24,630 research outputs found
Preconditioning for Allen-Cahn variational inequalities with non-local constraints
The solution of Allen-Cahn variational inequalities with mass constraints is of interest
in many applications. This problem can be solved both in its scalar and vector-valued form as a
PDE-constrained optimization problem by means of a primal-dual active set method. At the heart
of this method lies the solution of linear systems in saddle point form. In this paper we propose the
use of Krylov-subspace solvers and suitable preconditioners for the saddle point systems. Numerical
results illustrate the competitiveness of this approach
Preconditioning for Allen-Cahn variational inequalities with non-local constraints
The solution of Allen-Cahn variational inequalities with mass constraints is of interest in many applications. This problem can be solved both in its scalar and vector-valued form as a PDE-constrained optimization problem by means of a primal-dual active set method. At the heart of this method lies the solution of linear systems in saddle point form. In this paper we propose the use of Krylov-subspace solvers and suitable preconditioners for the saddle point systems. Numerical results illustrate the competitiveness of this approach
On some aspects of the geometry of differential equations in physics
In this review paper, we consider three kinds of systems of differential
equations, which are relevant in physics, control theory and other applications
in engineering and applied mathematics; namely: Hamilton equations, singular
differential equations, and partial differential equations in field theories.
The geometric structures underlying these systems are presented and commented.
The main results concerning these structures are stated and discussed, as well
as their influence on the study of the differential equations with which they
are related. Furthermore, research to be developed in these areas is also
commented.Comment: 21 page
On the probability density function of baskets
The state price density of a basket, even under uncorrelated Black-Scholes
dynamics, does not allow for a closed from density. (This may be rephrased as
statement on the sum of lognormals and is especially annoying for such are used
most frequently in Financial and Actuarial Mathematics.) In this note we
discuss short time and small volatility expansions, respectively. The method
works for general multi-factor models with correlations and leads to the
analysis of a system of ordinary (Hamiltonian) differential equations.
Surprisingly perhaps, even in two asset Black-Scholes situation (with its flat
geometry), the expansion can degenerate at a critical (basket) strike level; a
phenomena which seems to have gone unnoticed in the literature to date.
Explicit computations relate this to a phase transition from a unique to more
than one "most-likely" paths (along which the diffusion, if suitably
conditioned, concentrates in the afore-mentioned regimes). This also provides a
(quantifiable) understanding of how precisely a presently out-of-money basket
option may still end up in-the-money.Comment: Appeared in: Large Deviations and Asymptotic Methods in Finance,
Springer proceedings in Mathematics & Statistics, Editors: Friz, P.K.,
Gatheral, J., Gulisashvili, A., Jacquier, A., Teichmann, J., 2015, with minor
typos remove
Spectral/hp element methods: recent developments, applications, and perspectives
The spectral/hp element method combines the geometric flexibility of the
classical h-type finite element technique with the desirable numerical
properties of spectral methods, employing high-degree piecewise polynomial
basis functions on coarse finite element-type meshes. The spatial approximation
is based upon orthogonal polynomials, such as Legendre or Chebychev
polynomials, modified to accommodate C0-continuous expansions. Computationally
and theoretically, by increasing the polynomial order p, high-precision
solutions and fast convergence can be obtained and, in particular, under
certain regularity assumptions an exponential reduction in approximation error
between numerical and exact solutions can be achieved. This method has now been
applied in many simulation studies of both fundamental and practical
engineering flows. This paper briefly describes the formulation of the
spectral/hp element method and provides an overview of its application to
computational fluid dynamics. In particular, it focuses on the use the
spectral/hp element method in transitional flows and ocean engineering.
Finally, some of the major challenges to be overcome in order to use the
spectral/hp element method in more complex science and engineering applications
are discussed
Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods
In this paper, we discuss a general multiscale model reduction framework
based on multiscale finite element methods. We give a brief overview of related
multiscale methods. Due to page limitations, the overview focuses on a few
related methods and is not intended to be comprehensive. We present a general
adaptive multiscale model reduction framework, the Generalized Multiscale
Finite Element Method. Besides the method's basic outline, we discuss some
important ingredients needed for the method's success. We also discuss several
applications. The proposed method allows performing local model reduction in
the presence of high contrast and no scale separation
Numerical model of solid phase transformations governed by nucleation and growth. Microstructure development during isothermal crystallization
A simple numerical model which calculates the kinetics of crystallization
involving randomly distributed nucleation and isotropic growth is presented.
The model can be applied to different thermal histories and no restrictions are
imposed on the time and the temperature dependencies of the nucleation and
growth rates. We also develop an algorithm which evaluates the corresponding
emerging grain size distribution. The algorithm is easy to implement and
particularly flexible making it possible to simulate several experimental
conditions. Its simplicity and minimal computer requirements allow high
accuracy for two- and three-dimensional growth simulations. The algorithm is
applied to explore the grain morphology development during isothermal
treatments for several nucleation regimes. In particular, thermal nucleation,
pre-existing nuclei and the combination of both nucleation mechanisms are
analyzed. For the first two cases, the universal grain size distribution is
obtained. The high accuracy of the model is stated from its comparison to
analytical predictions. Finally, the validity of the
Kolmogorov-Johnson-Mehl-Avrami model is verified for all the cases studied
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