Mathematik

The persistent mutual information (PMI) is a complexity measure for stochastic processes. It is related to well-known complexity measures like excess entropy or statistical complexity. Essentially it is a variation of the excess entropy so that it can be interpreted as a specific measure of system internal memory. The PMI was first introduced in 2010 by Ball, Diakonova and MacKay as a measure for (strong) emergence. In this paper we define the PMI mathematically and investigate the relation to excess entropy and statistical complexity. In particular we prove that the excess entropy is an upper bound of the PMI. Furthermore we show some properties of the PMI and calculate it explicitly for some example processes. We also discuss to what extend it is a measure for emergence and compare it with alternative approaches used to formalize emergence.

Inferring the causal direction and causal effect between two discrete random variables X and Y from a finite sample is often a crucial problem and a challenging task. However, if we have access to observational and interventional data, it is possible to solve that task. If X is causing Y, then it does not matter if we observe an effect in Y by observing changes in X or by intervening actively on X. This invariance principle creates a link between observational and interventional distributions in a higher dimensional probability space. We embed distributions that originate from samples of X and Y into that higher dimensional space such that the embedded distribution is closest to the distributions that follow the invariance principle, with respect to the relative entropy. This allows us to calculate the best information-theoretic approximation for a given empirical distribution, that follows an assumed underlying causal model. We show that this information-theoretic approximation to ...

The persistent mutual information (PMI) is a complexity measure for stochastic processes. It is related to well-known complexity measures like excess entropy or statistical complexity. Essentially it is a variation of the excess entropy so that it can be interpreted as a specific measure of system internal memory. The PMI was first introduced in 2010 by Ball, Diakonova and MacKay as a measure for (strong) emergence. In this paper we define the PMI mathematically and investigate the relation to excess entropy and statistical complexity. In particular we prove that the excess entropy is an upper bound of the PMI. Furthermore we show some properties of the PMI and calculate it explicitly for some example processes. We also discuss to what extend it is a measure for emergence and compare it with alternative approaches used to formalize emergence.

Mathematical models use information from past observations to generate predictions about the future. If two models make identical predictions the one that needs less information from the past to do this is preferred. It is already known that certain classical models (certain Hidden Markov Models called ǫ-machines which are often optimal classical models) are not in general the preferred ones. We extend this result and show that even optimal classical models (models with minimal internal entropy) in general are not the best possible models (called ideal models). Instead of optimal classical models we can construct quantum models which are significantly better but not yet the best possible ones (i.e. they have a strictly smaller internal entropy). In this paper we show conditions when the internal entropies between classical models and specific quantum models coincide. Furthermore it turns out that this situation appears very rarely. An example shows that our results hold only for the specific quantum model construction and in general not for alternative constructions. Furthermore another example shows that classical models with minimal internal entropy need not to be related to quantum models with minimal internal entropy.

Mathematical models use information from past observations to generate predictions about the future. If two models make identical predictions the one that needs less information from the past to do this is preferred. It is already known that certain classical models (certain Hidden Markov Models called ǫ-machines which are often optimal classical models) are not in general the preferred ones. We extend this result and show that even optimal classical models (models with minimal internal entropy) in general are not the best possible models (called ideal models). Instead of optimal classical models we can construct quantum models which are significantly better but not yet the best possible ones (i.e. they have a strictly smaller internal entropy). In this paper we show conditions when the internal entropies between classical models and specific quantum models coincide. Furthermore it turns out that this situation appears very rarely. An example shows that our results hold only for the specific quantum model construction and in general not for alternative constructions. Furthermore another example shows that classical models with minimal internal entropy need not to be related to quantum models with minimal internal entropy.

