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Rough Semimartingales and -Variation Estimates for Martingale transforms
No full textWe establish a new scale of p-variation estimates for martingale paraproducts, martingale transforms and Itô integrals, of relevance in rough paths theory, stochastic and harmonic analysis. As an application, we introduce rough semimartingales, a common generalization of classical semimartingales and (controlled) rough paths and their integration theory
Stochastic augmented Lagrangian method in shape spaces
Get PDFIn this paper, we present a stochastic Augmented Lagrangian approach on (possibly infinite-dimensional) Riemannian manifolds to solve stochastic optimization problems with a finite number of deterministic constraints. We investigate the convergence of the method, which is based on a stochastic approximation approach with random stopping combined with an iterative procedure for updating Lagrange multipliers. The algorithm is applied to a multi-shape optimization problem with geometric constraints and demonstrated numerically
Second-order sufficient conditions for sparse optimal control of singular Allen--Cahn systems with dynamic boundary conditions
Get PDFIn this paper we study the optimal control of a parabolic initial-boundary value problem of Allen--Cahn type with dynamic boundary conditions. Phase field systems of this type govern the evolution of coupled diffusive phase transition processes with nonconserved order parameters that occur in a container and on its surface, respectively. It is assumed that the nonlinear functions driving the physical processes within the bulk and on the surface 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 leading to sparsity of optimal controls. For such cases, we derive second-order sufficient conditions for locally optimal controls
Magnetic Resonance Brain Imaging - Modeling and Data Analysis Using R: Code and Data
No full textThis is a collection of R code and additional data that can be used to execute all examples from the monograph "Magnetic Resonance Brain Imaging - Modeling and Data Analysis Using R" by Jörg Polzehl and Karsten Tabelow. The neuroimaging data used in this book is not provided here, but can be downloaded from the respective repositories using the code given in the Data Chapter. The additional data given here can be used to circumvent computationally expensive calculations and quickly get the results given in the book
Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging
No full textThis handbook gathers together the state of the art on mathematical models and algorithms for imaging and vision. Its emphasis lies on rigorous mathematical methods, which represent the optimal solutions to a class of imaging and vision problems, and on effective algorithms, which are necessary for the methods to be translated to practical use in various applications. Viewing discrete images as data sampled from functional surfaces enables the use of advanced tools from calculus, functions and calculus of variations, and nonlinear optimization, and provides the basis of high-resolution imaging through geometry and variational models. Besides, optimization naturally connects traditional model-driven approaches to the emerging data-driven approaches of machine and deep learning. No other framework can provide comparable accuracy and precision to imaging and vision. Written by leading researchers in imaging and vision, the chapters in this handbook all start with gentle introductions, which make this work accessible to graduate students. For newcomers to the field, the book provides a comprehensive and fast-track introduction to the content, to save time and get on with tackling new and emerging challenges. For researchers, exposure to the state of the art of research works leads to an overall view of the entire field so as to guide new research directions and avoid pitfalls in moving the field forward and looking into the next decades of imaging and information services. This work can greatly benefit graduate students, researchers, and practitioners in imaging and vision; applied mathematicians; medical imagers; engineers; and computer scientists
Two Entropic Finite Volume Schemes for a Nernst–Planck–Poisson System with Ion Volume Constraints
No full textModeling and simulation of ion transport in electrolytes is an important tool to investigate electrochemical devices as well as biological systems at the cell scale. Well designed models follow first principles of non-equilibrium thermodynamics and include the fact that ions have a finite size. It is highly desirable that these properties are valid as well for discretized models. In this contribution, we present two numerical fluxes for two-point flux finite volume schemes which fulfill these requirements. We review recent results on entropic behavior and convergence. Concluding, we present first simulation results for biological ion channels
High-Probability Bounds for Stochastic Optimization and Variational Inequalities: the Case of Unbounded Variance
No full textDuring recent years the interest of optimization and machine learning communities in high-probability convergence of stochastic optimization methods has been growing. One of the main reasons for this is that high-probability complexity bounds are more accurate and less studied than in-expectation ones. However, SOTA high-probability non-asymptotic convergence results are derived under strong assumptions such as the boundedness of the gradient noise variance or of the objective's gradient itself. In this paper, we propose several algorithms with high-probability convergence results under less restrictive assumptions. In particular, we derive new high-probability convergence results under the assumption that the gradient/operator noise has bounded central α-th moment for α∈(1,2] in the following setups: (i) smooth non-convex / Polyak-Lojasiewicz / convex / strongly convex / quasi-strongly convex minimization problems, (ii) Lipschitz / star-cocoercive and monotone / quasi-strongly monotone variational inequalities. These results justify the usage of the considered methods for solving problems that do not fit standard functional classes studied in stochastic optimization
Bias and multiscale correction methods for variational state estimation
Get PDFData assimilation performance can be significantly impacted by biased noise in observations, altering the signal magnitude and introducing fast oscillations or discontinuities when the system lacks smoothness. To mitigate these issues, this paper employ variational state estimation using the so-called parametrized-background data-weak method. This approach relies on a background manifold parametrized by a set of constraints, enabling the state estimation by solving a minimization problem on a reduced-order background model, subject to constraints imposed by the input measurements. The proposed formulation incorporates a novel bias correction mechanism and a manifold decomposition that handles rapid oscillations by treating them as slow-decaying modes based on a two-scale splitting of the classical reconstruction algorithm. The method is validated in different examples, including the assimilation of biased synthetic data, discontinuous signals, and Doppler ultrasound data obtained from experimental measurements
POD-ROMs for Incompressible Flows including Snapshots of the Temporal Derivative of the Full Order Solution: Error Bounds for the Pressure
No full textReduced order methods (ROMs) for the incompressible Navier–Stokes equations, based on proper orthogonal decomposition (POD), are studied that include snapshots which approach the temporal derivative of the velocity from a full order mixed finite element method (FOM). In addition, the set of snapshots contains the mean velocity of the FOM. Both the FOM and the POD-ROM are equipped with a grad-div stabilization. A velocity error analysis for this method can be found already in the literature. The present paper studies two different procedures to compute approximations to the pressure and proves error bounds for the pressure that are independent of inverse powers of the viscosity. Numerical studies support the analytic results and compare both methods
Rectifiable paths with polynomial log-signature are straight lines
Get PDFThe signature of a rectifiable path is a tensor series in the tensor algebra whose coefficients are definite iterated integrals of the path. The signature characterises the path up to a generalised form of reparametrisation. It is a classical result of K. T. Chen that the log-signature (the logarithm of the signature) is a Lie series. A Lie series is polynomial if it has finite degree. We show that the log-signature is polynomial if and only if the path is a straight line up to reparametrisation. Consequently, the log-signature of a rectifiable path either has degree one or infinite support. Though our result pertains to rectifiable paths, the proof uses results from rough path theory, in particular that the signature characterises a rough path up to reparametrisation