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The influence of natural image transformations on receptive field responses is crucial for modelling visual operations in computer vision and biological vision. In this regard, covariance properties with respect to geometric image transformations in the earliest layers of the visual hierarchy are essential for expressing robust image operations and for formulating invariant visual operations at higher levels. This paper defines and proves a joint covariance property under compositions of spatial scaling transformations, spatial affine transformations, Galilean transformations and temporal scaling transformations, which makes it possible to characterize how different types of image transformations interact with each other. Specifically, the derived relations show how the receptive field parameters need to be transformed, in order to match the output from spatio-temporal receptive fields with the underlying spatio-temporal image transformations.

In this work we extend the shifted Laplacian approach to the elastic Helmholtz equation. The shifted Laplacian multigrid method is a common preconditioning approach for the discretized acoustic Helmholtz equation. In some cases, like geophysical seismic imaging, one needs to consider the elastic Helmholtz equation, which is harder to solve: it is three times larger and contains a nullity-rich grad-div term. These properties make the solution of the equation more difficult for multigrid solvers. The key idea in this work is combining the shifted Laplacian with approaches for linear elasticity. We provide local Fourier analysis and numerical evidence that the convergence rate of our method is independent of the Poisson's ratio. Moreover, to better handle the problem size, we complement our multigrid method with the domain decomposition approach, which works in synergy with the local nature of the shifted Laplacian, so we enjoy the advantages of both methods without sacrificing performance. We demonstrate the efficiency of our solver on 2D and 3D problems in heterogeneous media.

Finding the distribution of the velocities and pressures of a fluid (by solving the Navier-Stokes equations) is a principal task in the chemical, energy, and pharmaceutical industries, as well as in mechanical engineering and the design of pipeline systems. With existing solvers, such as OpenFOAM and Ansys, simulations of fluid dynamics in intricate geometries are computationally expensive and require re-simulation whenever the geometric parameters or the initial and boundary conditions are altered. Physics-informed neural networks are a promising tool for simulating fluid flows in complex geometries, as they can adapt to changes in the geometry and mesh definitions, allowing for generalization across different shapes. We present a hybrid quantum physics-informed neural network that simulates laminar fluid flows in 3D Y-shaped mixers. Our approach combines the expressive power of a quantum model with the flexibility of a physics-informed neural network, resulting in a 21% higher accuracy compared to a purely classical neural network. Our findings highlight the potential of machine learning approaches, and in particular hybrid quantum physics-informed neural network, for complex shape optimization tasks in computational fluid dynamics. By improving the accuracy of fluid simulations in complex geometries, our research using hybrid quantum models contributes to the development of more efficient and reliable fluid dynamics solvers.

In this study, we consider the numerical solution of the Neumann initial boundary value problem for the wave equation in 2D domains. Employing the Laguerre transform with respect to the temporal variable, we effectively transform this problem into a series of Neumann elliptic problems. The development of a fundamental sequence for these elliptic equations provides us with the means to introduce modified double layer potentials. Consequently, we are able to derive a sequence of boundary hypersingular integral equations as a result of this transformation. To discretize the system of equations, we apply the Maue transform and implement the Nystr\"om method with trigonometric quadrature techniques. To demonstrate the practical utility of our approach, we provide numerical examples.

There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability between finite element method and computer aided design (CAD) software. However, these approaches have difficulty when the domain has singularities since the solution at the singularity may be multivalued. This paper develops a novel numerical approach to solve elliptic PDEs on real, closed, connected, orientable, and almost smooth algebraic curves and surfaces. Our method integrates numerical algebraic geometry, differential geometry, and a finite difference scheme which is demonstrated on several examples.