We consider a divining rod as an exemplaric network of elastic homogeneous strings. If the lengths of the strings are rationally dependent, it is known that even approximate controllability by a single boundary control fails, whenever the other two simple nodes satisfy the same boundary condition. In this paper we give a positive answer to the question whether exact controllability for some class of initial/final data holds, if the individual lengths of the strings are no longer rationally dependent. In order to do this, we resort to a special class of algebraic numbers, namely Roth's class. However, we do not use Fourier series expansions but rather d'Alembert's formula to analyze the propagation of the effect of the controllers along the network. *

A new method for the efficient solution of free material optimization problems is introduced. The method extends the sequential convex programming (SCP) concept to a class of optimization problems with matrix variables. The basic idea of the new method is to approximate the original optimization problem by a sequence of sub-problems, in which nonlinear functions (defined in matrix variables) are approximated by blockseparable, convex functions. The subproblems are semidefinite programs with a favorable structure, which can be efficiently solved by existing SDP software. The new method is shown to be globally convergent. The article is concluded by a series of numerical experiments demonstrating the effectiveness of the generalized SCP approach. 1 real-world applications. Nevertheless, the dual semidefinite approach has two major disadvantages. First of all, the computational complexity of the method depends cubically on the number of load cases . This makes the approach unpractical for 3D problems with more than a few (typically 3-5) load cases. Moreover, it is almost impossible to apply the dual approach to extended (multi-disciplinary) FMO problems. This is a serious drawback, as additional constraints as, for instance, displacement-based constraints play an important role in many real-world applications (compare [9, 13]). A direct treatment of the primal problem seems to avoid both of these difficulties, but unfortunately no successful algorithmic concept has been found for the solution of this problem so far.

Today, the prevention and treatment of voice disorders is an ever-increasing health concern. Since many occupations rely on verbal communication, vocal health is necessary just to maintain one's livelihood. Commonly applied models to study vocal fold vibrations and air flow distributions are self sustained physical models of the larynx composed of artificial silicone vocal folds. Choosing appropriate mechanical parameters for these vocal fold models while considering simplifications due to manufacturing restrictions is difficult but crucial for achieving realistic behavior. In the present work, a combination of experimental and numerical approaches to compute material parameters for synthetic vocal fold models is presented. The material parameters are derived from deformation behaviors of excised human larynges. The resulting deformations are used as reference displacements for a tracking functional to be optimized. Material optimization was applied to three-dimensional vocal fold models based on isotropic and transverse-isotropic material laws, considering both a layered model with homogeneous material properties on each layer and an inhomogeneous model. The best results exhibited a transversal-isotropic inhomogeneous (i.e., not producible) model. For the homogeneous model (three layers), the transversal-isotropic material parameters were also computed for each layer yielding deformations similar to the measured human vocal fold deformations.

In this article, we present the Free Material Optimization (FMO) problem for plates and shells based on Naghdi’s shell model. In FMO – a branch of structural optimization – we search for the ultimately best material properties in a given design domain loaded by a set of given forces. The optimization variable is the full material tensor at each point of the design domain. We give a basic formulation of the problem and prove existence of an optimal solution. Lagrange duality theory allows to identify the basic problem as the dual of an infinite-dimensional convex nonlinear semidefinite program. After discretization by the finite element method the latter problem can be solved using a nonlinear SDP code. The article is concluded by a few numerical studies.

The goal of this paper is to formulate and solve free material optimization problems with constraints on the smallest eigenfrequency of the optimal structure. A natural formulation of this problem as linear semidefinite program turns out to be numerically intractable. As an alternative, we propose a new approach, which is based on a nonlinear semidefinite low-rank approximation of the semidefinite dual. We introduce an algorithm based on this approach and analyze its convergence properties. The article is concluded by numerical experiments proving the effectiveness of the new approach.

We present a mathematical framework for the so-called multidisciplinary free material optimization (MDFMO) problems, a branch of structural optimization in which the full material tensor is considered as a design variable. We extend the original problem statement by a class of generic constraints depending either on the design or on the state variables. Examples are, for instance, local stress or displacement constraints. We show the existence of optimal solutions for this generalized FMO problem and discuss convergent approximation schemes based on the finite element method.

