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We consider a coupled system of two elliptic PDEs, where the elliptic term in the first equation shares the properties of the $p(x)$-Laplacian with discontinuous exponent, while in the second equation we have to deal with an a~priori $L^1$ term on the right hand side. Such a system of equations is suitable for the description of various electrothermal effects, in particular those, where the non-Ohmic behavior can change dramatically with respect to the spatial variable. We prove the existence of a weak solution under very weak assumptions on the data and also under general structural assumptions on the constitutive equations of the model. The main difficulty consists in the fact that we have to overcome simultaneously two obstacles - the discontinuous variable exponent (which limits the use of standard methods) and the $L^1$ right hand side of the heat equation. Our existence proof based on Galerkin approximation is highly constructive and therefore seems to be suitable also for numerical purposes.
Trajectory- or mesh-based methods for analyzing the dynamical behavior of large
molecules tend to be impractical due to the curse of dimensionality - their computational cost increases
exponentially with the size of the molecule. We propose a method to break the curse by a
novel square root approximation of transition rates, Monte Carlo quadrature and a discretization
approach based on solving linear programs. With randomly sampled points on the molecular energy
landscape and randomly generated discretizations of the molecular conguration space as our initial
data, we construct a matrix describing the transition rates between adjacent discretization regions.
This transition rate matrix yields a Markov State Model of the molecular dynamics. We use Perron
cluster analysis and coarse-graining techniques in order to identify metastable sets in conguration
space and approximate the transition rates between the metastable sets. Application of our method
to a simple energy landscape on a two-dimensional conguration space provides proof of concept and
an example for which we compare the performance of dierent discretizations. We show that the
computational cost of our method grows only polynomially with the size of the molecule. However,
nding discretizations of higher-dimensional conguration spaces in which metastable sets can be
identied remains a challenge.
The chemical master equation is a fundamental equation in chemical kinetics. It underlies the classical reaction-rate equations and takes stochastic effects into account. In this paper we give a simple argument showing that the solutions of a large class of chemical master equations are bounded in weighted $\ell_1$-spaces and possess high-order moments. This class includes all equations in which no reactions between two or more already present molecules and further external reactants occur that add mass to the system. As an illustration for the implications of this kind of regularity, we analyze the effect of truncating the state space. This leads to an error analysis for the finite state projections of the chemical master equation, an approximation that forms the basis of many numerical methods.
The present work deals with the resolution of the Poisson equation in a bounded domain made of a thin and periodic layer of finite length placed into a homogeneous medium.
We provide and justify a high order asymptotic expansion which takes into account the boundary layer effect occurring in the vicinity of the periodic layer as well as the corner singularities appearing in the neighborhood of the extremities of the layer. Our approach combines mixes
the method of matched asymptotic expansions and the method of periodic surface homogenization.
We study non-isothermal nucleation and growth phase transformations, which are described by a generalized Avrami model for the phase transition coupled with an energy balance to account for recalescence
effects. The main novelty of our work is the identification of temperature dependent nucleation rates. We prove that such rates can be uniquely identified from measurements in a subdomain and apply an optimal control approach to develop a numerical strategy for its computation.
Let $\sigma_t(x)$ denote the implied volatility at maturity t for a strike $K = S_0 e^{x t}$, where $x \in R$ and $S_0$ is the current value of the underlying. We show that $\sigma_t(x)$ has a uniform (in $x$) limit as maturity t tends to infinity, given by the formula \sigma_{\infty}(x) = \sqrt2 (h^*(x)^{1/2} + (h^*(x) − x)^{1/2}, for $x$ in some compact neighbourhood of zero in the class of affine stochastic volatility models. The function $h^*$ is the convex dual of the limiting cumulant generating function $h$ of the scaled log-spot process. We express $h$ in terms of the functional characteristics of the underlying model. The proof of the limiting formula rests on the large deviation behaviour of the scaled log-spot process as time tends to infinity. We apply our results to obtain the limiting smile for several classes of stochastic volatility models with jumps used in applications (e.g. Heston with state-independent jumps, Bates with state-dependent jumps and Barndorff-Nielsen-Shephard model).
In this article we show, that the binding kinetics of a molecular system can be
identied by a projection of a continuous process onto a nite number of macro states. We thus
interpret binding kinetics as a projection. When projecting onto non-overlapping macro states the
Markovianity is spoiled. As a consequence, the description of e.g. a receptor-ligand system by a two
state kinetics is not accurate. By assigning a degree of membership to each state, we abandon the
non-overlapping approach. This overlap is crucial for a correct mapping of binding eects by Markov
State Models with regard to their long time behavior. It enables us to describe the highly discussed
rebinding eect, where the spatial arrangement of the system has the be included. By introducing
a \degree of fuzziness" we have an indicator for the strength of the rebinding eect, such that the
minimal rebinding eect can be derived from an optimization problem. The fuzziness also includes
some new paradigms for molecular kinetics. These new model paradigms show good agreement with
experimental data.
