This paper proposes a level set-based topology optimization method for designing acoustic structures with viscous and thermal boundary layers in perspective. It is known that acoustic waves propagating in a narrow channel are damped by viscous and thermal boundary layers. To estimate these viscothermal effects, we first introduce a sequential linearized Navier-Stokes model based on three weakly coupled Helmholtz equations for viscous, thermal, and acoustic pressure fields. Then, the optimization problem is formulated, where a sound-absorbing structure comprising air and an isothermal rigid medium is targeted, and its sound absorption coefficient is set as an objective function. The adjoint variable method and the concept of the topological derivative are used to obtain design sensitivity. A level set-based topology optimization method is used to solve the optimization problem. Two-dimensional numerical examples are provided to support the validity of the proposed method. In addition, the mechanisms that lead to the high absorption coefficient of the optimized design are discussed.
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We consider the numerical solution of the real time equilibrium Dyson equation, which is used in calculations of the dynamical properties of quantum many-body systems. We show that this equation can be written as a system of coupled, nonlinear, convolutional Volterra integro-differential equations, for which the kernel depends self-consistently on the solution. As is typical in the numerical solution of Volterra-type equations, the computational bottleneck is the quadratic-scaling cost of history integration. However, the structure of the nonlinear Volterra integral operator precludes the use of standard fast algorithms. We propose a quasilinear-scaling FFT-based algorithm which respects the structure of the nonlinear integral operator. The resulting method can reach large propagation times, and is thus well-suited to explore quantum many-body phenomena at low energy scales. We demonstrate the solver with two standard model systems: the Bethe graph, and the Sachdev-Ye-Kitaev model.
Accelerated degradation tests are used to provide accurate estimation of lifetime properties of highly reliable products within a relatively short testing time. There data from particular tests at high levels of stress (e.\,g.\ temperature, voltage, or vibration) are extrapolated, through a physically meaningful model, to obtain estimates of lifetime quantiles under normal use conditions. In this work, we consider repeated measures accelerated degradation tests with multiple stress variables, where the degradation paths are assumed to follow a linear mixed effects model which is quite common in settings when repeated measures are made. We derive optimal experimental designs for minimizing the asymptotic variance for estimating the median failure time under normal use conditions when the time points for measurements are either fixed in advance or are also to be optimized.
We introduce and analyze various Regularized Combined Field Integral Equations (CFIER) formulations of time-harmonic Navier equations in media with piece-wise constant material properties. These formulations can be derived systematically starting from suitable coercive approximations of Dirichlet-to-Neumann operators (DtN), and we present a periodic pseudodifferential calculus framework within which the well posedness of CIER formulations can be established. We also use the DtN approximations to derive and analyze Optimized Schwarz (OS) methods for the solution of elastodynamics transmission problems. The pseudodifferential calculus we develop in this paper relies on careful singularity splittings of the kernels of Navier boundary integral operators which is also the basis of high-order Nystr\"om quadratures for their discretizations. Based on these high-order discretizations we investigate the rate of convergence of iterative solvers applied to CFIER and OS formulations of scattering and transmission problems. We present a variety of numerical results that illustrate that the CFIER methodology leads to important computational savings over the classical CFIE one, whenever iterative solvers are used for the solution of the ensuing discretized boundary integral equations. Finally, we show that the OS methods are competitive in the high-frequency high-contrast regime.
Approximate linear programs (ALPs) are well-known models based on value function approximations (VFAs) to obtain policies and lower bounds on the optimal policy cost of discounted-cost Markov decision processes (MDPs). Formulating an ALP requires (i) basis functions, the linear combination of which defines the VFA, and (ii) a state-relevance distribution, which determines the relative importance of different states in the ALP objective for the purpose of minimizing VFA error. Both these choices are typically heuristic: basis function selection relies on domain knowledge while the state-relevance distribution is specified using the frequency of states visited by a heuristic policy. We propose a self-guided sequence of ALPs that embeds random basis functions obtained via inexpensive sampling and uses the known VFA from the previous iteration to guide VFA computation in the current iteration. Self-guided ALPs mitigate the need for domain knowledge during basis function selection as well as the impact of the initial choice of the state-relevance distribution, thus significantly reducing the ALP implementation burden. We establish high probability error bounds on the VFAs from this sequence and show that a worst-case measure of policy performance is improved. We find that these favorable implementation and theoretical properties translate to encouraging numerical results on perishable inventory control and options pricing applications, where self-guided ALP policies improve upon policies from problem-specific methods. More broadly, our research takes a meaningful step toward application-agnostic policies and bounds for MDPs.
