In this technical note, we consider a dynamic linear, cantilevered rectangular plate. The evolutionary PDE model is given by the fourth order plate dynamics (via the spatial biharmonic operator) with clamped-free-free-free boundary conditions. We additionally consider damping/dissipation terms, as well as non-conservative lower order terms arising in various applications. Dynamical numerical simulations are achieved by way of a finite difference spatial approximation with a MATLAB time integrator. The rectangular geometry allows the use of standard 2D spatial finite differences, while the high spatial order of the problem and mixed clamped-free type boundary conditions present challenges. Dynamic energies are also computed. The relevant code is presented, with discussion of the model and context.
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We consider an optimal control problem for the steady-state Kirchhoff equation, a prototype for nonlocal partial differential equations, different from fractional powers of closed operators. Existence and uniqueness of solutions of the state equation, existence of global optimal solutions, differentiability of the control-to-state map and first-order necessary optimality conditions are established. The aforementioned results require the controls to be functions in $H^1$ and subject to pointwise upper and lower bounds. In order to obtain the Newton differentiability of the optimality conditions, we employ a Moreau-Yosida-type penalty approach to treat the control constraints and study its convergence. The first-order optimality conditions of the regularized problems are shown to be Newton diffentiable, and a generalized Newton method is detailed. A discretization of the optimal control problem by piecewise linear finite elements is proposed and numerical results are presented.
Three algorithm are proposed to evaluate volume potentials that arise in boundary element methods for elliptic PDEs. The approach is to apply a modified fast multipole method for a boundary concentrated volume mesh. If $h$ is the meshwidth of the boundary, then the volume is discretized using nearly $O(h^{-2})$ degrees of freedom, and the algorithm computes potentials in nearly $O(h^{-2})$ complexity. Here nearly means that logarithmic terms of $h$ may appear. Thus the complexity of volume potentials calculations is of the same asymptotic order as boundary potentials. For sources and potentials with sufficient regularity the parameters of the algorithm can be designed such that the error of the approximated potential converges at any specified rate $O(h^p)$. The accuracy and effectiveness of the proposed algorithms are demonstrated for potentials of the Poisson equation in three dimensions.
We show that any nonzero polynomial in the ideal generated by the $r \times r$ minors of an $n \times n$ matrix $X$ can be used to efficiently approximate the determinant. For any nonzero polynomial $f$ in this ideal, we construct a small depth-three $f$-oracle circuit that approximates the determinant of size $\Theta(r^{1/3})$ in the sense of border complexity. For many classes of algebraic circuits, this implies that every nonzero polynomial in the ideal generated by $r \times r$ minors is at least as hard to approximately compute as the determinant of size $\Theta(r^{1/3})$. We also prove an analogous result for the Pfaffian of a $2n \times 2n$ skew-symmetric matrix and the ideal generated by Pfaffians of $2r \times 2r$ principal submatrices. This answers a recent question of Grochow about complexity in polynomial ideals in the setting of border complexity. We give several applications of our result, two of which are highlighted below. $\bullet$ We prove super-polynomial lower bounds for Ideal Proof System refutations computed by low-depth circuits. This extends the recent breakthrough low-depth circuit lower bounds of Limaye, Srinivasan, and Tavenas to the setting of proof complexity. For many natural circuit classes, we show that the approximative proof complexity of our hard instance is governed by the approximative circuit complexity of the determinant. $\bullet$ We construct new hitting set generators for polynomial-size low-depth circuits. For any $\varepsilon > 0$, we construct generators with seed length $O(n^\varepsilon)$ that attain a near-optimal tradeoff between their seed length and degree, and are computable by low-depth circuits of near-linear size (with respect to the size of their output). This matches the seed length of the generators recently obtained by Limaye, Srinivasan, and Tavenas, but improves on the generator's degree and circuit complexity.
Understanding the time-varying structure of complex temporal systems is one of the main challenges of modern time series analysis. In this paper, we show that every uniformly-positive-definite-in-covariance and sufficiently short-range dependent non-stationary and nonlinear time series can be well approximated globally by a white-noise-driven auto-regressive (AR) process of slowly diverging order. To our best knowledge, it is the first time such a structural approximation result is established for general classes of non-stationary time series. A high dimensional $\mathcal{L}^2$ test and an associated multiplier bootstrap procedure are proposed for the inference of the AR approximation coefficients. In particular, an adaptive stability test is proposed to check whether the AR approximation coefficients are time-varying, a frequently-encountered question for practitioners and researchers of time series. As an application, globally optimal short-term forecasting theory and methodology for a wide class of locally stationary time series are established via the method of sieves.
