266 research outputs found
Implicit-explicit predictor-corrector methods combined with improved spectral methods for pricing European style vanilla and exotic options
In this paper we present a robust numerical method to solve several types of European style option pricing problems. The governing equations are described by variants of Black-Scholes partial differential equations (BS-PDEs) of the reaction-diffusion-advection type. To discretise these BS-PDEs numerically, we use the spectral methods in the asset (spatial) direction and couple them with a third-order implicit-explicit predictor-corrector (IMEX-PC) method for the discretisation in the time direction. The use of this high-order time integration scheme sustains the better accuracy of the spectral methods for which they are well-known. Our spectral method consists of a pseudospectral formulation of the BS-PDEs by means of an improved Lagrange formula. On the other hand, in the IMEX-PC methods, we integrate the diffusion terms implicitly whereas the reaction and advection terms are integrated explicitly. Using this combined approach, we first solve the equations for standard European options and then extend this approach to digital options, butterfly spread options, and European calls in the Heston model. Numerical experiments illustrate that our approach is highly accurate and very efficient for pricing financial options such as those described above
A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces
In this paper we present a high-order kernel method for numerically solving
diffusion and reaction-diffusion partial differential equations (PDEs) on
smooth, closed surfaces embedded in . For two-dimensional
surfaces embedded in , these types of problems have received
growing interest in biology, chemistry, and computer graphics to model such
things as diffusion of chemicals on biological cells or membranes, pattern
formations in biology, nonlinear chemical oscillators in excitable media, and
texture mappings. Our kernel method is based on radial basis functions (RBFs)
and uses a semi-discrete approach (or the method-of-lines) in which the surface
derivative operators that appear in the PDEs are approximated using
collocation. The method only requires nodes at "scattered" locations on the
surface and the corresponding normal vectors to the surface. Additionally, it
does not rely on any surface-based metrics and avoids any intrinsic coordinate
systems, and thus does not suffer from any coordinate distortions or
singularities. We provide error estimates for the kernel-based approximate
surface derivative operators and numerically study the accuracy and stability
of the method. Applications to different non-linear systems of PDEs that arise
in biology and chemistry are also presented
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Structure-preserving variational schemes for fourth order nonlinear partial differential equations with a Wasserstein gradient flow structure
There is a growing interest in studying nonlinear partial differential equations which constitute gradient flows in the Wasserstein metric and related structure preserving variational discretisations. In this thesis, we focus on the fourth order Derrida-Lebowitz-Speer-Spohn (DLSS) equation, the thin film equation, as well as other fourth order examples. We adapt the minimising movement schemes from implicit Euler (BDF1) to higher order schemes, i.e. backward difference formulae and diagonally implicit Runge-Kutta (DIRK) methods.
We prove numerical convergence of discrete solutions of the DIRK2 scheme using a comparison principle type approach with semi-convex based conditions. With basic assumptions including semi-convexity of our energy, verifying that the energy is monotonic in time normally yields convergence of its discrete solution for decreasing time step. However, as in the BDF2 example, for the DIRK2 scheme considered here the energy was not verified to be monotonic (it might be), yet with additional assumptions, convergence is obtained as well as other basic properties of gradient flows.
We propose fully discrete schemes which preserve positivity for the DLSS equation, the Thin Film equation and other nonlinear partial differential equations. We present results of numerical experiments confirming improved rates of convergence for higher order schemes. Furthermore, numerical results with non-constant time steps are presented, improving the efficiency of the proposed schemes
Differential-Algebraic Equations
Differential-Algebraic Equations (DAE) are today an independent field of research, which is gaining in importance and becoming of increasing interest for applications and mathematics itself. This workshop has drawn the balance after about 25 years investigations of DAEs and the research aims of the future were intensively discussed
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SciCADE 95: International conference on scientific computation and differential equations
This report consists of abstracts from the conference. Topics include algorithms, computer codes, and numerical solutions for differential equations. Linear and nonlinear as well as boundary-value and initial-value problems are covered. Various applications of these problems are also included
Order reduction of semilinear differential matrix and tensor equations
In this thesis, we are interested in approximating, by model order reduction, the solution to large-scale matrix- or tensor-valued semilinear Ordinary Differential Equations (ODEs). Under specific hypotheses on the linear operators and the considered domain, these ODEs often stem from the space discretization on a tensor basis of semilinear Partial Differential Equations (PDEs) with a dimension greater than or equal to two.