This work is devoted to the theoretical and numerical analysis of a two-species chemotaxis- Navier-Stokes system with Lotka-Volterra competitive kinetics in a bounded domain of Rd, d = 2, 3. First, we study the existence of global weak solutions and establish a regularity criterion which provides sufficient conditions to ensure the strong regularity of the weak solutions. After, we propose a finite element numerical scheme in which we use a splitting technique obtained by introducing an auxiliary variable given by the gradient of the chemical concentration and applying an inductive strategy, in order to deal with the chemoattraction terms in the two-species equations and prove optimal error estimates. For this scheme, we study the well-posedness and derive some uniform estimates for the discrete variables required in the convergence analysis. Finally, we present some numerical simulations oriented to verify the good behavior of our scheme, as well as to check numerically the optimal error estimates proved in our theoretical analysis.
We present a new enriched Galerkin (EG) scheme for the Stokes equations based on piecewise linear elements for the velocity unknowns and piecewise constant elements for the pressure. The proposed EG method augments the conforming piecewise linear space for velocity by adding an additional degree of freedom which corresponds to one discontinuous linear basis function per element. Thus, the total number of degrees of freedom is significantly reduced in comparison with standard conforming, non-conforming, and discontinuous Galerkin schemes for the Stokes equation. We show the well-posedness of the new EG approach and prove that the scheme converges optimally. For the solution of the resulting large-scale indefinite linear systems we propose robust block preconditioners, yielding scalable results independent of the discretization and physical parameters. Numerical results confirm the convergence rates of the discretization and also the robustness of the linear solvers for a variety of test problems.
One of the major issues in the computational mechanics is to take into account the geometrical complexity. To overcome this difficulty and to avoid the expensive mesh generation, geometrically unfitted methods, i.e. the numerical methods using the simple computational meshes that do not fit the boundary of the domain, and/or the internal interfaces, have been widely developed. In the present work, we investigate the performances of an unfitted method called $\phi$-FEM that converges optimally and uses classical finite element spaces so that it can be easily implemented using general FEM libraries. The main idea is to take into account the geometry thanks to a level set function describing the boundary or the interface. Up to now, the $\phi$-FEM approach has been proposed, tested and substantiated mathematically only in some simplest settings: Poisson equation with Dirichlet/Neumann/Robin boundary conditions. Our goal here is to demonstrate its applicability to some more sophisticated governing equations arising in the computational mechanics. We consider the linear elasticity equations accompanied by either pure Dirichlet boundary conditions or by the mixed ones (Dirichlet and Neumann boundary conditions co-existing on parts of the boundary), an interface problem (linear elasticity with material coefficients abruptly changing over an internal interface), a model of elastic structures with cracks, and finally the heat equation. In all these settings, we derive an appropriate variant of $\phi$-FEM and then illustrate it by numerical tests on manufactured solutions. We also compare the accuracy and efficiency of $\phi$-FEM with those of the standard fitted FEM on the meshes of similar size, revealing the substantial gains that can be achieved by $\phi$-FEM in both the accuracy and the computational time.
One- and multi-dimensional stochastic Maxwell equations with additive noise are considered in this paper. It is known that such system can be written in the multi-symplectic structure, and the stochastic energy increases linearly in time. High order discontinuous Galerkin methods are designed for the stochastic Maxwell equations with additive noise, and we show that the proposed methods satisfy the discrete form of the stochastic energy linear growth property and preserve the multi-symplectic structure on the discrete level. Optimal error estimate of the semi-discrete DG method is also analyzed. The fully discrete methods are obtained by coupling with symplectic temporal discretizations. One- and two-dimensional numerical results are provided to demonstrate the performance of the proposed methods, and optimal error estimates and linear growth of the discrete energy can be observed for all cases.
In this article, I introduce the differential equation model and review their frequentist and Bayesian computation methods. A numerical example of the FitzHugh-Nagumo model is given.
High-order clustering aims to identify heterogeneous substructures in multiway datasets that arise commonly in neuroimaging, genomics, social network studies, etc. The non-convex and discontinuous nature of this problem pose significant challenges in both statistics and computation. In this paper, we propose a tensor block model and the computationally efficient methods, \emph{high-order Lloyd algorithm} (HLloyd), and \emph{high-order spectral clustering} (HSC), for high-order clustering. The convergence guarantees and statistical optimality are established for the proposed procedure under a mild sub-Gaussian noise assumption. Under the Gaussian tensor block model, we completely characterize the statistical-computational trade-off for achieving high-order exact clustering based on three different signal-to-noise ratio regimes. The analysis relies on new techniques of high-order spectral perturbation analysis and a "singular-value-gap-free" error bound in tensor estimation, which are substantially different from the matrix spectral analyses in the literature. Finally, we show the merits of the proposed procedures via extensive experiments on both synthetic and real datasets.
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