We propose a novel Hadamard integrator for the self-adjoint time-dependent wave equation in an inhomogeneous medium. First, we create a new asymptotic series based on the Gelfand-Shilov function, dubbed Hadamard's ansatz, to approximate the Green's function of the time-dependent wave equation. Second, incorporating the leading term of Hadamard's ansatz into the Kirchhoff-Huygens representation, we develop an original Hadamard integrator for the Cauchy problem of the time-dependent wave equation and derive the corresponding Lagrangian formulation in geodesic polar coordinates. Third, to construct the Hadamard integrator in the Lagrangian formulation efficiently, we use a short-time ray tracing method to obtain wavefront locations accurately, and we further develop fast algorithms to compute Chebyshev-polynomial based low-rank representations of both wavefront locations and variants of Hadamard coefficients. Fourth, equipped with these low-rank representations, we apply the Hadamard integrator to efficiently solve time-dependent wave equations with highly oscillatory initial conditions, where the time step size is independent of the initial conditions. By judiciously choosing the medium-dependent time step, our new Hadamard integrator can propagate wave field beyond caustics implicitly and advance spatially overturning waves in time naturally. Moreover, since the integrator is independent of initial conditions, the Hadamard integrator can be applied to many different initial conditions once it is constructed. Both two-dimensional and three-dimensional numerical examples illustrate the accuracy and performance of the proposed method.
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Integration：Integration, the VLSI Journal。
Explanation：集成，VLSI雜志。
Publisher：Elsevier。
SIT：
The formation of shear shock waves in the brain has been proposed as one of the plausible explanations for deep intracranial injuries. In fact, such singular solutions emerge naturally in soft viscoelastic tissues under dynamic loading conditions. To improve our understanding of the mechanical processes at hand, the development of dedicated computational models is needed. The present study concerns three-dimensional numerical models of incompressible viscoelastic solids whose motion is analysed by means of shock-capturing finite volume methods. More specifically, we focus on the use of the artificial compressibility method, a technique that has been frequently employed in computational fluid dynamics. The material behaviour is deduced from the Fung--Simo quasi-linear viscoelasiticity theory (QLV) where the elastic response is of Yeoh type. We analyse the accuracy of the method and demonstrate its applicability for the study of nonlinear wave propagation in soft solids. The numerical results cover accuracy tests, shock formation and wave diffraction.
We couple the L1 discretization of the Caputo fractional derivative in time with the Galerkin scheme to devise a linear numerical method for the semilinear subdiffusion equation. Two important points that we make are: nonsmooth initial data and time-dependent diffusion coefficient. We prove the stability and convergence of the method under weak assumptions concerning regularity of the diffusivity. We find optimal pointwise in space and global in time errors, which are verified with several numerical experiments.
This paper presents a novel boundary integral equation (BIE) formulation for the two-dimensional time-harmonic water-waves problem. It utilizes a complex-scaled Laplace's free-space Green's function, resulting in a BIE posed on the infinite boundaries of the domain. The perfectly matched layer (PML) coordinate stretching that is used to render propagating waves exponentially decaying, allows for the effective truncation and discretization of the BIE unbounded domain. We show through a variety of numerical examples that, despite the logarithmic growth of the complex-scaled Laplace's free-space Green's function, the truncation errors are exponentially small with respect to the truncation length. Our formulation uses only simple function evaluations (e.g. complex logarithms and square roots), hence avoiding the need to compute the involved water-wave Green's function. Finally, we show that the proposed approach can also be used to find complex resonances through a \emph{linear} eigenvalue problem since the Green's function is frequency-independent.
Systems consisting of spheres rolling on elastic membranes have been used to introduce a core conceptual idea of General Relativity (GR): how curvature guides the movement of matter. However, such schemes cannot accurately represent relativistic dynamics in the laboratory because of the dominance of dissipation and external gravitational fields. Here we demonstrate that an ``active" object (a wheeled robot), which moves in a straight line on level ground and can alter its speed depending on the curvature of the deformable terrain it moves on, can exactly capture dynamics in curved relativistic spacetimes. Via the systematic study of the robot's dynamics in the radial and orbital directions, we develop a mapping of the emergent trajectories of a wheeled vehicle on a spandex membrane to the motion in a curved spacetime. Our mapping demonstrates how the driven robot's dynamics mix space and time in a metric, and shows how active particles do not necessarily follow geodesics in the real space but instead follow geodesics in a fiducial spacetime. The mapping further reveals how parameters such as the membrane elasticity and instantaneous speed allow the programming of a desired spacetime, such as the Schwarzschild metric near a non-rotating blackhole. Our mapping and framework facilitate creation of a robophysical analog to a general relativistic system in the laboratory at low cost that can provide insights into active matter in deformable environments and robot exploration in complex landscapes.
We present a multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty. The algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size depends on the number $N$ of samples used to discretized the probability space. We show that this reduced system can be solved with optimal $O(N)$ complexity. We test the multigrid method on three problems: a linear-quadratic problem, possibly with a local or a boundary control, for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and $L^1$-norm penalization on the control, in which the multigrid scheme is used within a semismooth Newton iteration; a risk-adverse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits excellent performances and robustness with respect to the parameters of interest.