The bulk of this thesis is devoted to the case where the discrete system is a matrix equation. We consider separately the cases of general Lipschitz continuous nonlinear functions and the Differential Riccati Equation (DRE) with a quadratic nonlinear term. In both settings, we construct a pair of left-right approximation spaces that leads to a reduced semilinear matrix differential equation with the same structure as the original problem, which can be more rapidly integrated with matrix-oriented integrators. For the DRE, under certain assumptions on the data, we show that a reduction process onto rational Krylov subspaces obtains significant computational and memory savings as opposed to current approaches.
In the more general setting, a challenging difference lies in selecting and constructing the two approximation bases to handle the nonlinear term effectively. In addition, the nonlinear term also needs to be approximated for efficiency. To this end, in the framework of the Proper Orthogonal Decomposition (POD) methodology and the Discrete Empirical Interpolation Method (DEIM), we derive a novel matrix-oriented reduction process leading to a practical, structure-aware low order approximation of the original problem.
In the final part of the thesis, we consider the multidimensional setting. Here we extend the matrix-oriented POD-DEIM algorithm to the tensor setting and illustrate how we can apply it to systems of such equations. Moreover, we discuss how to integrate the reduced-order model and, in particular, how to solve the resulting tensor-valued linear systems
Dynamics of Numerics & Spurious Behaviors in CFD Computations
The global nonlinear behavior of finite discretizations for constant time steps and fixed or adaptive grid spacings is studied using tools from dynamical systems theory. Detailed analysis of commonly used temporal and spatial discretizations for simple model problems is presented. The role of dynamics in the understanding of long time behavior of numerical integration and the nonlinear stability, convergence, and reliability of using time-marching approaches for obtaining steady-state numerical solutions in computational fluid dynamics (CFD) is explored. The study is complemented with examples of spurious behavior observed in steady and unsteady CFD computations. The CFD examples were chosen to illustrate non-apparent spurious behavior that was difficult to detect without extensive grid and temporal refinement studies and some knowledge from dynamical systems theory. Studies revealed the various possible dangers of misinterpreting numerical simulation of realistic complex flows that are constrained by available computing power. In large scale computations where the physics of the problem under study is not well understood and numerical simulations are the only viable means of solution, extreme care must be taken in both computation and interpretation of the numerical data. The goal of this paper is to explore the important role that dynamical systems theory can play in the understanding of the global nonlinear behavior of numerical algorithms and to aid the identification of the sources of numerical uncertainties in CFD
Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
The LifeV library: engineering mathematics beyond the proof of concept
LifeV is a library for the finite element (FE) solution of partial
differential equations in one, two, and three dimensions. It is written in C++
and designed to run on diverse parallel architectures, including cloud and high
performance computing facilities. In spite of its academic research nature,
meaning a library for the development and testing of new methods, one
distinguishing feature of LifeV is its use on real world problems and it is
intended to provide a tool for many engineering applications. It has been
actually used in computational hemodynamics, including cardiac mechanics and
fluid-structure interaction problems, in porous media, ice sheets dynamics for
both forward and inverse problems. In this paper we give a short overview of
the features of LifeV and its coding paradigms on simple problems. The main
focus is on the parallel environment which is mainly driven by domain
decomposition methods and based on external libraries such as MPI, the Trilinos
project, HDF5 and ParMetis.
Dedicated to the memory of Fausto Saleri.Comment: Review of the LifeV Finite Element librar
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