and correlation models, in this work, we study the extension of the Heston model by imposing stochastic correlations driven by a stochastic differential equation. We discuss the efficient algorithms for the extended Heston model by incorporating stochastic correlations. Our numerical experiments show that the proposed algorithms can efficiently provide highly accurate results for the extended Heston by including stochastic correlations. By investigating the effect of stochastic correlations on the implied volatility, we find that the performance of the Heston model can be proved by including stochastic correlations.

We investigate the hydrodynamic recovery of Lattice Boltzmann Method (LBM) by analyzing exact balance relations for energy and enstrophy derived from averaging the equations of motion on sub-volumes of different sizes. In the context of 2D isotropic homogeneous turbulence, we first validate this approach on decaying turbulence by comparing the hydrodynamic recovery of an ensemble of LBM simulations against the one of an ensemble of Pseudo-Spectral (PS) simulations. We then conduct a benchmark of LBM simulations of forced turbulence with increasing Reynolds number by varying the input relaxation times of LBM. This approach can be extended to the study of implicit subgrid-scale (SGS) models, thus offering a promising route to quantify the implicit SGS models implied by existing stabilization techniques within the LBM framework.

This article deals with artificial boundaries which you encounter when a large spatial domain is confined to a smaller computational domain. Such an artificial boundary condition should not preferably interact with the fluid at all. Standard boundary conditions, e.g., a pressure or velocity condition, result in unphysical reflections. So far, existing artificial boundary conditions for the lattice Boltzmann method (LBM) are transferred from macroscopic formulations.In this work we propose novel discrete artificial boundary conditions (ABCs) which are tailored on the LBM's mesoscopic level. They are derived directly for the chosen LBM with the aim of higher accuracy. We describe the idea of discrete ABCs in a three velocity (D1Q3) model governing the Navier-Stokes equations in one dimension. Numerical results finally demonstrate the superiority of our new boundary condition in terms of accuracy compared to previously used ABCs.

Mayfield’s numerical implementation of transparent boundary condition for the Schrodinger-type parabolic equations is revisited. An inaccuracy in the original proof of the conditional stability for the resulting scheme is pointed out. The highly unusual and impressive original result is reestablished and a new proof is presented. Some further remarks and estimates on the instability which occurs when the Mayfield condition is violated are given.

Force-gradient decomposition methods are used to improve the energy preservation of symplectic schemes applied to Hamiltonian systems. If the potential is composed of different parts with strongly varying dynamics, this multirate potential can be exploited by coupling force-gradient decomposition methods with splitting techniques for multi-time scale problems to further increase the accuracy of the scheme and reduce the computational costs. In this paper, we derive novel force-gradient nested methods and test them numerically. We apply them on the three-body problem, modified for a better observation of the advantageous properties, needed for the future research.

It is well known that the correlation between financial products or financial institutions, e.g. plays an essential role in pricing and evaluation of financial derivatives. Using simply a constant or deterministic correlation may lead to correlation risk, since market observations give evidence that correlation is not a deterministic quantity. In this work, we propose a new approach to model the correlation as a hyperbolic function of a stochastic process. Our general approach provides a stochastic correlation which is much more realistic to model real-world phenomena and could be used in many financial application fields. Furthermore, it is very flexible: any mean-reverting process with positive and negative values can be regarded and no additional parameter restrictions appear which simplifies the calibration procedure. As an example, we compute the price of a Quanto applying our new approach. Using our numerical results we discuss concisely the effect of considering stochastic correlation on pricing the Quanto.

This chapter deals with the modelling and calibration of stochastic correlations. Correlation plays essential role in pricing derivatives on multi-assets. Market observations give evidence that the correlation is hardly a deterministic quantity, however, a constant or deterministic correlation has been widely used, although it may lead to correlation risk. It has been recently proposed to model correlation by a stochastic process, similar to stochastic volatility process. In this chapter, we review the concept of stochastic correlation process including calibration via the transition density function and its application for pricing the European-style Quanto option. As an illustrating example, we compare the Quanto option prices between using constant and stochastic correlation and analyze the effect of considering stochastic correlations on pricing the Quanto option.

