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Splines over triangulations and splines over quadrangulations (tensor product splines) are two common ways to extend bivariate polynomials to splines. However, combination of both approaches leads to splines defined over mixed triangle and quadrilateral meshes using the isogeometric approach. Mixed meshes are especially useful for representing complicated geometries obtained e.g. from trimming. As (bi-)linearly parameterized mesh elements are not flexible enough to cover smooth domains, we focus in this work on the case of planar mixed meshes parameterized by (bi-)quadratic geometry mappings. In particular we study in detail the space of $C^1$-smooth isogeometric spline functions of general polynomial degree over two such mixed mesh elements. We present the theoretical framework to analyze the smoothness conditions over the common interface for all possible configurations of mesh elements. This comprises the investigation of the dimension as well as the construction of a basis of the corresponding $C^1$-smooth isogeometric spline space over the domain described by two elements. Several examples of interest are presented in detail.

Non-linear model predictive control (nMPC) is a powerful approach to control complex robots (such as humanoids, quadrupeds, or unmanned aerial manipulators (UAMs)) as it brings important advantages over other existing techniques. The full-body dynamics, along with the prediction capability of the optimal control problem (OCP) solved at the core of the controller, allows to actuate the robot in line with its dynamics. This fact enhances the robot capabilities and allows, e.g., to perform intricate maneuvers at high dynamics while optimizing the amount of energy used. Despite the many similarities between humanoids or quadrupeds and UAMs, full-body torque-level nMPC has rarely been applied to UAMs. This paper provides a thorough description of how to use such techniques in the field of aerial manipulation. We give a detailed explanation of the different parts involved in the OCP, from the UAM dynamical model to the residuals in the cost function. We develop and compare three different nMPC controllers: Weighted MPC, Rail MPC, and Carrot MPC, which differ on the structure of their OCPs and on how these are updated at every time step. To validate the proposed framework, we present a wide variety of simulated case studies. First, we evaluate the trajectory generation problem, i.e., optimal control problems solved offline, involving different kinds of motions (e.g., aggressive maneuvers or contact locomotion) for different types of UAMs. Then, we assess the performance of the three nMPC controllers, i.e., closed-loop controllers solved online, through a variety of realistic simulations. For the benefit of the community, we have made available the source code related to this work.

A grid-overlay finite difference method is proposed for the numerical approximation of the fractional Laplacian on arbitrary bounded domains. The method uses an unstructured simplicial mesh and an overlay uniform grid for the underlying domain and constructs the approximation based on a uniform-grid finite difference approximation and a data transfer from the unstructured mesh to the uniform grid. The method takes full advantage of both uniform-grid finite difference approximation in efficient matrix-vector multiplication via the fast Fourier transform and unstructured meshes for complex geometries. It is shown that its stiffness matrix is similar to a symmetric and positive definite matrix and thus invertible if the data transfer has full column rank and positive column sums. Piecewise linear interpolation is studied as a special example for the data transfer. It is proved that the full column rank and positive column sums of linear interpolation is guaranteed if the spacing of the uniform grid is smaller than or equal to a positive bound proportional to the minimum element height of the unstructured mesh. Moreover, a sparse preconditioner is proposed for the iterative solution of the resulting linear system for the homogeneous Dirichlet problem of the fractional Laplacian. Numerical examples demonstrate that the new method has similar convergence behavior as existing finite difference and finite element methods and that the sparse preconditioning is effective. Furthermore, the new method can readily be incorporated with existing mesh adaptation strategies. Numerical results obtained by combining with the so-called MMPDE moving mesh method are also presented.

With the advent of standards for deterministic network behavior, synthesizing network designs under delay constraints becomes the natural next task to tackle. Network Calculus (NC) has become a key method for validating industrial networks, as it computes formally verified end-to-end delay bounds. However, analyses from the NC framework have been designed to bound the delay of one flow at a time. Attempts to use classical analyses to derive a network configuration have shown that this approach is poorly suited to practical use cases. Consider finding a delay-optimal routing configuration: one model had to be created for each routing alternative, then each flow delay had to be bounded, and then the bounds had to be compared to the given constraints. To overcome this three-step process, we introduce Differential Network Calculus. We extend NC to allow the differentiation of delay bounds w.r.t. to a wide range of network parameters - such as flow paths or priority. This opens up NC to a class of efficient nonlinear optimization techniques that exploit the gradient of the delay bound. Our numerical evaluation on the routing and priority assignment problem shows that our novel method can synthesize flow paths and priorities in a matter of seconds, outperforming existing methods by several orders of magnitude.

The proximal Galerkin finite element method is a high-order, nonlinear numerical method that preserves the geometric and algebraic structure of bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free-boundary problems, enforce discrete maximum principles, and develop scalable, mesh-independent algorithms for optimal design. The paper begins with a derivation of the latent variable proximal point (LVPP) method: an unconditionally stable alternative to the interior point method. LVPP is an infinite-dimensional optimization algorithm that may be viewed as having an adaptive (Bayesian) barrier function that is updated with a new informative prior at each (outer loop) optimization iteration. One of the main benefits of this algorithm is witnessed when analyzing the classical obstacle problem. Therein, we find that the original variational inequality can be replaced by a sequence of semilinear partial differential equations (PDEs) that are readily discretized and solved with, e.g., high-order finite elements. Throughout this work, we arrive at several unexpected contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field density-based topology optimization. The complete latent variable proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis.