We consider the problem of exactly controlling the states of the de St. Venant equations from a given constant state to another constant state by applying nonlinear boundary controls. During this transition the solution stays in the class of C 1 -solutions. There are no restrictions on the distance between the initial state and the target state, so our result is a global controllability result for a nonlinear hyberbolic system.  2003 Éditions scientifiques et médicales Elsevier SAS MSC: 93C20; 93B05; 76B75

We consider a tree-like network of open channels with outflow at the root. Controls are exerted at the boundary nodes of the network except for the root. In each channel, the flow is modelled by the de St. Venant equations. The node conditions require the conservation of mass and the conservation of energy. We show that the states of the system can be controlled within the entire network in finite time from a stationary supercritical initial state to a given supercritical terminal state with the same orientation. During this transition, the states stay in the class of C1-functions, so no shocks occur. Copyright 2004 John Wiley & Sons, Ltd.

We study problems of boundary controllability with minimal L p -norm (p ∈ [2, ∞]) for the one-dimensional wave equation, where the state is controlled at both boundaries through Dirichlet or Neumann conditions. The problem is to reach a given terminal state and velocity in a given finite time, while minimizing the L p -norm of the controls. We give necessary and sufficient conditions for the solvability of this problem. We show as follows how this infinite-dimensional optimization problem can be transformed into a problem which is much simpler: The feasible set of the transformed problem is described by a finite number of simple pointwise equality constraints for the control function in the Dirichlet case while, in the Neumann case, an additional integral equality constraint appears. We provide explicit complete solutions of the problems for all p ∈ [2, ∞] in the Dirichlet case and solutions for some typical examples in the Neumann case.

The Workshop Optimal Control of Coupled Systems of PDE was held from April 17th -April 23rd, 2005 in the Mathematisches Forschungsinstitut Oberwolfach. The scientific program covered various topics such as controllability, feedback control, optimality conditions,analysis and control of Navier-Stokes equations, model reduction of large systems, optimal shape design, and applications in crystal growth, chemical reactions and aviation.

The Workshop Optimal Control of Coupled Systems of PDE was held from April 17th -April 23rd, 2005 in the Mathematisches Forschungsinstitut Oberwolfach. The scientific program covered various topics such as controllability, feedback control, optimality conditions,analysis and control of Navier-Stokes equations, model reduction of large systems, optimal shape design, and applications in crystal growth, chemical reactions and aviation.

In this work, we consider networks of so-called geometrically exact beams, namely, shearable beams that may undergo large motions. The corresponding mathematical model, commonly written in terms of displacements and rotations expressed in a fixed basis (Geometrically Exact Beam model, or GEB), has a quasilinear governing system. However, the model may also be written in terms of intrinsic variables expressed in a moving basis attached to the beam (Intrinsic GEB model, or IGEB) and while the number of equations is then doubled, the latter model has the advantage of being of first-order, hyperbolic and only semilinear. First, for any network, we show the existence and uniqueness of semi-global in time classical solutions to the IGEB model (i.e., for arbitrarily large time intervals, provided that the data are small enough). Then, for a specific network containing a cycle, we address the problem of local exact controllability of nodal profiles for the IGEB model-we steer the solution to satisfy given profiles at one of the multiple nodes by means of controls applied at the simple nodes-by using the constructive method of Zhuang, Leugering and Li [Exact boundary controllability of nodal profile for Saint-Venant system on a network with loops, in J. Math. Pures Appl., 2018]. Afterwards, for any network, we show that the existence of a unique classical solution to the IGEB network implies the same for the corresponding GEB network, by using that these two models are related by a nonlinear transformation. In particular, this allows us to give corresponding existence, uniqueness and controllability results for the GEB network. Contents

A systematic approach to maximise estimates on the region of attraction in the exponential stabilisation of geometrically exact (nonlinear) beam models via boundary feedback is presented. Starting from recently established stability results based on Lyapunov arguments, the main contribution of the presented work is to maximise the analytically found bounds on the initial datum, for which local exponential stability is guaranteed, via search of (optimal) polynomial Lyapunov functionals using an iterative semi-definite programming approach.

We study a Dirichlet optimal design problem for a quasi-linear monotone p-biharmonic equation with control and state constraints. We take the coefficient of the p-biharmonic operator as a design variable in () BV Ω. In this article, we discuss the relaxation of such problem.