The STRIKE Computational Finance Toolbox (CFT) is one of the output results of the “ITN STRIKE—Novel Methods in Computational Finance” and is concerned with combining the research output of the network. For this purpose the implemented models (MATLAB and PYTHON) of the PhD students and postdocs are collected and user interfaces are developed for a convenient use of those programs. Two different interfaces have been implemented. The first interface combines those submissions, which are dealing with one-dimensional spatial models, e.g. a problem settings with one stock price. With the second user interface two-dimensional programs can be used, which consist of problem settings with two spatial variables, e.g. one stock price and it’s volatility, or two stock prices.

In this chapter we combine high-order-compact (HOC) and alternating-direction-implicit (ADI) schemes for pricing basket options in a sparse grid setting HOC schemes exploit the structure of the underlying partial differential equation to obtain a high order of consistency while employing a compact stencil. As time discretisation we propose an efficient ADI splitting to derive a stable scheme. The combination technique is used to construct the so called sparse grid solution, which leads to a significant reduction of necessary grid points and thus to a lower computational effort.

We consider the computational efficiency of the backward vs. forward approaches and compare these with the respective ones resulting from a parametric reduced order model, whose speed-up can be put to good use in the calibration of the underlying dynamics. We apply a global Proper Orthogonal Decomposition in the time domain to obtain the reduced basis and the Modified Craig-Sneyd ADI and Chang-Cooper schemes to numerically solve the partial differential equations. The numerical results are presented for the Black-Scholes and Heston models.

We study a novel class of numerical integrators, the adapted nested force-gradient schemes, used within the molecular dynamics step of the Hybrid Monte Carlo (HMC) algorithm. We test these methods in the Schwinger model on the lattice, a well known benchmark problem. We derive the analytical basis of nested force-gradient type methods and demonstrate the advantage of the proposed approach, namely reduced computational costs compared with other numerical integration schemes in HMC., Comment: 13 pages, 2 figures

This minisymposium was an activity of the ECMI Special Interest Group on Computational Finance [2]. The SIG was launched at ECMI-2014 in Taormina (June 9–13, 2014) and (together with the ITN STRIKE network [3]) organized several sessions of a minisymposium in Computational Finance. The corresponding reporting can be found in [1]. At ECMI-2016 we brought together again twelve speakers.

In this chapter model order reduction (MOR) and the forward-backward duality are combined to generate forward and backward reduced models. We show that both resulting models are numerically efficient models and can in most situations reduce the computational effort in comparison with the full order models, when applying ADI and BDF2 time discretization schemes on a centered second-order and Chang-Cooper spatial discretizations, respectively. For the MOR part, a Proper Orthogonal Decomposition approach was taken.

In this chapter we discuss Alternating direction explicit (ADE) methods for one-dimensional convection diffusion equations and their numerical analysis including stability, consistency and convergence results. We propose ADE schemes for the two and three dimensional linear Black-Scholes pricing model. Our implemented methodology can be easily extended to higher dimensions. Extension for nonlinear Frey and Patie model is included at the end of this chapter where the nonlinear part is treated explicitly (This chapter is a shortened version of the papers (Buckova et al., Acta Math Univ Comen 84:309–325, 2015; Buckova et al., AIP Conf Proc 1773:030001, 2016.)).

We study the numerical solution of a Boundary Value problem for second order quadratic differential equations which arises in the numerical prediction of meteorological parameters. In the present work, we use finite differences and focus on the numerical solution of the resulting nonlinear system. More precisely, we apply classical Newton’s and Quasi-Newton methods paying attention to the special sparse form of the Jacobian matrix and modify appropriately the LU factorization in order to reduce significantly the required floating point operations. Furthermore, we implement and study in depth the behavior of all the proposed procedures in respect of their accuracy, stability and complexity, using data from South East Mediterranean Sea. All the methods are tested with a variety of initial values and their performance is presented and discussed leading to interesting results on the sensitivity of the selected starting point.