Adversarially robust streaming algorithms are required to process a stream of elements and produce correct outputs, even when each stream element can be chosen depending on earlier algorithm outputs. As with classic streaming algorithms, which must only be correct for the worst-case fixed stream, adversarially robust algorithms with access to randomness can use significantly less space than deterministic algorithms. We prove that for the Missing Item Finding problem in streaming, the space complexity also significantly depends on how adversarially robust algorithms are permitted to use randomness. (In contrast, the space complexity of classic streaming algorithms does not depend as strongly on the way randomness is used.) For Missing Item Finding on streams of length $r$ with elements in $\{1,...n\}$, and $\le 1/\text{poly}(n)$ error, we show that when $r = O(2^{\sqrt{\log n}})$, "random seed" adversarially robust algorithms, which only use randomness at initialization, require $r^{\Omega(1)}$ bits of space, while "random tape" adversarially robust algorithms, which may make random decisions at any time, may use $O(\text{polylog}(r))$ random bits. When $r = \Theta(\sqrt{n})$, "random tape" adversarially robust algorithms need $r^{\Omega(1)}$ space, while "random oracle" adversarially robust algorithms, which can read from a long random string for free, may use $O(\text{polylog}(r))$ space. The space lower bound for the "random seed" case follows, by a reduction given in prior work, from a lower bound for pseudo-deterministic streaming algorithms given in this paper.
Full waveform inversion (FWI) updates the subsurface model from an initial model by comparing observed and synthetic seismograms. Due to high nonlinearity, FWI is easy to be trapped into local minima. Extended domain FWI, including wavefield reconstruction inversion (WRI) and extended source waveform inversion (ESI) are attractive options to mitigate this issue. This paper makes an in-depth analysis for FWI in the extended domain, identifying key challenges and searching for potential remedies towards practical applications. WRI and ESI are formulated within the same mathematical framework using Lagrangian-based adjoint-state method with a special focus on time-domain formulation using extended sources, while putting connections between classical FWI, WRI and ESI: both WRI and ESI can be viewed as weighted versions of classic FWI. Due to symmetric positive definite Hessian, the conjugate gradient is explored to efficiently solve the normal equation in a matrix free manner, while both time and frequency domain wave equation solvers are feasible. This study finds that the most significant challenge comes from the huge storage demand to store time-domain wavefields through iterations. To resolve this challenge, two possible workaround strategies can be considered, i.e., by extracting sparse frequencial wavefields or by considering time-domain data instead of wavefields for reducing such challenge. We suggest that these options should be explored more intensively for tractable workflows.
As an optical processor, a Diffractive Deep Neural Network (D2NN) utilizes engineered diffractive surfaces designed through machine learning to perform all-optical information processing, completing its tasks at the speed of light propagation through thin optical layers. With sufficient degrees-of-freedom, D2NNs can perform arbitrary complex-valued linear transformations using spatially coherent light. Similarly, D2NNs can also perform arbitrary linear intensity transformations with spatially incoherent illumination; however, under spatially incoherent light, these transformations are non-negative, acting on diffraction-limited optical intensity patterns at the input field-of-view (FOV). Here, we expand the use of spatially incoherent D2NNs to complex-valued information processing for executing arbitrary complex-valued linear transformations using spatially incoherent light. Through simulations, we show that as the number of optimized diffractive features increases beyond a threshold dictated by the multiplication of the input and output space-bandwidth products, a spatially incoherent diffractive visual processor can approximate any complex-valued linear transformation and be used for all-optical image encryption using incoherent illumination. The findings are important for the all-optical processing of information under natural light using various forms of diffractive surface-based optical processors.
In this paper, we consider the variable-order time fractional mobile/immobile diffusion (TF-MID) equation in two-dimensional spatial domain, where the fractional order $\alpha(t)$ satisfies $0<\alpha_{*}\leq \alpha(t)\leq \alpha^{*}<1$. We combine the quadratic spline collocation (QSC) method and the $L1^+$ formula to propose a QSC-$L1^+$ scheme. It can be proved that, the QSC-$L1^+$ scheme is unconditionally stable and convergent with $\mathcal{O}(\tau^{\min{\{3-\alpha^*-\alpha(0),2\}}} + \Delta x^{2}+\Delta y^{2})$, where $\tau$, $\Delta x$ and $\Delta y$ are the temporal and spatial step sizes, respectively. With some proper assumptions on $\alpha(t)$, the QSC-$L1^+$ scheme has second temporal convergence order even on the uniform mesh, without any restrictions on the solution of the equation. We further construct a novel alternating direction implicit (ADI) framework to develop an ADI-QSC-$L1^+$ scheme, which has the same unconditionally stability and convergence orders. In addition, a fast implementation for the ADI-QSC-$L1^+$ scheme based on the exponential-sum-approximation (ESA) technique is proposed. Moreover, we also introduce the optimal QSC method to improve the spatial convergence to fourth-order. Numerical experiments are attached to support the theoretical analysis, and to demonstrate the effectiveness of the proposed schemes.
We present a priori error estimates for a multirate time-stepping scheme for coupled differential equations. The discretization is based on Galerkin methods in time using two different time meshes for two parts of the problem. We aim at surface coupled multiphysics problems like two-phase flows. Special focus is on the handling of the interface coupling to guarantee a coercive formulation as key to optimal order error estimates. In a sequence of increasing complexity, we begin with the coupling of two ordinary differential equations, coupled heat conduction equation, and finally a coupled Stokes problem. For this we show optimal multi-rate estimates in velocity and a suboptimal result in pressure. The a priori estimates prove that the multirate method decouples the two subproblems exactly. This is the basis for adaptive methods which can choose optimal lattices for the respective subproblems.
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