Dynamical Systems Gradient method for solving nonlinear equations with monotone operators

A version of the Dynamical Systems Gradient Method for solving ill-posed nonlinear monotone operator equations is studied in this paper. A discrepancy principle is proposed and justified. A numerical experiment was carried out with the new stopping rule. Numerical experiments show that the proposed stopping rule is efficient. Equations with monotone operators are of interest in many applications.Comment: 2 figure

Asymptotic stability of solutions to abstract differential equations

An evolution problem for abstract differential equations is studied. The typical problem is: $$\dot{u}=A(t)u+F(t,u), \quad t\geq 0; \,\, u(0)=u_0;\quad \dot{u}=\frac {du}{dt}\qquad (*)$$ Here $A(t)$ is a linear bounded operator in a Hilbert space $H$, and $F$ is a nonlinear operator, $\|F(t,u)\|\leq c_0\|u\|^p,\,\,p>1$, $c_0, p=const>0$. It is assumed that Re$(A(t)u,u)\leq -\gamma(t)\|u\|^2$ $\forall u\in H$, where $\gamma(t)>0$, and the case when $\lim_{t\to \infty}\gamma(t)=0$ is also considered. An estimate of the rate of decay of solutions to problem (*) is given. The derivation of this estimate uses a nonlinear differential inequality

Stability of solutions to some evolution problem

Large time behavior of solutions to abstract differential equations is studied. The corresponding evolution problem is: $$\dot{u}=A(t)u+F(t,u)+b(t), \quad t\ge 0; \quad u(0)=u_0. \qquad (*)$$ Here $\dot{u}:=\frac {du}{dt}$, $u=u(t)\in H$, $t\in \R_+:=[0,\infty)$, $A(t)$ is a linear dissipative operator: Re$(A(t)u,u)\le -\gamma(t)(u,u)$, $\gamma(t)\ge 0$, $F(t,u)$ is a nonlinear operator, $\|F(t,u)\|\le c_0\|u\|^p$, $p>1$, $c_0,p$ are constants, $\|b(t)\|\le \beta(t),$ $\beta(t)\ge 0$ is a continuous function. Sufficient conditions are given for the solution $u(t)$ to problem (*) to exist for all $t\ge0$, to be bounded uniformly on $\R_+$, and a bound on $\|u(t)\|$ is given. This bound implies the relation $\lim_{t\to \infty}\|u(t)\|=0$ under suitable conditions on $\gamma(t)$ and $\beta(t)$. The basic technical tool in this work is the following nonlinear inequality: $$ \dot{g}(t)\leq -\gamma(t)g(t)+\alpha(t,g(t))+\beta(t),\ t\geq 0;\quad g(0)=g_0. $

A nonlinear inequality and evolution problems

Assume that $g(t)\geq 0$, and $$\dot{g}(t)\leq -\gamma(t)g(t)+\alpha(t,g(t))+\beta(t),\ t\geq 0;\quad g(0)=g_0;\quad \dot{g}:=\frac{dg}{dt},$$ on any interval $[0,T)$ on which $g$ exists and has bounded derivative from the right, $\dot{g}(t):=\lim_{s\to +0}\frac{g(t+s)-g(t)}{s}$. It is assumed that $\gamma(t)$, and $\beta(t)$ are nonnegative continuous functions of $t$ defined on $\R_+:=[0,\infty)$, the function $\alpha(t,g)$ is defined for all $t\in \R_+$, locally Lipschitz with respect to $g$ uniformly with respect to $t$ on any compact subsets$[0,T]$, $T<\infty$, and non-decreasing with respect to $g$, $\alpha(t,g_1)\geq \alpha(t,g_2)$ if $g_1\ge g_2$. If there exists a function $\mu(t)>0$, $\mu(t)\in C^1(\R_+)$, such that $$\alpha\left(t,\frac{1}{\mu(t)}\right)+\beta(t)\leq \frac{1}{\mu(t)}\left(\gamma(t)-\frac{\dot{\mu}(t)}{\mu(t)}\right),\quad \forall t\ge 0;\quad \mu(0)g(0)\leq 1,$$ then $g(t)$ exists on all of $\R_+$, that is $T=\infty$, and the following estimate holds: $$0\leq g(t)\le \frac 1{\mu(t)},\quad \forall t\geq 0.$$ If $\mu(0)g(0)< 1$, then $0\leq g(t)< \frac 1{\mu(t)},\quad \forall t\geq 0.$ A discrete version of this result is obtained. The nonlinear inequality, obtained in this paper, is used in a study of the Lyapunov stability and asymptotic stability of solutions to differential equations in finite and infinite-dimensional spaces

DSM of Newton type for solving operator equations F(u) = f with minimal smoothness assumptions on F.

