Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics
Not a member yet
7600 research outputs found
Sort by
Synchronization Between Kerr Cavity Solitons and Broad Laser Pulse Injection
No full textThe synchronization of a soliton frequency comb in a Kerr cavity with pulsed laser injection is studied numerically. A neutral delay differential equation is used to model the light dynamics in the cavity. This model allows for the investigation of both cases where the pulse repetition period is close to the cavity round-trip time and where the repetition period of the injection pulses is close to a rational fraction M/N of the round-trip time. It is demonstrated that solitons can exist in this latter case, provided that the injection pulses are of a higher amplitude, which is directly proportional to the number M. Furthermore, it is shown that the synchronization range of the solitons is also proportional to the number M . The solitons excited by pulses with a period slightly different from the M : N -resonance can be destabilized by the Andronov--Hopf bifurcation, which occurs when the injection level at the soliton position decreases to M times the injection amplitude corresponding to the saddle-node bifurcation in a model equation with uniform injection
Optimality Conditions for Sparse Optimal Control of Viscous Cahn-Hilliard Systems with Logarithmic Potential
No full textIn this paper we study the optimal control of a parabolic initial-boundary value problem of viscous Cahn--Hilliard type with zero Neumann boundary conditions. Phase field systems of this type govern the evolution of diffusive phase transition processes with conserved order parameter. It is assumed that the nonlinear functions driving the physical processes within the spatial domain are double-well potentials of logarithmic type whose derivatives become singular at the boundary of their respective domains of definition. For such systems, optimal control problems have been studied in the past. We focus here on the situation when the cost functional of the optimal control problem contains a nondifferentiable term like the L1-norm, which leads to sparsity of optimal controls. For such cases, we establish first-order necessary and second-order sufficient optimality conditions for locally optimal controls. In the approach to second-order sufficient conditions, the main novelty of this paper, we adapt a technique introduced by Casas et al. in the paper (SIAM J Control Optim 53:2168–2202, 2015). In this paper, we show that this method can also be successfully applied to systems of viscous Cahn--Hilliard type with logarithmic nonlinearity. Since the Cahn--Hilliard system corresponds to a fourth-order partial differential equation in contrast to the second-order systems investigated before, additional technical difficulties have to be overcome
Towards a knowledge graph for models and algorithms in applied mathematics
Get PDFMathematical models and algorithms are an essential part of mathematical research data, as they are epistemically grounding numerical data. In order to represent models and algorithms as well as their relationship semantically to make this research data FAIR, two previously distinct ontologies were merged and extended, becoming a living knowledge graph. The link between the two ontologies is established by introducing computational tasks, as they occur in modeling, corresponding to algorithmic tasks. Moreover, controlled vocabularies are incorporated and a new class, distinguishing base quantities from specific use case quantities, was introduced. Also, both models and algorithms can now be enriched with metadata. Subject-specific metadata is particularly relevant here, such as the symmetry of a matrix or the linearity of a mathematical model. This is the only way to express specific workflows with concrete models and algorithms, as the feasible solution algorithm can only be determined if the mathematical properties of a model are known. We demonstrate this using two examples from different application areas of applied mathematics. In addition, we have already integrated over 250 research assets from applied mathematics into our knowledge graph
Regularity for non-autonomous parabolic equations with right-hand side singular measures involved
Get PDFThis article provides a theory for non-autonomous parabolic equations the right hand side of which includes singular measures - depending on the time parameter - on the spatial domain. In two space dimensions all bounded Radon measures are admissable as such. In higher dimensions the focus is on measures whose support is concentrated on l-sets in the sense of Jonsson and Wallin. It is shown that they may interpreted as elements from a Sobolev space W. So the right hand side is considered as an element from a W-valued Lebesgue space on the time interval. Having this at hand, previous results on maximal (non-autonomous) maximal parabolic regularity apply and show that the solution lies in the corresponding space of maximal parabolic regularity. In contrast to other work in this field we only require absolute minimal smothness for the data of the problem: the domain, the coefficients - and mixed boundary conditions are allowed. Under minimally stronger assumptions we even show the Hölder property in space and time. Overall, this work contains an interplay of geometric measure theory with advanced parabolic theory which delivers as much parabolic regularity for the solution as one can maximally expect
