426 research outputs found
Reduced Order Optimal Control of the Convective FitzHugh-Nagumo Equation
In this paper, we compare three model order reduction methods: the proper
orthogonal decomposition (POD), discrete empirical interpolation method (DEIM)
and dynamic mode decomposition (DMD) for the optimal control of the convective
FitzHugh-Nagumo (FHN) equations. The convective FHN equations consists of the
semi-linear activator and the linear inhibitor equations, modeling blood
coagulation in moving excitable media. The semilinear activator equation leads
to a non-convex optimal control problem (OCP). The most commonly used method in
reduced optimal control is POD. We use DEIM and DMD to approximate efficiently
the nonlinear terms in reduced order models. We compare the accuracy and
computational times of three reduced-order optimal control solutions with the
full order discontinuous Galerkin finite element solution of the convection
dominated FHN equations with terminal controls. Numerical results show that POD
is the most accurate whereas POD-DMD is the fastest
A local energy-based discontinuous Galerkin method for fourth order semilinear wave equations
This paper generalizes the earlier work on the energy-based discontinuous
Galerkin method for second-order wave equations to fourth-order semilinear wave
equations. We first rewrite the problem into a system with a second-order
spatial derivative, then apply the energy-based discontinuous Galerkin method
to the system. The proposed scheme, on the one hand, is more computationally
efficient compared with the local discontinuous Galerkin method because of
fewer auxiliary variables. On the other hand, it is unconditionally stable
without adding any penalty terms, and admits optimal convergence in the
norm for both solution and auxiliary variables. In addition, the
energy-dissipating or energy-conserving property of the scheme follows from
simple, mesh-independent choices of the interelement fluxes. We also present a
stability and convergence analysis along with numerical experiments to
demonstrate optimal convergence for certain choices of the interelement fluxes
An energy-based discontinuous Galerkin method for the nonlinear Schr\"odinger equation with wave operator
This work develops an energy-based discontinuous Galerkin (EDG) method for
the nonlinear Schr\"odinger equation with the wave operator. The focus of the
study is on the energy-conserving or energy-dissipating behavior of the method
with some simple mesh-independent numerical fluxes we designed. We establish
error estimates in the energy norm that require careful selection of a test
function for the auxiliary equation involving the time derivative of the
displacement variable. A critical part of the convergence analysis is to
establish the L2 error bounds for the time derivative of the approximation
error in the displacement variable by using the equation that determines its
mean value. Using a specially chosen test function, we show that one can create
a linear system for the time evolution of the unknowns even when dealing with
nonlinear properties in the original problem. Extensive numerical experiments
are provided to demonstrate the optimal convergence of the scheme in the L2
norm with our choices of the numerical flux
Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview
Over the past few decades, there has been substantial interest in evolution
equations that involving a fractional-order derivative of order
in time, due to their many successful applications in
engineering, physics, biology and finance. Thus, it is of paramount importance
to develop and to analyze efficient and accurate numerical methods for reliably
simulating such models, and the literature on the topic is vast and fast
growing. The present paper gives a concise overview on numerical schemes for
the subdiffusion model with nonsmooth problem data, which are important for the
numerical analysis of many problems arising in optimal control, inverse
problems and stochastic analysis. We focus on the following aspects of the
subdiffusion model: regularity theory, Galerkin finite element discretization
in space, time-stepping schemes (including convolution quadrature and L1 type
schemes), and space-time variational formulations, and compare the results with
that for standard parabolic problems. Further, these aspects are showcased with
illustrative numerical experiments and complemented with perspectives and
pointers to relevant literature.Comment: 24 pages, 3 figure
Recommended from our members
Small Collaboration: Numerical Analysis of Electromagnetic Problems (hybrid meeting)
The classical theory of electromagnetism describes the interaction of electrically charged particles through electromagnetic forces, which are carried by the electric and magnetic fields. The propagation of the electromagnetic fields can be described by Maxwell's equations. Solving Maxwell's equations numerically is a challenging problem which appears in many different technical applications. Difficulties arise for instance from material interfaces or if the geometrical features are much larger than or much smaller than a typical wavelength. The spatial discretization needs to combine good geometrical flexibility with a relatively high order of accuracy.
The aim of this small-scale, week-long interactive mini-workshop jointly organized by the University of Duisburg-Essen and the University of Twente, and kindly hosted at the MFO, is to bring together experts in non-standard and mixed finite elements methods with experts in the field of electromagnetism
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