This paper is a review of the authors’ results on the DSM (Dynamical Systems Method) for solving operator equation (*) F(u) = f. It is assumed that (*) is solvable. The novel feature of the results is the minimal assumption on the smoothness of F. It is assumed that F is continuously Fr´echet differentiable, but no smoothness assumptions on F0(u) are imposed. The DSM for solving equation (*) is developed. Under weak assumptions global existence of the solution u(t) is proved, the existence of u(1) is established, and the relation F(u(1)) = f is obtained. The DSM is developed for a stable solution of equation (*) when noisy data f are given, kf − f k

Stress-Constrained Topology Optimization with Application to the Design of Electrical Machines

Zweitveröffentlichung, ursprünglich veröffentlicht: Jonas Holley: Stress-Constrained Topology Optimization with Application to the Design of Electrical Machines. München: Verlag Dr. Hut, 2023, 199 Seiten, Dissertation Humboldt-Universität Berlin (2023). ISBN 978-3-8439-5378-8Während des Designprozesses physischer Gegenstände stellt die mechanische Stabilität in nahezu jedem Anwendungsbereich eine essentielle Anforderung dar. Stabilität kann mittels geeigneter Kriterien, die auf dem mechanischen Spannungstensor basieren, mathematisch quantifiziert werden. Dies dient dem Ziel der Vermeidung von Schädigung in jedem Punkt innerhalb des Gegenstands. Die vorliegende Arbeit behandelt die Entwicklung einer Methode zur Lösung von Designoptimierungsproblemen mit punktweisen Spannungsrestriktionen. Zunächst wird eine Regularisierung des Optimierungsproblems eingeführt, die einen zentralen Baustein für den Erfolg einer Lösungsmethode darstellt. Nach der Analyse des Problems hinsichtlich der Existenz von Lösungen wird ein Gradientenabstiegsverfahren basierend auf einer impliziten Designdarstellung und dem Konzept des topologischen Gradienten entwickelt. Da der entwickelte Ansatz eine Methode im Funktionenraum darstellt, ist die numerische Realisierung ein entscheidender Schritt in Richtung der praktischen Anwendung. Die Diskretisierung der Zustandsgleichung und der adjungierten Gleichung bildet die Basis für eine endlich-dimensionale Version des Optimierungsverfahrens. Im letzten Teil der Arbeit werden numerische Experimente durchgeführt, um die Leistungsfähigkeit des entwickelten Algorithmus zu bewerten. Zunächst wird das Problem des minimalen Volumens unter punktweisen Spannungsrestriktionen anhand der L-Balken Geometrie untersucht. Ein Schwerpunkt wird hierbei auf die Untersuchung der Regularisierung gelegt. Danach wird das multiphysikalische Design einer elektrischen Maschine adressiert. Zusätzlich zu den punktweisen Restriktionen an die mechanischen Spannungen wird die Maximierung des mittleren Drehmoments berücksichtigt, um das elektromagnetische Verhalten der Maschine zu optimieren. Der Erfolg der numerischen Tests demonstriert das Potential der entwickelten Methode in der Behandlung realistischer industrieller Problemstellungen.In the process of designing a physical object, the mechanical stability is an essential requirement in nearly every area of application. Stability can be quantified mathematically by suitable criteria based on the stress tensor, aiming at the prevention of damage in each point within the physical object. This thesis deals with the development of a framework for the solution of optimal design problems with pointwise stress constraints. First, a regularization of the optimal design problem is introduced. This perturbation of the original problem represents a central element for the success of a solution method. After analyzing the perturbed problem with respect to the existence of solutions, a line search type gradient descent scheme is developed based on an implicit design representation via a level set function. The core of the optimization method is provided by the topological gradient, which quantifies the effect of an infinitesimal small topological perturbation of a given design on an objective functional. Since the developed approach is a method in function space, the numerical realization is a crucial step towards its practical application. The discretization of the state and adjoint equation provide the basis for developing a finite-dimensional version of the optimization scheme. In the last part of the thesis, numerical experiments are conducted in order to assess the performance of the developed algorithm. First, the stress-constrained minimum volume problem for the L-Beam geometry is addressed. An emphasis is put on examining the effect of the proposed regularization. Afterwards, the multiphysical design of an electrical machine is addressed. In addition to the pointwise constraints on the mechanical stress, the maximization of the mean torque is considered in order to improve the electromagnetic performance of the machine. The success of the numerical tests demonstrate the potential of the developed design method in dealing with real industrial problems

