Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics
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Simulation of Single-Electron Shuttling for Spin-Qubit Transport in a SiGe Quantum Bus
No full textSpin qubits in gate-defined semiconductor quantum dots (QDs) are one of the major candidates for the realization of fault-tolerant universal quantum computers. Ongoing advances in the growth of SiGe heterostructures with isotopically purified 28Si quantum wells have enabled exceptionally long coherence times. Moreover, the compatibility with industry standard fabrication technology opens up excellent prospects for scaling up SiGe-based quantum processors to very large numbers of qubits. Recently, small-scale devices have been demonstrated, which execute one- and two-qubit logic gates as well as initialization and read-out operations with high fidelity using all-electrical control. The wiring and interconnection of large arrays of tunnel-coupled QDs, however, is a challenging problem as numerous control signals must be routed from external sources to every QD [1]. While control lines can be stacked in multiple layers, there are clear limitations in view of geometric constraints. A possible solution to this fan-out problem is partitioning of the qubit register into smaller QD arrays interconnected by coherent quantum links. Ref. [2] describes the design of such a scalable quantum bus, which allows to shuttle electrons using moving QDs along a one-dimensional channel in a conveyor belt mode, see Fig. 1. The quantum bus design provides sufficient space for QD wiring and classical on-chip control electronic
Building Ontologies and Knowledge Graphs for Mathematics and its Applications
No full textOntologies and knowledge graphs for mathematical algorithms and modelsare presented, that have been developed by the Mathematical Research Data Initiative.This enables FAIR data handling in mathematics and the applied disciplines. Moreover,challenges of harmonization during the ontology development are discussed
Algorithms for euclidean-regularised optimal transport
Get PDFThis paper addresses the Optimal Transport problem, which is regularized by the square of Euclidean -norm. It offers theoretical guarantees regarding the iteration complexities of the Sinkhorn--Knopp algorithm, Accelerated Gradient Descent, Accelerated Alternating Minimisation, and Coordinate Linear Variance Reduction algorithms. Furthermore, the paper compares the practical efficiency of these methods and their counterparts when applied to the entropy-regularized Optimal Transport problem. This comparison is conducted through numerical experiments carried out on the MNIST dataset
Nutrient control for a viscous Cahn–Hilliard–Keller–Segel model with logistic source describing tumor growth
Get PDFIn this paper, we address a distributed control problem for a system of partial differential equations describing the evolution of a tumor that takes the biological mechanism of chemotaxis into account. The system describing the evolution is obtained as a nontrivial combination of a Cahn--Hilliard type system accounting for the segregation between tumor cells and healthy cells, with a Keller--Segel type equation accounting for the evolution of a nutrient species and modeling the chemotaxis phenomenon. First, we develop a robust mathematical background that allows us to analyze an associated optimal control problem. This analysis forced us to select a source term of logistic type in the nutrient equation and to restrict the analysis to the case of two space dimensions. Then, the existence of an optimal control and first-order necessary conditions for optimality are established
Degree Distributions in Recursive Trees with Fitnesses
No full textWe study a general model of recursive trees where vertices are equipped with independent weights and at each time-step a vertex is sampled with probability proportional to its fitness function (a function of its weight and degree) and connects to new-coming vertices. Under a certain technical assumption, applying the theory of Crump-Mode-Jagers branching processes, we derive formulas for the almost sure limiting distribution of the proportion of vertices with a given degree and weight, and proportion of edges with endpoint having a certain weight. As an application of this theorem, we prove rigorously observations of Bianconi related to the evolving Cayley tree in []. We also study the process in depth when the technical condition can fail in the particular case when the fitness function is affine, a model we call "generalised preferential attachment with fitness". We show that this model can exhibit condensation where a positive proportion of edges accumulate around vertices with maximal weight, or, more drastically, have a degenerate limiting degree distribution where the entire proportion of edges accumulate around these vertices. Finally, we prove stochastic convergence for the degree distribution under a different assumption of a strong law of large numbers for the partition function associated with the process
