SHOT-R: A next generation algorithm for particle kinematics analysis

Analysis of live-imaging experiments is crucial to decipher a plethora of cellular mechanisms within physiological and pathological contexts. Kymograph, i.e. graphical representations of particle spatial position over time, and single particle tracking (SPT) are the currently available tools to extract information on particle transport and velocity. However, the spatiotemporal approximation applied in particle trajectory reconstruction with those methods intrinsically prevents an accurate analysis of particle kinematics and of instantaneous behaviours. Here, we present SHOT-R, a novel numerical method based on polynomial reconstruction of 4D (3D+time) particle trajectories. SHOT-R, contrary to other tools, computes bona fide instantaneous and directional velocity, and acceleration. Thanks to its high order continuous reconstruction it allows, for the first time, kinematics analysis of co-trafficked particles. Overall, SHOT-R is a novel, versatile, and physically reliable numerical method that achieves all-encompassing particle kinematics studies at unprecedented accuracy on any live-imaging experiment where the spatiotemporal coordinates can be retrieved.

Joining Forces for Reconstruction Inverse Problems

A spectral inverse problem concerns the reconstruction of parameters of a parent graph from prescribed spectral data of subgraphs. Also referred to as the P–NP Isomorphism Problem, Reconstruction or Exact Graph Matching, the aim is to seek sets of parameters to determine a graph uniquely. Other related inverse problems, including the Polynomial Reconstruction Problem (PRP), involve the recovery of graph invariants. The PRP seeks to extract the spectrum of a graph from the deck of cards each showing the spectrum of a vertex-deleted subgraph. We show how various algebraic methods join forces to reconstruct a graph or its invariants from a minimal set of restricted eigenvalue-eigenvector information of the parent graph or its subgraphs. We show how functions of the entries of eigenvectors of the adjacency matrix A of a graph can be retrieved from the spectrum of eigenvalues of A. We establish that there are two subclasses of disconnected graphs with each card of the deck showing a common eigenvalue. These could occur as possible counter examples to the positive solution of the PRP.

Graphs Having Most of Their Eigenvalues Shared by a Vertex Deleted Subgraph

Let G be a simple graph and {1,2,…,n} be its vertex set. The polynomial reconstruction problem asks the question: given a deck P(G) containing the n characteristic polynomials of the vertex deleted subgraphs G−1, G−2, …, G−n of G, can ϕ(G,x), the characteristic polynomial of G, be reconstructed uniquely? To date, this long-standing problem has only been solved in the affirmative for some specific classes of graphs. We prove that if there exists a vertex v such that more than half of the eigenvalues of G are shared with those of G−v, then this fact is recognizable from P(G), which allows the reconstruction of ϕ(G,x). To accomplish this, we make use of determinants of certain walk matrices of G. Our main result is used, in particular, to prove that the reconstruction of the characteristic polynomial from P(G) is possible for a large subclass of disconnected graphs, strengthening a result by Sciriha and Formosa.

Alternating Polynomial Reconstruction Method for Hyperbolic Conservation Laws

We propose a new multi-moment numerical solver for hyperbolic conservation laws by using the alternating polynomial reconstruction approach. Unlike existing multi-moment schemes, our approach updates model variables by implementing two polynomial reconstructions alternately. First, Hermite interpolation reconstructs the solution within the cell by matching the point-based variables containing both physical values and their spatial derivatives. Then the reconstructed solution is updated by the Euler method. Second, we solve a constrained least-squares problem to correct the updated solution to preserve the conservation laws. Our method enjoys the advantages of a compact numerical stencil and high-order accuracy. Fourier analysis also indicates that our method allows a larger CFL number compared with many other high-order schemes. By adding a proper amount of artificial viscosity, shock waves and other discontinuities can also be computed accurately and sharply without solving an approximated Riemann problem.

Two-way nesting ocean models with different vertical coordinates

<p>The NEMO platform possesses a versatile block-structured refinement capacity thanks to the AGRIF library. It is however restricted up to versions 4.0x, to the horizontal direction only. In the present work, we explain how we extended the nesting capabilities to the vertical direction, a feature which can appear, in some circumstances, as beneficial as refining the horizontal grid.</p><p>Doing so is not a new concept per se, except that we consider here the general case of child and parent grids with possibly different vertical coordinate systems, hence not logically defined from each other as in previous works. This enables connecting together for instance z (geopotential), s (terrain following) or eventually ALE (Arbitrary Lagrangian Eulerian) coordinate systems. In any cases, two-way exchanges are enabled, which is the other novel aspect tackled here.  </p><p>Considering the vertical nesting procedure itself, we describe the use of high order conservative and monotone polynomial reconstruction operators to remap from parent to child grids and vice versa. Test cases showing the feasibility of the approach are presented, with particular attention on the connection of s and z grids in the context of gravity flow modelling. This work can be considered as a preliminary step towards the application of the vertical nesting concept over major overflow regions in global realistic configurations. The numerical representation of these areas is indeed known to be particularly sensitive to the vertical coordinate formulation. More generally, this work illustrates the typical methodology from the development to the validation of a new feature in the NEMO model.</p>

Continuous Regularized Least Squares Polynomial Approximation on the Sphere

In this paper, we consider the problem of polynomial reconstruction of smooth functions on the sphere from their noisy values at discrete nodes on the two-sphere. The method considered in this paper is a weighted least squares form with a continuous regularization. Preliminary error bounds in terms of regularization parameter, noise scale, and smoothness are proposed under two assumptions: the mesh norm of the data point set and the perturbation bound of the weight. Condition numbers of the linear systems derived by the problem are discussed. We also show that spherical tϵ-designs, which can be seen as a generalization of spherical t-designs, are well applied to this model. Numerical results show that the method has good performance in view of both the computation time and the approximation quality.

ICCM2016: Multi-Patch-Based B-Spline Method for Solid and Structure

In this paper, the solution domain is divided into multi-patches on which B-spline basis functions are used for approximation. The different B-spline patches are connected by a transition region which is described by several elements. The basis functions in different B-spline patches are modified in the transition region to ensure the basic polynomial reconstruction condition and the compatibility of displacements and their gradients. This new method is applied to the stress analysis of 2D elasticity problems in order to investigate its performance. Numerical results show that the present method is accurate and stable.

Open-boundary spectral and flux-balance Vlasov simulation

Simulations of one-dimensional Vlasov–Maxwell solutions with non-periodic boundary conditions are discussed. Results obtained using a recently developed filtered flux-balance simulation system are compared to those obtained using a filtered, Fourier–Fourier transformed system. Excellent agreement is confirmed except for the appearance of the Gibbs phenomenon on the discontinuous simulated solutions of the transformed system. Recovery of the flux-balance results from the Fourier transformed results using the inverse polynomial reconstruction method is demonstrated.