479 research outputs found
Compressive Space-Time Galerkin Discretizations of Parabolic Partial Differential Equations
We study linear parabolic initial-value problems in a space-time variational
formulation based on fractional calculus. This formulation uses "time
derivatives of order one half" on the bi-infinite time axis. We show that for
linear, parabolic initial-boundary value problems on , the
corresponding bilinear form admits an inf-sup condition with sparse tensor
product trial and test function spaces. We deduce optimality of compressive,
space-time Galerkin discretizations, where stability of Galerkin approximations
is implied by the well-posedness of the parabolic operator equation. The
variational setting adopted here admits more general Riesz bases than previous
work; in particular, no stability in negative order Sobolev spaces on the
spatial or temporal domains is required of the Riesz bases accommodated by the
present formulation. The trial and test spaces are based on Sobolev spaces of
equal order with respect to the temporal variable. Sparse tensor products
of multi-level decompositions of the spatial and temporal spaces in Galerkin
discretizations lead to large, non-symmetric linear systems of equations. We
prove that their condition numbers are uniformly bounded with respect to the
discretization level. In terms of the total number of degrees of freedom, the
convergence orders equal, up to logarithmic terms, those of best -term
approximations of solutions of the corresponding elliptic problems.Comment: 26 page
Weak convergence for a spatial approximation of the nonlinear stochastic heat equation
We find the weak rate of convergence of the spatially semidiscrete finite
element approximation of the nonlinear stochastic heat equation. Both
multiplicative and additive noise is considered under different assumptions.
This extends an earlier result of Debussche in which time discretization is
considered for the stochastic heat equation perturbed by white noise. It is
known that this equation has a solution only in one space dimension. In order
to obtain results for higher dimensions, colored noise is considered here,
besides white noise in one dimension. Integration by parts in the Malliavin
sense is used in the proof. The rate of weak convergence is, as expected,
essentially twice the rate of strong convergence.Comment: 19 page
Discontinuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity
An integro-differential equation, modeling dynamic fractional order
viscoelasticity, with a Mittag-Leffler type convolution kernel is considered. A
discontinuous Galerkin method, based on piecewise constant polynomials is
formulated for temporal semidiscretization of the problem. Stability estimates
of the discrete problem are proved, that are used to prove optimal order a
priori error estimates. The theory is illustrated by a numerical example.Comment: 16 pages, 2 figure
Analytical solution for heat conduction due to a moving Gaussian heat flux with piecewise constant parameters
We provide an analytical solution of the heat equation in the half-space
subject to a moving Gaussian heat flux with piecewise constant parameters. The
solution is of interest in powder bed fusion applications where these
parameters can be used to control the conduction of heat due to a scanning beam
of concentrated energy. The analytical solution is written in a dimensionless
form as a sum of integrals over (dimensionless) time. For the numerical
computation of these integrals we suggest a quadrature scheme that utilizes
pre-calculated look-up tables for the required quadrature orders. Such a scheme
is efficient because the required quadrature orders are strongly dependent on
the parameters in the heat flux. The possibilities of using the obtained
computational technique for the control and optimization of powder bed fusion
processes are discussed
Optimal closing of a pair trade with a model containing jumps
A pair trade is a portfolio consisting of a long position in one asset and a
short position in another, and it is a widely applied investment strategy in
the financial industry. Recently, Ekstr\"om, Lindberg and Tysk studied the
problem of optimally closing a pair trading strategy when the difference of the
two assets is modelled by an Ornstein-Uhlenbeck process. In this paper we study
the same problem, but the model is generalized to also include jumps. More
precisely we assume that the above difference is an Ornstein-Uhlenbeck type
process, driven by a L\'evy process of finite activity. We prove a verification
theorem and analyze a numerical method for the associated free boundary
problem. We prove rigorous error estimates, which are used to draw some
conclusions from numerical simulations.Comment: 17 pages, 4 figures
Strong convergence of a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation
We consider the stochastic Cahn-Hilliard equation driven by additive Gaussian
noise in a convex domain with polygonal boundary in dimension . We
discretize the equation using a standard finite element method in space and a
fully implicit backward Euler method in time. By proving optimal error
estimates on subsets of the probability space with arbitrarily large
probability and uniform-in-time moment bounds we show that the numerical
solution converges strongly to the solution as the discretization parameters
tend to zero.Comment: 25 page
On a randomized backward Euler method for nonlinear evolution equations with time-irregular coefficients
In this paper we introduce a randomized version of the backward Euler method,
that is applicable to stiff ordinary differential equations and nonlinear
evolution equations with time-irregular coefficients. In the finite-dimensional
case, we consider Carath\'eodory type functions satisfying a one-sided
Lipschitz condition. After investigating the well-posedness and the stability
properties of the randomized scheme, we prove the convergence to the exact
solution with a rate of in the root-mean-square norm assuming only that
the coefficient function is square integrable with respect to the temporal
parameter.
These results are then extended to the numerical solution of
infinite-dimensional evolution equations under monotonicity and Lipschitz
conditions. Here we consider a combination of the randomized backward Euler
scheme with a Galerkin finite element method. We obtain error estimates that
correspond to the regularity of the exact solution. The practicability of the
randomized scheme is also illustrated through several numerical experiments.Comment: 37 pages, 3 figure
Full discretisation of semi-linear stochastic wave equations driven by multiplicative noise
A fully discrete approximation of the semi-linear stochastic wave equation
driven by multiplicative noise is presented. A standard linear finite element
approximation is used in space and a stochastic trigonometric method for the
temporal approximation. This explicit time integrator allows for mean-square
error bounds independent of the space discretisation and thus do not suffer
from a step size restriction as in the often used St\"ormer-Verlet-leap-frog
scheme. Furthermore, it satisfies an almost trace formula (i.e., a linear drift
of the expected value of the energy of the problem). Numerical experiments are
presented and confirm the theoretical results
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