Port-Hamiltonian (PH) systems provide a framework for modeling, analysis and control of complex dynamical systems, where the complexity might result from multi-physical couplings, non-trivial domains and diverse nonlinearities. A major benefit of the PH representation is the explicit formulation of power interfaces, so-called ports, which allow for a power-preserving interconnection of subsystems to compose flexible multibody systems in a modular way. In this work, we present a PH representation of geometrically exact strings with nonlinear material behaviour. Furthermore, using structure-preserving discretization techniques a corresponding finite-dimensional PH state space model is developed. Applying mixed finite elements, the semi-discrete model retains the PH structure and the ports (pairs of velocities and forces) on the discrete level. Moreover, discrete derivatives are used in order to obtain an energy-consistent time-stepping method. The numerical properties of the newly devised model are investigated in a representative example. The developed PH state space model can be used for structure-preserving simulation and model order reduction as well as feedforward and feedback control design.
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In multivariate time series analysis, the coherence measures the linear dependency between two-time series at different frequencies. However, real data applications often exhibit nonlinear dependency in the frequency domain. Conventional coherence analysis fails to capture such dependency. The quantile coherence, on the other hand, characterizes nonlinear dependency by defining the coherence at a set of quantile levels based on trigonometric quantile regression. Although quantile coherence is a more powerful tool, its estimation remains challenging due to the high level of noise. This paper introduces a new estimation technique for quantile coherence. The proposed method is semi-parametric, which uses the parametric form of the spectrum of the vector autoregressive (VAR) model as an approximation to the quantile spectral matrix, along with nonparametric smoothing across quantiles. For each fixed quantile level, we obtain the VAR parameters from the quantile periodograms, then, using the Durbin-Levinson algorithm, we calculate the preliminary estimate of quantile coherence using the VAR parameters. Finally, we smooth the preliminary estimate of quantile coherence across quantiles using a nonparametric smoother. Numerical results show that the proposed estimation method outperforms nonparametric methods. We show that quantile coherence-based bivariate time series clustering has advantages over the ordinary VAR coherence. For applications, the identified clusters of financial stocks by quantile coherence with a market benchmark are shown to have an intriguing and more accurate structure of diversified investment portfolios that may be used by investors to make better decisions.
Optimal design is a critical yet challenging task within many applications. This challenge arises from the need for extensive trial and error, often done through simulations or running field experiments. Fortunately, sequential optimal design, also referred to as Bayesian optimization when using surrogates with a Bayesian flavor, has played a key role in accelerating the design process through efficient sequential sampling strategies. However, a key opportunity exists nowadays. The increased connectivity of edge devices sets forth a new collaborative paradigm for Bayesian optimization. A paradigm whereby different clients collaboratively borrow strength from each other by effectively distributing their experimentation efforts to improve and fast-track their optimal design process. To this end, we bring the notion of consensus to Bayesian optimization, where clients agree (i.e., reach a consensus) on their next-to-sample designs. Our approach provides a generic and flexible framework that can incorporate different collaboration mechanisms. In lieu of this, we propose transitional collaborative mechanisms where clients initially rely more on each other to maneuver through the early stages with scant data, then, at the late stages, focus on their own objectives to get client-specific solutions. Theoretically, we show the sub-linear growth in regret for our proposed framework. Empirically, through simulated datasets and a real-world collaborative material discovery experiment, we show that our framework can effectively accelerate and improve the optimal design process and benefit all participants.
This paper considers the computation of the matrix exponential $\mathrm{e}^A$ with numerical quadrature. Although several quadrature-based algorithms have been proposed, they focus on (near) Hermitian matrices. In order to deal with non-Hermitian matrices, we use another integral representation including an oscillatory term and consider applying the double exponential (DE) formula specialized to Fourier integrals. The DE formula transforms the given integral into another integral whose interval is infinite, and therefore it is necessary to truncate the infinite interval. In this paper, to utilize the DE formula, we analyze the truncation error and propose two algorithms. The first one approximates $\mathrm{e}^A$ with the fixed mesh size which is a parameter in the DE formula affecting the accuracy. Second one computes $\mathrm{e}^A$ based on the first one with automatic selection of the mesh size depending on the given error tolerance.
In this article we propose two finite element schemes for the Navier-Stokes equations, based on a reformulation that involves differential operators from the de Rham sequence and an advection operator with explicit skew-symmetry in weak form. Our first scheme is obtained by discretizing this formulation with conforming FEEC (Finite Element Exterior Calculus) spaces: it preserves the pointwise divergence free constraint of the velocity, its total momentum and its energy, in addition to being pressure robust. Following the broken-FEEC approach, our second scheme uses fully discontinuous spaces and local conforming projections to define the discrete differential operators. It preserves the same invariants up to a dissipation of energy to stabilize numerical discontinuities. For both schemes we use a middle point time discretization which preserve these invariants at the fully discrete level and we analyse its well-posedness in terms of a CFL condition. Numerical test cases performed with spline finite elements allow us to verify the high order accuracy of the resulting numerical methods, as well as their ability to handle general boundary conditions.