In this paper we introduce a joint model for the electricity spot and the emission market in Germany. The model is used to simulate the effects of changing market parameters and policy implications. We propose a structural approach to connect fundamental factors, such as demand, capacity, fuel prices and emission allowances, to the price of electricity. Based on empirical bid behavior of electricity producers in day-ahead auctions we develop a novel CO2 emission rate model. In order to introduce a feedback of CO2 allowances to the actual emission of greenhouse gases, a forward–backward stochastic differential equation is used to deduce a partial differential equation to price emission allowances. The resulting equation will be solved by an alternating direction implicit finite difference scheme.

When using a lattice Boltzmann method on an unbounded (or very large) domain one has to confine this spatial domain to a computational domain. This is realized by introducing so-called artificial boundary conditions. Until recently, characteristic boundary conditions for the Euler equations were considered and adapted to the lattice Boltzmann method. In this work we propose novel discrete artificial boundary conditions which are derived directly for the chosen lattice Boltzmann model, i.e., on the discrete level. They represent the first exact artificial boundary conditions for lattice Boltzmann methods. Doing so, we avoid any detour of considering continuous equations and obtain boundary conditions that are perfectly adapted to the chosen numerical scheme. We illustrate the idea for a one dimensional, two velocity (D1Q2) lattice Boltzmann method and show how the computational efficiency can be increased by a finite memory approach. Analytical investigations and numerical results finally demonstrate the advantages of our new boundary condition compared to previously used artificial boundary conditions.

For numerically solving fluid dynamics problems efficiently one is often facing the problem of having to confine the computational domain to a small domain of interest introducing so-called non-reflecting boundary conditions (NRBCs). In this work we address the problem of supplying NRBCs in fluid simulations in two space dimensions using the lattice Boltzmann method (LBM): so-called characteristic boundary conditions are revisited and transferred to the framework of lattice Boltzmann simulations. Numerical tests show clearly that the unwanted unphysical reflections can be reduced significantly by applying our newly developed methods. Hereby the key idea is to transfer and generalize Thompson's boundary conditions originally developed for the nonlinear Euler equations of gas dynamics to the setting of lattice Boltzmann methods. Finally, we give strong numerical evidence that the proposed methods possess a long-time stability property.

In this paper we will derive an SIR model describing vaccination as well as waning immunity and propose a finite difference scheme for its solution together with some qualitative results. For the modeling of the waning immunity we assume a statistical distribution for the level of antibodies depending on the time lapsed since individual's full recovery or vaccination. We arrive at a system of two ODEs and two PDEs that we reduce to a model of just two ODEs and a few algebraic equations. Next, we propose and implement an efficient numerical scheme to solve this reduced model, based on finite differences. To illustrate our findings we provide graphical results and discuss some qualitative properties of the solutions. Additionally, we derive formulas for the basic reproduction number R 0 and the effective reproduction number R ( t ) of the reduced model and show the behavior of solutions for examples with R 0 > 1 and R 0 1 .

This work presents a comparison study of different numerical methods to solve Black-Scholes-type partial differential equations (PDE) in the convection-dominated case, i.e., for European options, if the ratio of the risk-free interest rate and the squared volatility-known in fluid dynamics as Péclet number-is high. For Asian options, additional similar problems arise when the “spatial” variable, the stock price, is close to zero.Here we focus on three methods: the exponentially fitted scheme, a modification of Wang’s finite volume method specially designed for the Black-Scholes equation, and the Kurganov-Tadmor scheme for a general convection-diffusion equation, that is applied for the first time to option pricing problems. Special emphasis is put in the Kurganov-Tadmor because its flexibility allows the simulation of a great variety of types of options and it exhibits quadratic convergence. For the reduction technique proposed by Wilmott, a put-call parity is presented based on the similarity reduction and the put-call parity expression for Asian options. Finally, we present experiments and comparisons with different (non)linear Black-Scholes PDEs.

Student academic underachievement is a concern of paramount importance in Europe, where around 15% of the students in the last high school courses do not achieve the minimum knowledge academic requirement. In this paper, we propose a model based on a system of differential equations to study the dynamics of the students academic performance in the German region of North Rhine-Westphalia. This approach is supported by the idea that both, good and bad study habits, are a mixture of personal decisions and influence of classmates. This model allows us to forecast the student academic performance by means of confidence intervals over the next few years., This work has been partially supported by the Spanish Ministry of Economy and Competitiveness grant MTM2009-08587 and Universitat Politecnica de Valencia grant PAID06-11-2070.