Given two points A,B in the plane, the locus of all points P for which the angles at A and B in the triangle A,B,P have a constant sum is a circular arc, by Thales' theorem. We show that the difference of these angles is kept a constant by points P on a hyperbola (albeit with foci different from A and B). Whereas hyperbolae are well-known to maintain a constant difference between the distances to their foci, the above angle property seems not to be widely known. The question was motivated by recent work by Alegr\'ia et al. and De Berg et al. on Voronoi diagrams of turning rays.
Quality of Life (QOL) outcomes are important in the management of chronic illnesses. In studies of efficacies of treatments or intervention modalities, QOL scales multi-dimensional constructs are routinely used as primary endpoints. The standard data analysis strategy computes composite (average) overall and domain scores, and conducts a mixed-model analysis for evaluating efficacy or monitoring medical conditions as if these scores were in continuous metric scale. However, assumptions of parametric models like continuity and homoscedastivity can be violated in many cases. Furthermore, it is even more challenging when there are missing values on some of the variables. In this paper, we propose a purely nonparametric approach in the sense that meaningful and, yet, nonparametric effect size measures are developed. We propose estimator for the effect size and develop the asymptotic properties. Our methods are shown to be particularly effective in the presence of some form of clustering and/or missing values. Inferential procedures are derived from the asymptotic theory. The Asthma Randomized Trial of Indoor Wood Smoke data will be used to illustrate the applications of the proposed methods. The data was collected from a three-arm randomized trial which evaluated interventions targeting biomass smoke particulate matter from older model residential wood stoves in homes that have children with asthma.
We introduce a novel minimal order hybrid Discontinuous Galerkin (HDG) and a novel mass conserving mixed stress (MCS) method for the approximation of incompressible flows. For this we employ the $H(\operatorname{div})$-conforming linear Brezzi-Douglas-Marini space and the lowest order Raviart-Thomas space for the approximation of the velocity and the vorticity, respectively. Our methods are based on the physically correct diffusive flux $-\nu \varepsilon(u)$ and provide exactly divergence-free discrete velocity solutions, optimal (pressure robust) error estimates and a minimal number of coupling degrees of freedom. For the stability analysis we introduce a new Korn-like inequality for vector-valued element-wise $H^1$ and normal continuous functions. Numerical examples conclude the work where the theoretical findings are validated and the novel methods are compared in terms of condition numbers with respect to discrete stability parameters.
We investigate time-adaptive Magnus-type integrators for the numerical approximation of a Mott transistor. The rapidly attenuating electromagnetic field calls for adaptive choice of the time steps. As a basis for step selection, asymptotically correct defect-based estimators of the local error are employed. We analyze the error of the numerical approximation in the presence of the unsmooth external potential and demonstrate the advantages of the adaptive approach.
Identification methods for dynamic networks typically require prior knowledge of the network and disturbance topology, and often rely on solving poorly scalable non-convex optimization problems. While methods for estimating network topology are available in the literature, less attention has been paid to estimating the disturbance topology, i.e., the (spatial) noise correlation structure and the noise rank in a filtered white noise representation of the disturbance signal. In this work we present an identification method for dynamic networks, in which an estimation of the disturbance topology precedes the identification of the full dynamic network with known network topology. To this end we extend the multi-step Sequential Linear Regression and Weighted Null Space Fitting methods to deal with reduced rank noise, and use these methods to estimate the disturbance topology and the network dynamics in the full measurement situation. As a result, we provide a multi-step least squares algorithm with parallel computation capabilities and that rely only on explicit analytical solutions, thereby avoiding the usual non-convex optimizations involved. Consequently we consistently estimate dynamic networks of Box Jenkins model structure, while keeping the computational burden low. We provide a consistency proof that includes path-based data informativity conditions for allocation of excitation signals in the experimental design. Numerical simulations performed on a dynamic network with reduced rank noise clearly illustrate the potential of this method.
We present a numerical stability analysis of the immersed boundary(IB) method for a special case which is constructed so that Fourier analysis is applicable. We examine the stability of the immersed boundary method with the discrete Fourier transforms defined differently on the fluid grid and the boundary grid. This approach gives accurate theoretical results about the stability boundary since it takes the effects of the spreading kernel of the immersed boundary method on the numerical stability into account. In this paper, the spreading kernel is the standard 4-point IB delta function. A three-dimensional incompressible viscous flow and a no-slip planar boundary are considered. The case of a planar elastic membrane is also analyzed using the same analysis framework and it serves as an example of many possible generalizations of our theory. We present some numerical results and show that the observed stability behaviors are consistent with what are predicted by our theory.
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