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Data-driven methods for quantitative imaging
Get PDFIn the field of quantitative imaging, the image information at a pixel or voxel in an underlying domain entails crucial information about the imaged matter. This is particularly important in medical imaging applications, such as quantitative Magnetic Resonance Imaging (qMRI), where quantitative maps of biophysical parameters can characterize the imaged tissue and thus lead to more accurate diagnoses. Such quantitative values can also be useful in subsequent, automatized classification tasks in order to discriminate normal from abnormal tissue, for instance. The accurate reconstruction of these quantitative maps is typically achieved by solving two coupled inverse problems which involve a (forward) measurement operator, typically ill-posed, and a physical process that links the wanted quantitative parameters to the reconstructed qualitative image, given some underlying measurement data. In this review, by considering qMRI as a prototypical application, we provide a mathematically-oriented overview on how data-driven approaches can be employed in these inverse problems eventually improving the reconstruction of the associated quantitative maps
Polynomial Volterra processes
Get PDFWe study the class of continuous polynomial Volterra processes, which we define as solutions to stochas- tic Volterra equations driven by a continuous semimartingale with affine drift and quadratic diffusion matrix in the state of the Volterra process. To demonstrate the versatility of possible state spaces within our framework, we construct polynomial Volterra processes on the unit ball. This construction is based on a stochastic invariance principle for stochastic Volterra equations with possibly singular kernels. Similarly to classical polynomial processes, polynomial Volterra processes allow for tractable expressions of the mo- ments in terms of the unique solution to a system of deterministic integral equations, which reduce to a system of ODEs in the classical case. By applying this observation to the moments of the finite-dimensional distributions we derive a uniqueness result for polynomial Volterra processes. Moreover, we prove that the moments are polynomials with respect to the initial condition, another crucial property shared by classical polynomial processes. The corresponding coefficients can be interpreted as a deterministic dual process and solve integral equations dual to those verified by the moments themselves. Additionally, we obtain a representation of the moments in terms of a pure jump process with killing, which corresponds to another non-deterministic dual process
Weighted mesh algorithms for general Markov decision processes: Convergence and tractability
Get PDFWe introduce a mesh-type approach for tackling discrete-time, finite-horizon Markov Decision Processes (MDPs) characterized by state and action spaces that are general, encompassing both finite and infinite (yet suitably regular) subsets of Euclidean space. In particular, for bounded state and action spaces, our algorithm achieves a computational complexity that is tractable in the sense of Novak & Wozniakowski, and is polynomial in the time horizon. For unbounded state space the algorithm is ``semi-tractable'' in the sense that the complexity is proportional to ε -c with some dimension independent c ≥ 2, for achieving an accuracy ε and polynomial in the time horizon with degree linear in the underlying dimension. As such the proposed approach has some flavor of the randomization method by Rust which deals with infinite horizon MDPs and uniform sampling in compact state space. However, the present approach is essentially different due to the finite horizon and a simulation procedure due to general transition distributions, and more general in the sense that it encompasses unbounded state space. To demonstrate the effectiveness of our algorithm, we provide illustrations based on Linear-Quadratic Gaussian (LQG) control problems
A generalized -convergence concept for a type of equilibrium problems
No full textA novel generalization of Γ-convergence applicable to a class of equilibrium problems is studied. After the introduction of the latter, a variety of its applications is discussed. The existence of equilibria with emphasis on Nash equilibrium problems is investigated. Subsequently, our Γ-convergence notion for equilibrium problems, generalizing the existing one from optimization, is introduced and discussed. The work ends with its application to a class of penalized generalized Nash equilibrium problems and quasi-variational inequalities
FSSH-2: Fewest Switches Surface Hopping with robust switching probability