Markov State Models (MSMs) are widely used to represent molecular
conformational changes as jump-like transitions between subsets of the conformational
state space. However, the simulation of peptide folding in explicit water is
usually said to be unsuitable for the MSM framework. In this article, we summarize
the theoretical background of MSMs and indicate that explicit water simulations do
not contradict these principles. The algorithmic framework of a meshless conformational
space discretization is applied to an explicit water system and the sampling
results are compared to a long-term molecular dynamics trajectory. The meshless
discretization approach is based on spectral clustering of stochastic matrices (MSMs)
and allows for a parallelization of MD simulations. In our example of Trialanine we
were able to compute the same distribution of a long term simulation in less computing
time.
This paper is concerned with the distributed optimal control of a
time-discrete Cahn--Hilliard/Navier--Stokes system with variable
densities.
It focuses on the double-obstacle potential which yields an optimal
control problem for a family of coupled systems in each time instance of a
variational inequality of fourth order and the Navier--Stokes equation.
By proposing a suitable time-discretization, energy estimates are proved
and the existence of solutions to the primal system and of optimal
controls is established for the original problem as well as for a family
of regularized problems. The latter correspond to Moreau--Yosida type
approximations of the double-obstacle potential. The consistency of these
approximations is shown and first order optimality conditions for the
regularized problems are derived. Through a limit process, a stationarity
system for the original problem is established which is related to a
function space version of C-stationarity.
We propose a composite step method, designed for equality constrained optimization with partial differential equations. Focus is laid on the construction of a globalization scheme, which is based on cubic regularization of the objective and an affine covariant damped Newton method for feasibility. We show finite termination of the inner loop and fast local convergence of the algorithm. We discuss preconditioning strategies for the iterative solution of the arising linear systems with projected conjugate gradient. Numerical results are shown for optimal control problems subject to a nonlinear heat equation and subject to nonlinear elastic equations arising from an implant design problem in craniofacial surgery.
In a strong constant electric field, a dielectric particle
immersed in a weakly conducting fluid exhibits spontaneous rotations. This phenomenon is known under the name of the Quincke effect. In the original setup the particle was suspended on a silk thread and performed torsional oscillations of remarkably high amplitude.
We derive the governing equations for this experiment, and
ascertain that onset of oscillations from the quiescent state corresponds to the supercritical Hopf bifurcation.
For the case of a soft thread, we characterize the regime of large-scale torsional relaxation oscillations: explicit estimates are derived for their period and amplitude,
effects of bifurcation delay are described. In a stronger electric field, these relaxation oscillations yield to small-scale erratic rotations of the pendulum.
We study a mechanical equilibrium problem for a material consisting of two components with different densities, which allows to change the outer shape by changing the interface between the subdomains. We formulate the shape design problem of compensating unwanted workpiece
changes by controlling the interface, employ regularity results for transmission problems for a rigorous derivation of optimality conditions based on the speed method, and conclude with some numerical results based on a spline approximation of the interface.
In this paper we propose a time discretization of a system of two parabolic equations describing diffusion-driven atom rearrangement in crystalline matter. The equations express the balances of microforces and microenergy; the two phase fields are the order parameter and the chemical potential. The initial and boundary-value problem for the evolutionary system is known to be well posed. Convergence of the discrete scheme to the solution of the continuous problem is proved by a careful development of uniform estimates, by weak compactness and a suitable treatment of
nonlinearities. Moreover, for the difference of discrete
and continuous solutions we prove an error estimate of
order one with respect to the time step.
This note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. The system arises from a model of two-species phase segregation
on an atomic lattice [22]; it consists of the balance equations of microforces and microenergy; the two unknowns are the order parameter rho and the chemical potential mu. Some recent results obtained for this class of problems is reviewed and, in the case of a nonconstant and nonlinear atom mobility, uniqueness and continuous dependence on the initial data are shown with the help of a new line of
argumentation developed in [12].
