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1
Content available remote A strong and weak approximation scheme for stochastic differential equations driven by a time-changed Brownian motion
Jum E., Kobayashi K.
Probability and Mathematical Statistics
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2016
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Vol. 36, Fasc. 2
201--220
EN
This paper establishes a discretization scheme for a large class of stochastic differential equations driven by a time-changed Brownian motion with drift, where the time change is given by a general inverse subordinator. The scheme involves two types of errors: one generated by application of the Euler-Maruyama scheme and the other ascribed to simulation of the inverse subordinator. With the two errors carefully examined, the orders of strong and weak convergence are established. In particular, an improved error estimate for the Euler-Maruyama scheme is derived, which is required to guarantee the strong convergence. Numerical examples are attached to support the convergence results.
2
Content available Dynamic programming approach to structural optimization problem - numerical algorithm
Fulmański P., Nowakowski A., Pustelnik J.
Opuscula Mathematica
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2014
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Vol. 34, no. 4
699--724
EN
In this paper a new shape optimization algorithm is presented. As a model application we consider state problems related to fluid mechanics, namely the Navier-Stokes equations for viscous incompressible fluids. The general approach to the problem is described. Next, transformations to classical optimal control problems are presented. Then, the dynamic programming approach is used and sufficient conditions for the shape optimization problem are given. A new numerical method to find the approximate value function is developed.
3
Content available remote A Dynamic Piezoelectric Contact Problem
Barboteu M., Sofonea M.
Machine Dynamics Problems
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2008
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Vol. 32, No. 1
23-32
EN
We consider a mathematical model, which describes the dynamic process of contact between a piezoelectric body and an electrically conductive foundation. The material's behavior is modeled with a nonlinear electro-viscoelastic constitutive law; the contact is frictionless and is described with the normal compliance condition and a regularized electrical conductivity condition. We state the variational formulation for the problem, and then we introduce a fully discrete scheme, based on the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the time derivatives. We implement this scheme in a numerical code and, in order to verify its accuracy, we present numerical simulations in the study of a two-dimensional test problem.
4
Content available remote Finite element error analysis for state-constrained optimal control of the Stokes equations
Los Reyes J. C. de, Meyer C., Vexler B.
Control and Cybernetics
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2008
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Vol. 37, no 2
251-284
EN
An optimal control problem for 2d and 3d Stokes equations is investigated with pointwise inequality constraints on the state and the control. The paper is concerned with the full discretization of the control problem allowing for different types of discretization of both the control and the state. For instance, piecewise linear and continuous approximations of the control are included in the present theory. Under certain assumptions on the L∞-error of the finite element discretization of the state, error estimates for the control are derived which can be seen to be optimal since their order of convergence coincides with the one of the interpolation error. The assumptions of the L∞-finite-eleinent-error can be verified for different numerical settings. Finally the results of two numerical experiments are presented.
5
Content available remote Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints
Meyer C.
Control and Cybernetics
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2008
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Vol. 37, no 1
51-83
EN
We consider a linear-quadratic elliptic optimal control problem with pointwise state constraints. The problem is fully discretized using linear ansatz functions for state and control. Based on a Slater-type argument, we investigate the approximation behavior for mesh size tending to zero. The obtained convergence order for the L²-error of the control and for H 1-error of the state is 1 - ε in the two-dimensional case and 1/2 - ε in three dimensions, provided that the domain satisfies certain regularity assumptions. In a second step, a state-constrained problem with additional control constraints is considered. Here, the control is discretized by constant ansatz functions. It is shown that the convergence theory can be adapted to this case yielding the same order of convergence. The theoretical findings are confirmed by numerical examples.
6
Content available remote A Model for Adhesive Frictional Contact
Sofonea M., Lerguet Z.
Machine Dynamics Problems
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2006
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Vol. 30, No. 2
158-168
EN
The aim of this paper is to present a mathematical model which describes the quasistatic process of adhesive frictional contact between a deformable body and an obstacle, the so-called foundation. The material's behavior is assumed to be elastic, with a nonlinear constitutive law; the adhesive contact is modelled with a surface variable, the bonding field, associated to the normal compliance condition and the static version of Coulomb's law of dry friction. We describe the assumptions which lead to the mathematical model of the process and derive a variational formulation of the problem; then, under a smallness assumption on the coefficient of friction, we prove the uniqueness of the solution for the model.
7
Content available Error estimates for the finite-element approximation of a semilinear elliptic control problem
Casas E., Troeltzsch F.
Control and Cybernetics
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2002
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Vol. 31, no 3
695-712
EN
We consider the finite-element approximation of a distributed optimal control problem governed by a semilinear elliptic partial differential equation, where pointwise constraints on the control are given. We prove the existence of local approximate solutions converging to a given local reference solution. Moreover, we derive error estimates for local solutions in the maximum norm.
8
Content available remote Numerical Analysis and Simulations of Quasistatic Frictionless Contact Problems
Fernandez Garcia J. R., Han W., Shillor M., Sofonea M.
International Journal of Applied Mathematics and Computer Science
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2001
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Vol. 11, no 1
205-222
EN
A summary of recent results concerning the modelling as well as the variational and numerical analysis of frictionless contact problems for viscoplastic materials are presented. The contact is modelled with the Signorini or normal compliance conditions. Error estimates for the fully discrete numerical scheme are described, and numerical simulations based on these schemes are reported.
9
Sieci neuronowe i eksperyment. Cz. 1
Polański Z.
LAB Laboratoria, Aparatura, Badania
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2000
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R. 5, nr 4
46-47
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