No full textThis study introduces the FSSH-2 scheme, a redefined and numerically stable adiabatic Fewest Switches Surface Hopping (FSSH) method. It reformulates the standard FSSH hopping probability without non-adiabatic coupling vectors and allows for numerical time integration with larger step sizes. The advantages of FSSH-2 are demonstrated by numerical experiments for five different model systems in one and two spatial dimensions with up to three electronic states
Optical mode calculation in large-area photonic crystal surface-emitting lasers
No full textWe discuss algorithms and numerical challenges in constructing and resolving spectral prob- lems for photonic crystal surface-emitting lasers (PCSELs) with photonic crystal layers and large (up to several tens of mm2) emission areas. We show that finite difference schemes created using coarse numerical meshes provide sufficient accuracy for several major (lowest-threshold) modes of particular device designs. Our technique is applied to the example of large-area all- semiconductor PCSELs, showing how it can be used to optimize device performance
Representation formulas and far-field behavior of time-periodic flow past a body
Get PDFThis paper is concerned with integral representations and asymptotic expansions of solutions to the time-periodic incompressible Navier-Stokes equations for fluid flow in the exterior of a rigid body that moves with constant velocity. Using the time-periodic Oseen fundamental solution, we derive representation formulas for solutions with suitable regularity. From these formulas, the decomposition of the velocity component of the fundamental solution into steady-state and purely periodic parts and their detailed decay rate in space, we deduce complete information on the asymptotic structure of the velocity and pressure fields
Weak solutions to a model for phase separation coupled with finite-strain viscoelasticity subject to external distortion
Get PDFWe study the coupling of a viscoelastic deformation governed by a Kelvin--Voigt model at equilibrium, based on the concept of second-grade nonsimple materials, with a plastic deformation due to volumetric swelling, described via a phase-field variable subject to a Cahn--Hilliard model expressed in a Lagrangian frame. Such models can be used to describe the time evolution of hydrogels in terms of phase separation within a deformable substrate. The equations are mainly coupled via a multiplicative decomposition of the deformation gradient into both contributions and via a Korteweg term in the Eulerian frame. To treat time-dependent Dirichlet conditions for the deformation, an auxiliary variable with fixed boundary values is introduced, which results in another multiplicative structure. Imposing suitable growth conditions on the elastic and viscous potentials, we construct weak solutions to this quasistatic model as the limit of time-discrete solutions to incremental minimization problems. The limit passage is possible due to additional regularity induced by the hyperelastic and viscous stresses
On De Giorgi’s lemma for variational interpolants in metric and Banach spaces
Get PDFVariational interpolants are an indispensable tool for the construction of gradient-flow solutions via the Minimizing Movement Scheme. De Giorgi's lemma provides the associated discrete energy-dissipation inequality. It was originally developed for metric gradient systems. Drawing from this theory we study the case of generalized gradient systems in Banach spaces, where a refined theory allows us to extend the validity of the discrete energy-dissipation inequality and to establish it as an equality. For the latter we have to impose the condition of radial differentiability of the dissipation potential. Several examples are discussed to show how sharp the results are
Theory of valley-splitting in Si/SiGe spin-qubits: Interplay of strain, resonances and random alloy disorder
Get PDFElectron spin-qubits in silicon-germanium (SiGe) heterostructures are a major candidate for the realization of scalable quantum computers due to their long spin coherence times and compatibility with existing semiconductor fabrication techniques. A critical challenge in strained Si/SiGe quantum wells (QWs) is the existence of two nearly degenerate conduction band minima that can lead to leakage of quantum information. To address this issue, various strategies have been explored to enhance the valley splitting (i.e., the energy gap between the two low-energy conduction band minima), such as sharp interfaces, oscillating germanium concentrations in the QW (wiggle wells), and shear strain engineering. In this work, we develop a comprehensive envelope-function theory augmented by the empirical pseudopotential method to incorporate the effects of disorder, strain, and resonances arising from the symmetries of silicon’s crystal structure. We apply our model to analyze common epitaxial profiles studied in the literature and compare our results with previous work. This framework provides an efficient tool for quantifying the interplay of these effects on the valley splitting, enabling complex epitaxial optimization in future work