Dynamical low-rank (DLR) approximation has gained interest in recent years as a viable solution to the curse of dimensionality in the numerical solution of kinetic equations including the Boltzmann and Vlasov equations. These methods include the projector-splitting and Basis-update & Galerkin DLR integrators, and have shown promise at greatly improving the computational efficiency of kinetic solutions. However, this often comes at the cost of conservation of charge, current and energy. In this work we show how a novel macro-micro decomposition may be used to separate the distribution function into two components, one of which carries the conserved quantities, and the other of which is orthogonal to them. We apply DLR approximation to the latter, and thereby achieve a clean and extensible approach to a conservative DLR scheme which retains the computational advantages of the base scheme. Moreover, our decomposition is compatible with the projector-splitting integrator, and can therefore access second-order accuracy in time via a Strang splitting scheme. We describe a first-order integrator which can exactly conserve charge and either current or energy, as well as a second-order accurate integrator which exactly conserves charge and energy. To highlight the flexibility of the proposed macro-micro decomposition, we implement a pair of velocity space discretizations, and verify the claimed accuracy and conservation properties on a suite of plasma benchmark problems.

We give an efficient perfect sampling algorithm for weighted, connected induced subgraphs (or graphlets) of rooted, bounded degree graphs. Our algorithm utilizes a vertex-percolation process with a carefully chosen rejection filter and works under a percolation subcriticality condition. We show that this condition is optimal in the sense that the task of (approximately) sampling weighted rooted graphlets becomes impossible in finite expected time for infinite graphs and intractable for finite graphs when the condition does not hold. We apply our sampling algorithm as a subroutine to give near linear-time perfect sampling algorithms for polymer models and weighted non-rooted graphlets in finite graphs, two widely studied yet very different problems. This new perfect sampling algorithm for polymer models gives improved sampling algorithms for spin systems at low temperatures on expander graphs and unbalanced bipartite graphs, among other applications.

Generalized linear models (GLMs) are popular for data-analysis in almost all quantitative sciences, but the choice of likelihood family and link function is often difficult. This motivates the search for likelihoods and links that minimize the impact of potential misspecification. We perform a large-scale simulation study on double-bounded and lower-bounded response data where we systematically vary both true and assumed likelihoods and links. In contrast to previous studies, we also study posterior calibration and uncertainty metrics in addition to point-estimate accuracy. Our results indicate that certain likelihoods and links can be remarkably robust to misspecification, performing almost on par with their respective true counterparts. Additionally, normal likelihood models with identity link (i.e., linear regression) often achieve calibration comparable to the more structurally faithful alternatives, at least in the studied scenarios. On the basis of our findings, we provide practical suggestions for robust likelihood and link choices in GLMs.

When the eigenvalues of the coefficient matrix for a linear scalar ordinary differential equation are of large magnitude, its solutions exhibit complicated behaviour, such as high-frequency oscillations, rapid growth or rapid decay. The cost of representing such solutions using standard techniques grows with the magnitudes of the eigenvalues. As a consequence, the running times of most solvers for ordinary differential equations also grow with these eigenvalues. However, a large class of scalar ordinary differential equations with slowly-varying coefficients admit slowly-varying phase functions that can be represented at a cost which is bounded independent of the magnitudes of the eigenvalues of the corresponding coefficient matrix. Here, we introduce a numerical algorithm for constructing slowly-varying phase functions which represent the solutions of a linear scalar ordinary differential equation. Our method's running time depends on the complexity of the equation's coefficients, but is bounded independent of the magnitudes of the equation's eigenvalues. Once the phase functions have been constructed, essentially any reasonable initial or boundary value problem for the scalar equation can be easily solved. We present the results of numerical experiments showing that, despite its greater generality, our algorithm is competitive with state-of-the-art methods for solving highly-oscillatory second order differential equations. We also compare our method with Magnus-type exponential integrators and find that our approach is orders of magnitude faster in the high-frequency regime.

By establishing an interesting connection between ordinary Bell polynomials and rational convolution powers, some composition and inverse relations of Bell polynomials as well as explicit expressions for convolution roots of sequences are obtained. Based on these results, a new method is proposed for calculation of partial Bell polynomials based on prime factorization. It is shown that this method is more efficient than the conventional recurrence procedure for computing Bell polynomials in most cases, requiring far less arithmetic operations. A detailed analysis of the computation complexity is provided, followed by some numerical evaluations.