The Cox–Ingersoll–Ross CIR model has been a benchmark in finance for many years because of its analytical and structural tractability. The wide applications and extensions of the CIR model requires to evaluate the cumulative distribution function CDF of the integrated CIR process in financial modelling. Usually the characteristic function of the integrated CIR process is known analytically and one can use the option pricing method of Carr and Madan to transform it into the corresponding CDF. This characteristic function is defined via complex logarithms which often leads to numerical instabilities when it is integrated using the inverse Fourier transform. Typically, this instability is expected to be apparent for wide ranges of model parameters. In this work, we adapt the recent approach by Kahl and Jackel for the Heston model to deal with such instability problems. Our new strategy allows to construct a very robust routine to determine numerically a highly accurate CDF of the integrated CIR process for almost any choices of parameters.

In this note we derive, using the subequation method, a new nonstandard finite difference scheme (NSFD) for a class of convection-diffusion equations having constant coefficients. Despite the fact that this scheme has nonlinear denominator functions of the step sizes (even for linear PDEs), it has a couple of favourable properties: it is explicit and due to its construction it reproduces important properties of the solution of the parabolic PDE. This proposed method conserves, by construction, the positivity of the solution if one choses a right combination of spatial and temporal step sizes and hence it is perfectly suited for solving for example air pollution problems or the Black-Scholes equation for the valuation of standard options, since it avoids negative values for the calculated prices. Finally, we illustrate the usefulness of this newly proposed method on a classical benchmark example from the literature.

Recently, Galapon [E.A. Galapon, Delta-convergent sequences that vanish at the support of the limit Dirac delta function, J. Phys. A 42 (2009) 175–201] has posed the question of the existence of delta-convergent sequences that vanish at the support of the limit Dirac delta function and gave an example of sequences of this type. It is a sequence of even functions that do not have a compact support. Motivated by the question, in this short note we develop some results concerning delta sequences and show more examples of delta sequences of the type with or without compact support and that are even or not even.

This work deals with the stochastic modelling of correlation in finance. It is well known that the correlation between financial products, financial institutions, e.g., plays an essential role in pricing and evaluation of financial derivatives. Using simply a constant or deterministic correlation may lead to correlation risk, since market observations give evidence that the correlation is hardly a deterministic quantity. For example, we illustrate this issue with the analysis of correlation between daily returns time series of S&P Index and Euro/USD exchange rates. The approach of modelling the correlation as a hyperbolic function of a stochastic process has been recently proposed. Here, we review this novel concept and generalize this approach to derive stochastic correlation processes (SCP) from a hyperbolic transformation of the modified Ornstein-Uhlenbeck process. We determine a transition density function of this SCP in closed form which could be used easily to calibrate SCP models to historical data. As an illustrating example of our new approach, we compute the price of a quantity adjusting option (Quanto) and discuss concisely the effect of considering stochastic correlation on pricing the Quanto.

Correlation plays an essential role in many problems of finance and economics, such as pricing financial products and hedging strategies, since it models the degree of relationship between, e.g., financial products and financial institutions. However, usually for simplicity the correlation coefficient is assumed to be a constant in many models, although financial quantities are correlated in a strongly nonlinear way in the real market. This work provides a new time-dependent correlation function, which can be easily used to construct dynamically (time-dependent) correlated Brownian motions and flexibly incorporated in many financial models. The aim of using our time-dependent correlation function is to reasonably choose additional parameters to increase the fitting quality on the one hand, but also add an economic concept on the other hand. As examples, we illustrate the applications of dynamic correlation in the Heston model. From our numerical results we conclude that the Heston model extended by incorporating time-dependent correlations can provide a better volatility smile than the pure Heston model.

The Fichera theory was first proposed in 1960 by Gaetano Fichera and later developed by Olejnik and Radkevic in 1973. It turned out to be very useful for establishing the well-posedness of initial boundary value problems for parabolic partial differential equations degenerating to hyperbolic ones at the boundary.