Filter off: No filter for categories. Modify filter
October 4 to 7, 2022: The workshop focuses on mathematical modeling, analysis and computer simulation of complex physical and biological processes and systems. The objective is to create an interdisciplinary workshop, which brings together international experts working on mathematical models from various fields. The aim is to get new innovative ideas and create synergies between the scientific research topics. Program: http://www-m6.ma.tum.de/~turova/html/MMSC2022_program.pdf; Details can be found at https://easychair.org/cfp/MMSC-2022.
Geometric singular perturbation theory provides a powerful mathematical framework for the analysis of stationary multiple time-scale systems possessing a critical manifold, particularly when combined with a method of desingularization known as blow-up. The theory for oscillatory multiple time-scale systems which possess a limit cycle manifold instead of (or in addition to) a critical manifold is less developed, particularly in the non-normally hyperbolic regime. In this talk, it is shown how the blow-up method can be applied to analyse the global oscillatory transition near a regular folded limit cycle manifold in a class of three time-scale systems with two small parameters. Furthermore, the applicability of the blow-up method to fast-slow maps is discussed via an embedding-based approach.
The Laplacian matrix of an undirected graph with positive edge weights encodes graph properties in matrix form. In this talk, we will discuss how Laplacian matrices appear prominently in multiple applications in statistics and machine learning. Our interest in Laplacian matrices originates in graphical models for extremes. For Hüsler--Reiss distributions, which are considered as an analogue of Gaussians in extreme value theory, they characterize an extremal notion of multivariate total positivity of order 2 (MTP2). This leads to a consistent estimation procedure with a typically sparse graphical structure. Furthermore, the underlying convex optimization problem under Laplacian constraints allows for a simple block descent algorithm that we implemented in R. An active area of research in machine learning are Laplacian-constrained Gaussian graphical models. These models admit structure learning under various connectivity constraints. Multiple algorithms for these problems with different lasso-type penalties are available in the literature. A surprising appearance of Laplacian matrices is in the design of discrete choice experiments. Here, the Fisher information of a discrete choice design is a Laplacian matrix, which gives rise to a new approach for learning locally D-optimal designs.
We consider a class of optimal liquidation problems where the agent's transactions create both temporary and transient price impact driven by a Volterra-type propagator. We formulate these problems as minimization of revenue-risk functionals, where the agent also exploits available information on a progressively measurable price predicting signal. By using an infinite dimensional stochastic control approach, we characterize the value function in terms of a solution to a free-boundary L^2-valued backward stochastic differential equation and an operator-valued Riccati equation. We then derive explicitly the optimal trading strategy by solving these equations. Our results also cover the case of singular price impact kernels, such as the power-law kernel.
Topological insulators are novel materials which are insulators in their bulk and conductors along their boundary. They are characterized by a topological invariant which is a continuous map from the space of Hamiltonians to Z or Z_2. Identifying this invariant with a measurable quantity (such as electric conductivity) is then a macroscopic form of quantization. In this talk I will discuss the problem of defining the ambient topology for this space of Hamiltonians so that the invariant is indeed continuous (and hence locally constant), as well as proving that its path components are bijective to the codomain Z or Z_2, in the regime when Hamiltonians have strong disorder.
Many complex processes exhibit transient dynamics - intriguing or important dynamical behaviors over a relatively long but finite time period. A fundamental issue is to understand transient dynamics of different mechanisms. In this talk, we focus on a class of randomly perturbed processes arising in chemical reactions and population dynamics where species only persist over finite time periods and go to extinction in the long run. To capture such transient persistent dynamics, we use quasi-stationary distributions (QSDs) and study their noise-vanishing asymptotic. Special attention will be paid to essential differences between models with and without environmental noises. The talk ends up with some discussions.
The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid algorithm run on noisy intermediate-scale quantum devices. It was originally proposed to solve combinatorial optimization problems. Finding the optimal QAOA parameters that allow to extract the (approximate) solution of the input problem instance is often a computationally expensive task. It has been observed in simulations that the QAOA output value concentrates for problem instances sampled from a reasonable distribution at fixed QAOA parameters. An analytic proof of the heuristically observed concentration over instances of the QAOA would significantly reduce the computational cost for general problems by allowing the transfer of optimal parameters between instances of the same distribution. So far, this concentration over instances has been proven analytically only for instances with independently and identically distributed edges of the instance graph. In this talk I will be talking about our various approaches to analytically prove concentration over instances for more general problem distributions.
Under an endogenous binary treatment with heterogeneous effects and multiple instruments, we propose a two-step procedure for identifying complier groups with identical local average treatment effects (LATE) despite relying on distinct instruments, even if several instruments violate the identifying assumptions. Our procedure is based on the fact that the LATE is homogeneous for instruments which (i) satisfy the LATE assumptions (instrument validity and treatment monotonicity in the instrument) and (ii) generate identical complier groups in terms of treatment propensities given the respective instruments. Under the plurality assumption that within each set of instruments with identical treatment propensities, instruments truly satisfying the LATE assumptions are the largest group, our procedure permits identifying these true instruments in a data driven way. We also provide a simulation study investigating the finite sample properties of our approach and an empirical application investigating the effect of incarceration on recidivism in the US with judge assignments serving as instruments.
This talk is concerned with generalized differentiability properties of solution operators of elliptic obstacle-type variational inequalities. We prove that such operators are semismooth when considered as maps between suitable Lebesgue spaces and equipped with the strong-weak Bouligand differential as a generalized set-valued derivative. It is shown that this semismoothness allows to solve optimal control problems with H1-cost terms and one-sided pointwise control constraints by means of a semismooth Newton method. The q-superlinear convergence of the resulting algorithm is established in the infinite-dimensional setting and its mesh independence is demonstrated in numerical experiments. The talk concludes with comments on further applications of the derived results in the context of quasi-variational inequalities and the optimal control of contact problems.
The (generalized) Derivative Nonlinear Schrödinger (DNLS) equation can be derived as an envelope equation via multiple scaling perturbation analysis from dispersive wave systems. It occurs when the cubic coefficient for the associated NLS equation vanishes for the spa- tial wave number of the underlying slowly modulated wave packet. It is the purpose of this talk to explain that the DNLS equation makes correct predictions about the dynamics of a Klein-Gordon model with a cubic nonlinearity. We present two proofs for this fact. The first one is based on energy estimates and normal form transformations. The second proof is based on the use of modulational Gevrey spaces. New difficulties occur due to a total resonance and due to a second order resonance. This is joint work with Max Hess.
In this talk, Marta will describe her work creating computational models of cardiac electrophysiology. She will show examples of how cardiac modelling can help understand disease processes and design personalised treatments for arrhythmias.
Physics-Informed Neural Networks (PINNs) can solve systems of differential equations and identify system parameters that best reproduce experimental measurements. Marta will talk about her work using PINNs in cardiac electrophysiology, showcasing how PINNs can characterise anti-arrhythmic drug effects and localise heart scar.