We present a novel multilevel Monte Carlo approach for estimating quantities of interest for stochastic partial differential equations (SPDEs). Drawing inspiration from [Giles and Szpruch: Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without L\'evy area simulation, Annals of Appl. Prob., 2014], we extend the antithetic Milstein scheme for finite-dimensional stochastic differential equations to Hilbert space-valued SPDEs. Our method has the advantages of both Euler and Milstein discretizations, as it is easy to implement and does not involve intractable L\'evy area terms. Moreover, the antithetic correction in our method leads to the same variance decay in a MLMC algorithm as the standard Milstein method, resulting in significantly lower computational complexity than a corresponding MLMC Euler scheme. Our approach is applicable to a broader range of non-linear diffusion coefficients and does not require any commutative properties. The key component of our MLMC algorithm is a truncated Milstein-type time stepping scheme for SPDEs, which accelerates the rate of variance decay in the MLMC method when combined with an antithetic coupling on the fine scales. We combine the truncated Milstein scheme with appropriate spatial discretizations and noise approximations on all scales to obtain a fully discrete scheme and show that the antithetic coupling does not introduce an additional bias.

In this paper we propose a method to approximate the Gaussian function on ${\mathbb R}$ by a short cosine sum. We extend the differential approximation method proposed in [4,39] to approximate $\mathrm{e}^{-t^{2}/2\sigma}$ in the weighted space $L_2({\mathbb R}, \mathrm{e}^{-t^{2}/2\rho})$ where $\sigma, \, \rho >0$. We prove that the optimal frequency parameters $\lambda_1, \ldots , \lambda_{N}$ for this method in the approximation problem $ \min\limits_{\lambda_{1},\ldots, \lambda_{N}, \gamma_{1} \ldots \gamma_{N}}\|\mathrm{e}^{-\cdot^{2}/2\sigma} - \sum\limits_{j=1}^{N} \gamma_{j} \, {\mathrm e}^{\lambda_{j} \cdot}\|_{L_{2}({\mathbb R}, \mathrm{e}^{-t^{2}/2\rho})}$, are zeros of a scaled Hermite polynomial. This observation leads us to a numerically stable approximation method with low computational cost of $\mathit{O}(N^{3})$ operations. Furthermore, we derive a direct algorithm to solve this approximation problem based on a matrix pencil method for a special structured matrix. The entries of this matrix are determined by hypergeometric functions. For the weighted $L_{2}$-norm, we prove that the approximation error decays exponentially with respect to the length $N$ of the sum. An exponentially decaying error in the (unweighted) $L^{2}$-norm is achieved using a truncated cosine sum.

This paper proposes a decoupled numerical scheme of the time-dependent Ginzburg--Landau equations under the temporal gauge. For the magnetic potential and the order parameter, the discrete scheme adopts the second type Ned${\rm \acute{e}}$lec element and the linear element for spatial discretization, respectively; and a linearized backward Euler method and the first order exponential time differencing method for time discretization, respectively. The maximum bound principle (MBP) of the order parameter and the energy dissipation law in the discrete sense are proved. The discrete energy stability and MBP-preservation can guarantee the stability and validity of the numerical simulations, and further facilitate the adoption of an adaptive time-stepping strategy, which often plays an important role in long-time simulations of vortex dynamics, especially when the applied magnetic field is strong. An optimal error estimate of the proposed scheme is also given. Numerical examples verify the theoretical results of the proposed scheme and demonstrate the vortex motions of superconductors in an external magnetic field.

Conventional neural network elastoplasticity models are often perceived as lacking interpretability. This paper introduces a two-step machine-learning approach that returns mathematical models interpretable by human experts. In particular, we introduce a surrogate model where yield surfaces are expressed in terms of a set of single-variable feature mappings obtained from supervised learning. A postprocessing step is then used to re-interpret the set of single-variable neural network mapping functions into mathematical form through symbolic regression. This divide-and-conquer approach provides several important advantages. First, it enables us to overcome the scaling issue of symbolic regression algorithms. From a practical perspective, it enhances the portability of learned models for partial differential equation solvers written in different programming languages. Finally, it enables us to have a concrete understanding of the attributes of the materials, such as convexity and symmetries of models, through automated derivations and reasoning. Numerical examples have been provided, along with an open-source code to enable third-party validation.

Automated math word problem solvers based on neural networks have successfully managed to obtain 70-80\% accuracy in solving arithmetic word problems. However, it has been shown that these solvers may rely on superficial patterns to obtain their equations. In order to determine what information math word problem solvers use to generate solutions, we remove parts of the input and measure the model's performance on the perturbed dataset. Our results show that the model is not sensitive to the removal of many words from the input and can still manage to find a correct answer when given a nonsense question. This indicates that automatic solvers do not follow the semantic logic of math word problems, and may be overfitting to the presence of specific words.