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Joerg Kampen

Publications and source records attributed to Joerg Kampen.

32 records · Page 2Linked to original sources
arXiv2013

On the multivariate Burgers equation and the incompressible Navier-Stokes equation (Part II)

We consider global schemes with H2-estimates for the multivariate Burgers equation and the incompressible Navier-Stokes equation in its Leray projection form. We extend the definition of a global scheme and the global L2-theory of the multivariate Burgers equation to the incompressible Navier-Stokes equation, where we consider estimates of the gradient of the pressure in its Leray projection form in H1 at each time step of the scheme. The extended scheme has a simplified control function (simplified compared to [7]), which controls the growth of the Leray projection term. We use the fact that the data in our scheme are in H2 at each time step such that the data for the Poisson equation related to the elimination of pressure are in H1 at each approximation step. The upper bound of the growth of the velocity components is linear with respect to the time.

math.AP↗
arXiv2012

Global existence, regularity and a probabilistic scheme for a class of ultraparabolic Cauchy problems

In this paper we establish a constructive method in order to show global existence and regularity for a class of degenerate parabolic Cauchy problems which satisfy a weak Hoermander condition on a subset of the domain where the data are measurable and which have regular data on the complementary set of the domain. This result has practical incentives related to the computation of Greeks in reduced LIBOR market models, which are standard computable approximations of the HJM-description of interest rate markets. The method leads to a probabilistic scheme for the computation of the value function and its sensitivities based on Malliavin calculus. From a practical perspective the main contribution of the paper is an Monte-Carlo algorithm which includes weight corrections for paths which move in time into a region where a (weak) Hoermander condition holds.

math.AP↗
arXiv2012

On a class of singular solutions to the incompressible 3-D Euler equation

A class of singular 3D-velocity vector fields is constructed which satisfy the incompressible 3D-Euler equation. It is shown that such a solution scheme does not exist in dimension 2. The solutions constructed are bounded and smooth up to finite time where they become singular. Although the solution is smooth and bounded there seems to be no bound in L2 of the velocity field.

math.AP↗
arXiv2012

Constructive analysis of the Navier-Stokes equation

A global time-discretized scheme for the Navier-Stokes equation system in its Leray projection form is defined. It is shown that the scheme converges to a bounded global classical solution for smooth data which have polynomial decay at infinity. Furthermore, the algorithm proposed is extended to the situation of initial-boundary value problems. Algorithms constructed in a different context (cf. [4, 10, 5, 9]) may be used within the proposed scheme in order to compute the solution of Leray's form of the Navier-Stokes system. The main idea for global existence is to define a control function dynamically and show explicitly that the scheme which solves a controlled Navier-Stokes type equation can control the modulus of velocity and the first derivatives of velocity to be bounded. The method described here can be extended to Navier-Stokes equations on compact manifolds which is done in a subsequent paper.

math.AP↗
arXiv2012

Generalized Brownian motion from a logical point of view

We describe generalized Brownian motion related to parabolic equation systems from a logical point of view, i.e., as a generalization of Anderson's random walk. The connection to classical spaces is based on the Loeb measure. It seems that the construction of Roux in [11] is the only attempt in the literature to define generalized Brownian motion related to parabolic systems with coupled second order terms, where Lamé's equation of elastic mechanics is considered as an example. In this paper we provide an exact construction from a logical point of view in a more general situation. A Feynman-Kac formula for generalized Brownian motion is derived which is a useful tool in order to design probabilistic algorithms for Cauchy problems and initial-boundary value (of a class of) parabolic systems as well as for stationary boundary problems of (a class of) elliptic equation systems. The article includes a selfcontained introduction into all tools of nonstandard analysis needed, and which can be read with a minimum knowledge of logic in order to make the results available to a wider audience.

math.PR↗
arXiv2011

On the multivariate Burgers equation and the incompressible Navier-Stokes equation (Part I)

We provide a constructive global existence proof for the multivariate viscous Burgers equation system defined on the whole space or on a domain isomorphic to the n-torus and with time horizon up to infinity and C^{\infty}- data (satisfying some growth conditions if the problem is posed on the whole space). The proof is by a time discretized semiexplicit perturbative expansion in transformed coordinates where the convergence is guaranteed by certain a priori estimates. The scheme is useful in order to define computation for related equation systems of fluid dynamics.

math.AP↗
arXiv2010

On local analytic expansions of densities in the context of (micro-)hypoelliptic and classes of semi-elliptic equations

Explicit representations of densities for linear parabolic partial differential equations are useful in order to design computation schemes of high accuracy for a considerable class of diffusion models. Approximations of lower order based on the WKB-expansion have been used in order to compute Greeks in standard models of the interest rate market (cf. [2]). However, it turns out that for higher order approximations another related expansion leads to more accurate schemes. We compute a local explicit formula for a class of parabolic problems and determine a lower bound of the time horizon where it holds (given a certain bounded domain). Although the local analytic expansions hold only for strictly elliptic equations we show that the expansions can be used in order to design higher order schemes for various types of (micro)-hypoelliptic and semi-elliptic equations, e.g. the reduced market models considered in [7] or front fixing schemes for multivariate American derivatives [3].

