In this paper, a spectral method is formulated as a numerical solution for the stochastic Ginzburg-Landau equation driven by space-time white noise. The rates of pathwise convergence and convergence in expectation in ...
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We introduce a numerical approach to perform the effective (coarse-scale) bifurcation analysis of solutions of dissipative evolution equations with spatially varying coefficients. The advantage of this approach is tha...
We introduce a numerical approach to perform the effective (coarse-scale) bifurcation analysis of solutions of dissipative evolution equations with spatially varying coefficients. The advantage of this approach is that the `coarse model' (the averaged, effective equation) need not be explicitly constructed. The method only uses a time-integrator code for the detailed problem and judicious choices of initial data and integration times; the bifurcation computations are based on the so-called recursive projection method (Shroff and Keller 1993 SIAM J. Numer. Anal. 30 1099-120).
The selection of marker gene panels is critical for capturing the cellular and spatial heterogeneity in the expanding atlases of single-cell RNA sequencing and spatial transcriptomics data. We introduce geneCover, a l...
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A second moment turbulence closure model of the type used before for flows with density stratification, frame rotation and streamline curvature is augmented to describe MHD flows with small magnetic Reynolds number. I...
This paper proposes a novel approach for solving nonlinear partial differential equations (PDEs) with a quantum computer, the trainable embedding quantum physics informed neural network (TE-QPINN). We combine quantum ...
This paper proposes a novel approach for solving nonlinear partial differential equations (PDEs) with a quantum computer, the trainable embedding quantum physics informed neural network (TE-QPINN). We combine quantum machine learning (QML) with physics informed neural networks (PINNs) in a hybrid approach. By leveraging the advantages of classical and quantum computers, we can create algorithms that have a potential to be run on noisy intermediate-scale quantum devices (NISQ). We use feedforward neural networks (FNN) as problem-agnostic embedding functions, giving the used quantum circuit greater expressibility than previously introduced embedding. This expressibility allows us to solve a wide range of problems without using a problem specific ansatz. Additionally, we introduce a hybrid backpropagation algorithm that allows efficient updates of the used weights and biases in the FNN embedding functions. In this paper we showcase the capabilities of TE-QPINNs of a wide range of problems, including the two-dimensional Poisson, Burgers and Navier-Stokes equations. In direct comparison with classical PINNs, this approach showed an ability to achieve superior results while using the same number of parameters, highlighting their potential for more efficient optimization in high-dimensional parameter spaces, which could be transformative for future applications.
The question of whether all species in a multispecies community governed by differential equations can persist for all time is one of the most important in theoretical ecology. Criteria for this property vary widely, ...
The question of whether all species in a multispecies community governed by differential equations can persist for all time is one of the most important in theoretical ecology. Criteria for this property vary widely, asymptotic stability and global asymptotic stability being 2 of the conditions most widely used. Neither of these criteria appears to reflect intuitive concepts of persistence in a satisfactory manner: the 1st because it is only a local condition, the 2nd because it rules out cyclic behavior. A more realistic criterion is that of permanent coexistence, which essentially requires that there should be a region separated from the boundary (corresponding to a zero value of the population of at least 1 sp.) which all orbits enter and remain within. A mathematical technique for establishing permanent coexistence is illustrated by an application to the long-standing problem of predator-mediated coexistence in a 2-prey 1-predator community.
Predator mediated coexistence of 2 competing species with general frequency dependent switching in the predator is examined. The stability criterion used is permanent coexistence. This is a global criterion which ensu...
Predator mediated coexistence of 2 competing species with general frequency dependent switching in the predator is examined. The stability criterion used is permanent coexistence. This is a global criterion which ensures that eventually the species end up in a region M of phase space separated from the boundary (corresponding to extinction of at least one of the species), but which places no restriction on the behavior in M, and so allows, for example, the existence of a stable limit cycle. The principal determinant of survival turns out to be the strength of the switching when 1 prey is rare, the form of the switching elsewhere being irrelevant. Strong switching is a powerful influence for coexistence. In contrast with the conclusion of previous investigations, the influence of switching is complex, and under some circumstances weak switching can actually destroy coexistence in a system which without switching leads to survival of all species.
An FFT method for solving the discrete Poisson equation on a rectangle using a regular hexagonal grid is described and the results obtained for a model Dirichlet problem are compared with those obtained on a rectangul...
An FFT method for solving the discrete Poisson equation on a rectangle using a regular hexagonal grid is described and the results obtained for a model Dirichlet problem are compared with those obtained on a rectangular grid. For a given grid size the results demonstrate that the hexagonal method is more accurate, but rather less efficient, than the usual 5-point method, whereas for comparable accuracy to be achieved by both methods, the hexagonal method was found to be approximately 20 to 30 times faster than the 5-point method for the model problem.
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