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Topology optimization is a valuable tool in engineering, facilitating the design of optimized structures. However, topological changes often require a remeshing step, which can become challenging. In this work, we propose an isogeometric…
We study the problem of uncertainty quantification for the numerical solution of elliptic partial differential equation boundary value problems posed on domains with stochastically varying boundaries. We also use the uncertainty…
A longstanding problem related to floating-point implementation of numerical programs is to provide efficient yet precise analysis of output errors. We present a framework to compute lower bounds on largest absolute roundoff errors, for a…
Identifying the topology underlying a set of time series is useful for tasks such as prediction, denoising, and data completion. Vector autoregressive (VAR) model-based topologies capture dependencies among time series and are often…
We prove error estimates for the semi-implicit numerical scheme of sphere-constrained high-index saddle dynamics, which serves as a powerful instrument in finding saddle points and constructing the solution landscapes of constrained systems…
This paper introduces a novel method for the automatic detection and handling of nonlinearities in a generic transformation. A nonlinearity index that exploits second order Taylor expansions and polynomial bounding techniques is first…
Performing highly accurate simulations of droplet systems is a challenging problem. This is primarily due to the interface dynamics which is complicated further by the addition of surfactants. This paper presents a boundary integral method…
With the increasing amount of data being visualized in large information spaces, methods providing data-driven context have become indispensable. Off-screen visualization techniques, therefore, have been extensively researched for their…
This article deals with a particular class of shape and topology optimization problems: the optimized design is a region $G$ of the boundary $\partial \Omega$ of a given domain $\Omega$, which supports a particular type of boundary…
If the non-zero finite floating-point numbers are interpreted as point intervals, then the effect of rounding can be interpreted as computing one of the bounds of the result according to interval arithmetic. We give an interval…
We present a formal model developed to reason about topologies created by sensor ranges. This model is used to formalise the topological aspects of an existing counting algorithm to estimate the number of targets in the area covered by the…
We introduce numerical algebraic geometry methods for computing lower bounds on the reach, local feature size, and the weak feature size of the real part of an equidimensional and smooth algebraic variety using the variety's defining…
Topological invariants, including the Chern numbers, can topologically classify parameterized Hamiltonians. We find that topological invariants can be properly defined and calculated even if the parameter space is discrete, which is done by…
We introduce a method for training neural networks to perform image or volume segmentation in which prior knowledge about the topology of the segmented object can be explicitly provided and then incorporated into the training process. By…
Topological deep learning is a formalism that is aimed at introducing topological language to deep learning for the purpose of utilizing the minimal mathematical structures to formalize problems that arise in a generic deep learning…
If we wish to integrate a function $h|\Omega\subset\Re^{n}\to\Re$ along a single $T$-level surface of a function $\psi |\Omega\subset\Re^{n}\to\Re$, then a number of different methods for extracting finite elements appropriate to the…
To facilitate widespread adoption of automated engineering design techniques, existing methods must become more efficient and generalizable. In the field of topology optimization, this requires the coupling of modern optimization methods…
Let M either be a closed real analytic Riemannian manifold or a closed smooth Riemannian surface. We estimate from below the volume of a nodal domain component in an arbitrary ball provided that this component enters the ball deeply enough.
The standard approach to characterizing topological matter, computing topological invariants, fails when the symmetry protecting the topological phase is preserved only on average in a disordered system. Because topological invariants rely…
Neural ordinary differential equations (neural ODE) are powerful continuous-time machine learning models for depicting the behavior of complex dynamical systems, but their verification remains challenging due to limited reachability…