Generative modelling with tensor train approximations of Hamilton-Jacobi-Bellman equations
Get PDFSampling from probability densities is a common challenge in fields such as Uncertainty Quantification (UQ) and Generative Modelling (GM). In GM in particular, the use of reverse-time diffusion processes depending on the log-densities of Ornstein-Uhlenbeck forward processes are a popular sampling tool. In [5] the authors point out that these log-densities can be obtained by solution of a Hamilton-Jacobi-Bellman (HJB) equation known from stochastic optimal control. While this HJB equation is usually treated with indirect methods such as policy iteration and unsuper-vised training of black-box architectures like Neural Networks, we propose instead to solve the HJB equation by direct time integration, using compressed polynomials represented in the Tensor Train (TT) format for spatial discretization. Crucially, this method is sample-free, agnostic to normalization constants and can avoid the curse of dimensionality due to the TT compression. We provide a complete derivation of the HJB equation?s action on Tensor Train polynomials and demonstrate the performance of the proposed time-step-, rank- and degree-adaptive integration method on a nonlinear sampling task in 20 dimensions
Approximating Langevin Monte Carlo with ResNet-like neural network architectures
Get PDFWe sample from a given target distribution by constructing a neural network which maps samples from a simple reference, e.g. the standard normal distribution, to samples from the target. To that end, we propose using a neural network architecture inspired by the Langevin Monte Carlo (LMC) algorithm. Based on LMC perturbation results, we show approximation rates of the proposed architecture for smooth, log-concave target distributions measured in the Wasserstein-2 distance. The analysis heavily relies on the notion of sub-Gaussianity of the intermediate measures of the perturbed LMC process. In particular, we derive bounds on the growth of the intermediate variance proxies under different assumptions on the perturbations. Moreover, we propose an architecture similar to deep residual neural networks and derive expressivity results for approximating the sample to target distribution map
Optimal stopping with randomly arriving opportunities to stop
Get PDFWe develop methods to solve general optimal stopping problems with opportunities to stop that arrive randomly. Such problems occur naturally in applications with market frictions. Pivotal to our approach is that our methods operate on random rather than deterministic time scales. This enables us to convert the original problem into an equivalent discrete-time optimal stopping problem with natural number valued stopping times and a possibly infinite horizon. To numerically solve this problem, we design a random times least squares Monte Carlo method. We also analyze an iterative policy improvement procedure in this setting. We illustrate the efficiency of our methods and the relevance of randomly arriving opportunities in a few examples
Hyperbolic-parabolic normal form and local classical solutions for cross-diffusion systems with incomplete diffusion
No full textWe investigate degenerate cross-diffusion equations with a rank-deficient diffusion matrix that are considered to model populations which move as to avoid spatial crowding and have recently been found to arise in a mean-field limit of interacting stochastic particle systems. To date, their analysis in multiple space dimensions has been confined to the purely convective case with equal mobility coefficients. In this article, we introduce a normal form for an entropic class of such equations which reveals their structure of a symmetric hyperbolic-parabolic system. Due to the state-dependence of the range and kernel of the singular diffusive matrix, our way of rewriting the equations is different from that classically used for symmetric second-order systems with a nullspace invariance property. By means of this change of variables, we solve the Cauchy problem for short times and positive initial data in H^s(mathbbT^d) for s>d/2+1
Essential enhancements in Abelian networks: Continuity and uniform strict monotonicity
No full textWe prove that in wide generality the critical curve of the activated random walk model is a continuous function of the deactivation rate, and we provide a bound on its slope, which is uniform with respect to the choice of the graph. Moreover, we derive strict monotonicity properties for the probability of a wide class of “increasing” events, extending previous results of (Invent. Math. 188 (2012) 127–150). Our proof method is of independent interest and can be viewed as a reformulation of the ‘essential enhancements’ technique, which was introduced for percolation, in the framework of abelian networks