math.AP↗
arXiv2010

Determiniation of the homotopy class of maps on compact orientable surfaces of positive genus g with infinitely many periodic points (part I)

We consider the homotopical dynamics on compact orientable surfaces of positive genus g. We establish a sufficient and necessary algebraic criterion for homotopy classes with infinitely many periodic points of maps on such surfaces in terms of the characteristic polynomial of the matrix representing the correspondig homomorphism of the first homology group. Our methods differ from that of the literature, which typically uses Nielsen number theory, because Nielsen numbers of iterates are difficult to compute except in special cases where they can be easily defined in terms of the easily computable Lefschetz number. Instead we use some fixed point theory from the point of view of real algebraic geometry, and some intersection theory in the context of De Rham theory on surfaces and combine it with some observations of dynamical system theory which started from the work of Sarkovskii. However, in the special case of the homotopical dynamics of a two dimensional torus, where the Nielsen number is determined by the modulus of the Lefschetz number, we recover a result in [1].

math.DS↗
arXiv2010

Characteristic functions of affine processes via calculus of their operator symbols

The characteristic functions of multivariate Feller processes with generator of affine type, and with smooth symbol functions have an explicit representation in terms of power series with rational number coefficients and with monmoms consisting of powers of the the symbol functions and formal derivatives of the symbol functions. The power series repesentations are convergent globally in time and on bounded domains of arbitrary size. Generalized symbol functions can be derived leading to power series expansions which are convergent on arbitrary domains in special cases. The rational number coefficients can be efficiently computed by an integer recursion. As a numerical consequence characteristic functions of multivariate affine processes can be efficiently computed from the symbol function avoiding computation of the generalized Riccati equations (an observation first made recently in a more general context).

math.FA↗
arXiv2009

Global analytic expansion of solution for a class of linear parabolic systems with coupling of first order derivatives terms

We derive global analytic representations of fundamental solutions for a class of linear parabolic systems with full coupling of first order derivative terms where coefficient may depend on space and time. Pointwise convergence of the global analytic expansion is proved. This leads to analytic representations of solutions of initial-boundary problems of first and second type in terms of convolution integrals or convolution integrals and linear integral equations. The results have both analytical and numerical impact. Analytically, our representations of fundamental solutions of coupled parabolic systems may be used to define generalized stochastic processes. Moreover, some classical analytical results based on a priori estimates of elliptic equations are a simple corollary of our main result. Numerically, accurate, stable and efficient schemes for computation and error estimates in strong norms can be obtained for a considerable class of Cauchy- and initial-boundary problems of parabolic type. Furthermore, there are obvious and less obvious applications to finance and physics. Warning: The argument given in the current version is only valid in special cases (essentially the scalar case). A more involved argument is needed for systems and will be communicated soon in a replacement,

math.AP↗
arXiv2008

How to compute the length of a geodesic on a Riemannian manifold with small error in arbitrary Sobolev norms

We compute the length of geodesics on a Riemannian manifold by regular polynomial interpolation of the global solution of the eikonal equation related to the line element $ds^2=g_{ij}dx^idx^j$ of the manifold. Our algorithm approximates the length functional in arbitrarily strong Sobolev norms. Error estimates are obtained where the geometric information is used. It is pointed out how the algorithm can be used to get accurate approximation of solutions of parabolic partial differential equations leading obvious applications to finance and physics.

math.NA↗
arXiv2008

Regular polynomial interpolation and approximation of global solutions of linear partial differential equations

We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the 'limit' of the recursively constructed family of polynomials to the solution and error estimates are obtained from a priori estimates for some standard classes of linear partial differential equations, i.e. elliptic and hyperbolic equations. Another variation of the algorithm allows to construct polynomial interpolations which preserve systems of linear partial differential equations at the interpolation points. We show how this can be applied in order to compute higher order terms of WKB-approximations of fundamental solutions of a large class of linear parabolic equations. The error estimates are sensitive to the regularity of the solution. Our method is compatible with recent developments for solution of higher dimensional partial differential equations, i.e. (adaptive) sparse grids, and weighted Monte-Carlo, and has obvious applications to mathematical finance and physics.

math.NA↗
arXiv2008

Monte Carlo Greeks for financial products via approximative transition densities

In this paper we introduce efficient Monte Carlo estimators for the valuation of high-dimensional derivatives and their sensitivities (''Greeks''). These estimators are based on an analytical, usually approximative representation of the underlying density. We study approximative densities obtained by the WKB method. The results are applied in the context of a Libor market model.

q-fin.CP↗