Publications

The theory of rapid mixing random walks plays a fundamental role in the study of modern randomised algorithms. Usually, the mixing time is measured with respect to the worst initial position. It is well known that the presence of bottlenecks in a graph hampers mixing and, in particular, starting inside a small bottleneck significantly slows down the diffusion of the walk in the first steps of the process. The average mixing time is defined to be the mixing time starting at a uniformly random vertex and hence is not sensitive to the slow diffusion caused by these bottlenecks. In this paper we provide a general framework to show logarithmic average mixing time for random walks on graphs with small bottlenecks. The framework is especially effective on certain families of random graphs with heterogeneous properties. We demonstrate its applicability on two random models for which the mixing time was known to be of order (log n)2, speeding up the mixing to order logn. First, in the context of smoothed analysis on connected graphs, we show logarithmic average mixing time for randomly perturbed graphs of bounded degeneracy. A particular instance is the Newman-Watts small-world model. Second, we show logarithmic average mixing time for supercritically percolated expander graphs. When the host graph is complete, this application gives an alternative proof that the average mixing time of the giant component in the supercritical Erd˝os-Rényi graph is logarithmic.

Task specific hyperparameter tuning in reservoir computing is an open issue, and is of particular relevance for hardware implemented reservoirs. We investigate the influence of directly including externally controllable task specific timescales on the performance and hyperparameter sensitivity of reservoir computing approaches. We show that the need for hyperparameter optimisation can be reduced if timescales of the reservoir are tailored to the specific task. Our results are mainly relevant for temporal tasks requiring memory of past inputs, for example chaotic timeseries prediction. We consider various methods of including task specific timescales in the reservoir computing approach and demonstrate the universality of our message by looking at both time-multiplexed and spatially-multiplexed reservoir computing.

Current engineering design trends, such as lightweight machines and human-machine interaction, often lead to underactuated systems. Output trajectory tracking of such systems is a challenging control problem. Here, we use a two-design-degree of freedom control approach by combining funnel feedback control with feedforward control based on servo-constraints. We present experimental results to verify the approach and demonstrate that the addition of a feedforward controller mitigates drawbacks of the funnel controller. We also present new experimental results for the real-time implementation of a feedforward controller based on servo-constraints on a minimum phase system.

As a promising spectroscopic technique, surface enhanced raman spectroscopy (SERS) has been intensively used in bio/chemical sensing, attributing to its unique advantages of ultrasensitive and accurate detection of trace amounts of analytes, high specific fingerprint-like features, fast response, and noninvasive sensing. The robustness and consistency of SERS signals in practical analytical applications highly rely on the composition, structural morphology, and uniformity of SERS substrates. These factors play a pivotal role in determining the intensity and reproducibility of the detected signals. SERS substrates based on ordered nanostructures that are fabricated from anodic aluminum oxide (AAO)-template-assisted approaches are of significant interest due to their cost effectiveness, scalability, precise structural control, and exceptionally ordered features. In this review, recent progress in SERS substrates with high sensitivity and reproducibility prepared from AAO templates is highlighted. We emphasize the optimization strategies toward achieving efficient SERS-active substrates by fine-tuning the size, composition, and morphology of AAO-derived ordered nanostructures. Furthermore, we delve into the discussion of flexible and smart SERS substrates, while also exploring key aspects pertinent to further amplifying SERS signals. Overall, this review aims to offer insights into the future integration of the AAO templates technique with SERS, providing crucial perspectives for forthcoming research in this field.

Singularities in the fundamental solutions pose a mathematical challenge for all applications of the boundary element method, if the source and field point lie on the same element. To avoid complex and error-prone numerical procedures, analytical solutions for the integrals that arise are desirable. In this work, easy and efficiently to implement analytical solutions are presented for the fundamental solutions of the three-dimensional Stokes equation as well as Laplace’s equation. Explicit expressions are derived for general triangular elements using constant shape functions. In addition, options for extending to arbitrary polygonal elements are shown. In particular, the three cases that the incenter, the centroid or the vertices of the triangles are used as source points for the calculation are addressed. The impressive numerical efficiency of the method is demonstrated by explicit examples.