We study the complexity of randomized computation of integrals depending on a parameter, with integrands from Sobolev spaces. That is, for $r,d_1,d_2\in{\mathbb N}$, $1\le p,q\le \infty$, $D_1= [0,1]^{d_1}$, and $D_2= [0,1]^{d_2}$ we are given $f\in W_p^r(D_1\times D_2)$ and we seek to approximate $$ Sf=\int_{D_2}f(s,t)dt\quad (s\in D_1), $$ with error measured in the $L_q(D_1)$-norm. Our results extend previous work of Heinrich and Sindambiwe (J.\ Complexity, 15 (1999), 317--341) for $p=q=\infty$ and Wiegand (Shaker Verlag, 2006) for $1\le p=q<\infty$. Wiegand's analysis was carried out under the assumption that $W_p^r(D_1\times D_2)$ is continuously embedded in $C(D_1\times D_2)$ (embedding condition). We also study the case that the embedding condition does not hold. For this purpose a new ingredient is developed -- a stochastic discretization technique. The paper is based on Part I, where vector valued mean computation -- the finite-dimensional counterpart of parametric integration -- was studied. In Part I a basic problem of Information-Based Complexity on the power of adaption for linear problems in the randomized setting was solved. Here a further aspect of this problem is settled.
Beam search is a go-to strategy for decoding neural sequence models. The algorithm can naturally be viewed as a subset optimization problem, albeit one where the corresponding set function does not reflect interactions between candidates. Empirically, this leads to sets often exhibiting high overlap, e.g., strings may differ by only a single word. Yet in use-cases that call for multiple solutions, a diverse or representative set is often desired. To address this issue, we propose a reformulation of beam search, which we call determinantal beam search. Determinantal beam search has a natural relationship to determinantal point processes (DPPs), models over sets that inherently encode intra-set interactions. By posing iterations in beam search as a series of subdeterminant maximization problems, we can turn the algorithm into a diverse subset selection process. In a case study, we use the string subsequence kernel to explicitly encourage n-gram coverage in text generated from a sequence model. We observe that our algorithm offers competitive performance against other diverse set generation strategies in the context of language generation, while providing a more general approach to optimizing for diversity.
We propose a model to flexibly estimate joint tail properties by exploiting the convergence of an appropriately scaled point cloud onto a compact limit set. Characteristics of the shape of the limit set correspond to key tail dependence properties. We directly model the shape of the limit set using B\'ezier splines, which allow flexible and parsimonious specification of shapes in two dimensions. We then fit the B\'ezier splines to data in pseudo-polar coordinates using Markov chain Monte Carlo, utilizing a limiting approximation to the conditional likelihood of the radii given angles. By imposing appropriate constraints on the parameters of the B\'ezier splines, we guarantee that each posterior sample is a valid limit set boundary, allowing direct posterior analysis of any quantity derived from the shape of the curve. Furthermore, we obtain interpretable inference on the asymptotic dependence class by using mixture priors with point masses on the corner of the unit box. Finally, we apply our model to bivariate datasets of extremes of variables related to fire risk and air pollution.
Within the next decade the Laser Interferometer Space Antenna (LISA) is due to be launched, providing the opportunity to extract physics from stellar objects and systems, such as \textit{Extreme Mass Ratio Inspirals}, (EMRIs) otherwise undetectable to ground based interferometers and Pulsar Timing Arrays (PTA). Unlike previous sources detected by the currently available observational methods, these sources can \textit{only} be simulated using an accurate computation of the gravitational self-force. Whereas the field has seen outstanding progress in the frequency domain, metric reconstruction and self-force calculations are still an open challenge in the time domain. Such computations would not only further corroborate frequency domain calculations and models, but also allow for full self-consistent evolution of the orbit under the effect of the self-force. Given we have \textit{a priori} information about the local structure of the discontinuity at the particle, we will show how to construct discontinuous spatial and temporal discretisations by operating on discontinuous Lagrange and Hermite interpolation formulae and hence recover higher order accuracy. In this work we demonstrate how this technique in conjunction with well-suited gauge choice (hyperboloidal slicing) and numerical (discontinuous collocation with time symmetric) methods can provide a relatively simple method of lines numerical algorithm to the problem. This is the first of a series of papers studying the behaviour of a point-particle prescribing circular geodesic motion in Schwarzschild in the \textit{time domain}. In this work we describe the numerical machinery necessary for these computations and show not only our work is capable of highly accurate flux radiation measurements but it also shows suitability for evaluation of the necessary field and it's derivatives at the particle limit.
We present self-supervised geometric perception (SGP), the first general framework to learn a feature descriptor for correspondence matching without any ground-truth geometric model labels (e.g., camera poses, rigid transformations). Our first contribution is to formulate geometric perception as an optimization problem that jointly optimizes the feature descriptor and the geometric models given a large corpus of visual measurements (e.g., images, point clouds). Under this optimization formulation, we show that two important streams of research in vision, namely robust model fitting and deep feature learning, correspond to optimizing one block of the unknown variables while fixing the other block. This analysis naturally leads to our second contribution -- the SGP algorithm that performs alternating minimization to solve the joint optimization. SGP iteratively executes two meta-algorithms: a teacher that performs robust model fitting given learned features to generate geometric pseudo-labels, and a student that performs deep feature learning under noisy supervision of the pseudo-labels. As a third contribution, we apply SGP to two perception problems on large-scale real datasets, namely relative camera pose estimation on MegaDepth and point cloud registration on 3DMatch. We demonstrate that SGP achieves state-of-the-art performance that is on-par or superior to the supervised oracles trained using ground-truth labels.
Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently. In this tutorial-style overview, we highlight the important role of statistical models in enabling efficient nonconvex optimization with performance guarantees. We review two contrasting approaches: (1) two-stage algorithms, which consist of a tailored initialization step followed by successive refinement; and (2) global landscape analysis and initialization-free algorithms. Several canonical matrix factorization problems are discussed, including but not limited to matrix sensing, phase retrieval, matrix completion, blind deconvolution, robust principal component analysis, phase synchronization, and joint alignment. Special care is taken to illustrate the key technical insights underlying their analyses. This article serves as a testament that the integrated consideration of optimization and statistics leads to fruitful research findings.
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