The bracket geometry of statistics

In this thesis we build a geometric theory of Hamiltonian Monte Carlo, with an emphasis on symmetries and its bracket generalisations, construct the canonical geometry of smooth measures and Stein operators, and derive the complete recipe of measure-constraints preserving dynamics and diffusions on arbitrary manifolds. Specifically, we will explain the central role played by mechanics with symmetries to obtain efficient numerical integrators, and provide a general method to construct explicit integrators for HMC on geodesic orbit manifolds via symplectic reduction. Following ideas developed by Maxwell, Volterra, Poincaré, de Rham, Koszul, Dufour, Weinstein, and others, we will then show that any smooth distribution generates considerable geometric content, including ``musical" isomorphisms between multi-vector fields and twisted differential forms, and a boundary operator - the rotationnel, which, in particular, engenders the canonical Stein operator. We then introduce the ``bracket formalism" and its induced mechanics, a generalisation of Poisson mechanics and gradient flows that provides a general mechanism to associate unnormalised probability densities to flows depending on the score pointwise. Most importantly, we will characterise all measure-constraints preserving flows on arbitrary manifolds, showing the intimate relation between measure-preserving Nambu mechanics and closed twisted forms. Our results are canonical. As a special case we obtain the characterisation of measure-preserving bracket mechanical systems and measure-preserving diffusions, thus explaining and extending to manifolds the complete recipe of SGMCMC. We will discuss the geometry of Stein operators and extend the density approach by showing these are simply a reformulation of the exterior derivative on twisted forms satisfying Stokes' theorem. Combining the canonical Stein operator with brackets allows us to naturally recover the Riemannian and diffusion Stein operators as special cases. Finally, we shall introduce the minimum Stein discrepancy estimators, which provide a unifying perspective of parameter inference based on score matching, contrastive divergence, and minimum probability flow.Open Acces

Development of a SGM-based multi-view reconstruction framework for aerial imagery

Advances in the technology of digital airborne camera systems allow for the observation of surfaces with sampling rates in the range of a few centimeters. In combination with novel matching approaches, which estimate depth information for virtually every pixel, surface reconstructions of impressive density and precision can be generated. Therefore, image based surface generation meanwhile is a serious alternative to LiDAR based data collection for many applications. Surface models serve as primary base for geographic products as for example map creation, production of true-ortho photos or visualization purposes within the framework of virtual globes. The goal of the presented theses is the development of a framework for the fully automatic generation of 3D surface models based on aerial images - both standard nadir as well as oblique views. This comprises several challenges. On the one hand dimensions of aerial imagery is considerable and the extend of the areas to be reconstructed can encompass whole countries. Beside scalability of methods this also requires decent processing times and efficient handling of the given hardware resources. Moreover, beside high precision requirements, a high degree of automation has to be guaranteed to limit manual interaction as much as possible. Due to the advantages of scalability, a stereo method is utilized in the presented thesis. The approach for dense stereo is based on an adapted version of the semi global matching (SGM) algorithm. Following a hierarchical approach corresponding image regions and meaningful disparity search ranges are identified. It will be verified that, dependent on undulations of the scene, time and memory demands can be reduced significantly, by up to 90% within some of the conducted tests. This enables the processing of aerial datasets on standard desktop machines in reasonable times even for large fields of depth. Stereo approaches generate disparity or depth maps, in which redundant depth information is available. To exploit this redundancy, a method for the refinement of stereo correspondences is proposed. Thereby redundant observations across stereo models are identified, checked for geometric consistency and their reprojection error is minimized. This way outliers are removed and precision of depth estimates is improved. In order to generate consistent surfaces, two algorithms for depth map fusion were developed. The first fusion strategy aims for the generation of 2.5D height models, also known as digital surface models (DSM). The proposed method improves existing methods regarding quality in areas of depth discontinuities, for example at roof edges. Utilizing benchmarks designed for the evaluation of image based DSM generation we show that the developed approaches favorably compare to state-of-the-art algorithms and that height precisions of few GSDs can be achieved. Furthermore, methods for the derivation of meshes based on DSM data are discussed. The fusion of depth maps for 3D scenes, as e.g. frequently required during evaluation of high resolution oblique aerial images in complex urban environments, demands for a different approach since scenes can in general not be represented as height fields. Moreover, depths across depth maps possess varying precision and sampling rates due to variances in image scale, errors in orientation and other effects. Within this thesis a median-based fusion methodology is proposed. By using geometry-adaptive triangulation of depth maps depth-wise normals are extracted and, along the point coordinates are filtered and fused using tree structures. The output of this method are oriented points which then can be used to generate meshes. Precision and density of the method will be evaluated using established multi-view benchmarks. Beside the capability to process close range datasets, results for large oblique airborne data sets will be presented. The report closes with a summary, discussion of limitations and perspectives regarding improvements and enhancements. The implemented algorithms are core elements of the commercial software package SURE, which is freely available for scientific purposes