We consider viscoelastic solids undergoing thermal expansion and exhibiting hysteresis effects due to plasticity or phase transformations. Within the framework of generalized standard solids, the problem is described in a 3D setting by the momentum equilibrium equation, the flow rule describing the dependence of the stress on the strain history, and the heat transfer equation. Under appropriate regularity assumptions on the data, a local existence result for this thermodynamically consistent system is established, by combining existence results for ordinary differential equations in Banach spaces with a fixed-point argument. Then global estimates are obtained by using both the classical energy estimate and more specific techniques for the heat equation introduced by Boccardo and Gallouet. Finally a global existence result is derived.
The Stefan problem is coupled with a spatially inhomogeneous and anisotropic Gibbs-Thomson condition at the phase boundary. We show the long-time existence of weak solutions for the non-degenerate Stefan problem with a
spatially inhomogeneous and anisotropic Gibbs-Thomson law and a conditional existence result for the corresponding degenerate Stefan problem. To this end, approximate solutions are constructed by means of variational problems
for energy functionals with spatially inhomogeneous and anisotropic interfacial energy. By passing to the limit, we establish solutions of the Stefan problem with a spatially inhomogeneous and anisotropic Gibbs--Thomson law in a weak generalized BV-formulation.
Uncertainty is inevitable when solving science and engineering application problems. In the face of
uncertainty, it is essential to determine robust and risk-averse solutions. In this work,
we consider a class of PDE-constrained optimization problems in which the PDE coefficients
and inputs may be uncertain. We introduce two approximations for minimizing the
conditional value-at-risk for such PDE-constrained optimization problems. These approximations are based
on the primal and dual formulations of the conditional value-at-risk. For the primal problem,
we introduce a smooth approximation of the conditional value-at-risk in order to utilize
derivative-based optimization algorithms and to take advantage of the convergence properties
of quadrature-based discretizations. For this smoothed conditional value-at-risk, we prove
differentiability as well as consistency of our approximation. For the dual problem, we
regularize the inner maximization problem, rigorously derive optimality conditions, and demonstrate
the consistency of our approximation. Furthermore, we propose a fixed-point iteration that takes
advantage of the structure of the regularized optimality conditions and provides a means of calculating
worst-case probability distributions based on the given probability level. We conclude with numerical
results.
This paper is concerned with the distributed optimal control of a
time-discrete Cahn--Hilliard/Navier--Stokes system with variable
densities.
It focuses on the double-obstacle potential which yields an optimal
control problem for a family of coupled systems in each time instance of a
variational inequality of fourth order and the Navier--Stokes equation.
By proposing a suitable time-discretization, energy estimates are proved
and the existence of solutions to the primal system and of optimal
controls is established for the original problem as well as for a family
of regularized problems. The latter correspond to Moreau--Yosida type
approximations of the double-obstacle potential. The consistency of these
approximations is shown and first order optimality conditions for the
regularized problems are derived. Through a limit process, a stationarity
system for the original problem is established which is related to a
function space version of C-stationarity.
With symmetric local absorbing boundary conditions for the Helmholtz equation scattering problems can be solved on a truncated domain, where the outgoing radiation condition is approximated by a Dirichlet-to-Neumann map with higher tangential derivatives on its outer boundary. Feng's conditions are symmetric local absorbing boundary conditions, which are based on an asymptotic expansion of the coefficients of the exact Dirichlet-to-Neumann map for large radia of the circular outer boundary. In this article we analyse the well-posedness of variational formulations with symmetric local absorbing boundary conditions in general and show how the modelling error introduced by Feng's conditions depends on the radius of the truncated domain.
Adaptive Numerical Solution of Eigenvalue Problems arising from Finite Element Models. AMLS vs. AFEM
(2015)
We discuss adaptive numerical methods for the solution of eigenvalue problems arising either
from the finite element discretization of a partial differential equation (PDE) or from discrete finite element modeling.
When a model is described by a partial differential equation, the adaptive finite element method
starts from a coarse finite element mesh which, based on a posteriori error estimators, is adaptively refined
to obtain eigenvalue/eigenfunction approximations of prescribed accuracy. This method is well established for classes of elliptic PDEs,
but is still in its infancy for more complicated PDE models.
For complex technical systems, the typical approach is to directly derive finite element models
that are discrete in space and are combined with macroscopic models to describe certain phenomena like damping or friction.
In this case one typically starts with a fine uniform mesh and computes eigenvalues and eigenfunctions using
projection methods from numerical linear algebra that are often combined with the algebraic multilevel
substructuring to achieve an adequate performance.
These methods work well in practice but their convergence and error analysis is rather difficult.
We analyze the relationship between these two extreme approaches. Both approaches have their pros and cons which are discussed in detail.
Our observations are demonstrated with several numerical examples.