Existence and weak-strong uniqueness for damage systems in viscoelasticity
Get PDFIn this paper we investigate the existence of solutions and their weak-strong uniqueness property for a PDE system modelling damage in viscoelastic materials. In fact, we address two solution concepts, emphweak and emphstrong solutions. For the former, we obtain a global-in-time existence result, but the highly nonlinear character of the system prevents us from proving their uniqueness. For the latter, we prove local-in-time existence. Then, we show that the strong solution, as long as it exists, is unique in the class of weak solutions. This emphweak-strong uniqueness statement is proved by means of a suitable relative energy inequality
Hyperbolic relaxation of the chemical potential in the viscous Cahn--Hilliard equation
Get PDFIn this paper, we study a hyperbolic relaxation of the viscous Cahn–Hilliard system with zero Neumann boundary conditions. In fact, we consider a relaxation term involving the second time derivative of the chemical potential in the first equation of the system. We develop a well-posedness, continuous dependence and regularity theory for the initial-boundary value problem. Moreover, we investigate the asymptotic behavior of the system as the relaxation parameter tends to 0 and prove the convergence to the viscous Cahn–Hilliard system
Solvability and optimal control of a multi-species Cahn--Hilliard--Keller--Segel tumor growth model
Get PDFThis paper investigates an optimal control problem associated with a two-dimensional multispecies Cahn–Hilliard–Keller–Segel tumor growth model, which incorporates complex biological processes such as species diffusion, chemotaxis, angiogenesis, and nutrient consumption, resulting in a highly nonlinear system of nonlinear partial differential equations. The modeling derivation and corresponding analysis have been addressed in a previous contribution. Building on this foundation, the scope of this study involves investigating a distributed control problem with the goal of optimizing a tracking-type cost functional. This latter aims to minimize the deviation of tumor cell location from desired target configurations while penalizing the costs associated with implementing control measures, akin to introducing a suitable medication. Under appropriate mathematical assumptions, we demonstrate that sufficiently regular solutions exhibit continuous dependence on the control variable. Furthermore, we establish the existence of optimal controls and characterize the first-order necessary optimality conditions through a suitable variational inequality
Metaheuristic Algorithms for Enhancing Multicepstral Representation in Voice Spoofing Detection: An Experimental Approach
No full textThe problem of voice spoofing detection is critical for identity authentication within biometric systems. Among the existing countermeasures, those based on soft computing have received attention from researchers in the last few years. However, it is known that spoofing representation is only effective when many features are used, which limits its applicability due to the curse of dimensionality. Accordingly, we focus on strategies to reduce the dimensionality of multicepstral features while maintaining reasonable accuracy in distinguishing between real and spoofed voices. Given the complexity of voice data, identifying and prioritizing the features with the highest information content is of utmost relevance. The study utilized four metaheuristic algorithms-GA, DA, PSO, and GWO for dimension reduction. The findings indicate that all algorithms, particularly GWO, exceed baseline performance levels. This demonstrates their efficacy in detecting voice spoofing. Moreover, it was found that certain combinations of cepstral coefficients when applied with principal component analysis projection, notably enhanced the model’s performance of voice spoofing detection
WAW 2024
No full textThis book constitutes the refereed proceedings of the 19th International Workshop on Modelling and Mining Networks, WAW 2024, held in Warsaw, Poland, during June 3–6, 2024. The 12 full papers presented in this book were carefully reviewed and selected from 19 submissions. The aim of this workshop was to further the understanding of networks that arise in theoretical as well as applied domains. The goal was also to stimulate the development of high-performance and scalable algorithms that exploit these networks
A Parameter Robust Reconstruction Nonconforming Virtual Element Method for the Incompressible Poroelasticity Model
No full textA reconstruction nonconforming virtual element method for the incompressible poroelasticity model is developed and analyzed. We investigate to determine the divergence-free displacement in incompressible poroelasticity model on polygons. The presented method mainly involves the parameter robust for μ also can be considered as the real pressure robustness for the virtual element. Using the space on the polygons to build the reconstruction operator, the method can be applied on general polygonal meshes, satisfy pressure robustness and overcome Poisson locking in incompressible model. The results are corroborated by theoretical derivations as well as numerical results