A Self-organizing Adaptive-resolution Particle Method with Anisotropic Kernels

AbstractAdaptive-resolution particle methods reduce the computational cost for problems that develop a wide spectrum of length scales in their solution. Concepts from self-organization can be used to determine suitable particle distributions, sizes, and numbers at runtime. If the spatial derivatives of the function strongly depend on the direction, the computational cost and the required number of particles can be further reduced by using anisotropic particles. Anisotropic particles have ellipsoidal influence regions (shapes) that are locally aligned with the direction of smallest variation of the function. We present a framework that allows consistent evaluation of linear differential operators on arbitrary distributions of anisotropic particles. We further extend the concept of particle self-organization to anisotropic particles, where also the directions and magnitudes of anisotropy are self-adapted. We benchmark the accuracy and efficiency of the method in a number of 2D and 3D test cases

Overcoming data sparsity

Unilever is currently designing and testing recommendation algorithms that would make recommendations about products to online customers given the customer ID and the current content of their basket. Unilever collected a large amount of purchasing data that demonstrates that most of the items (around 80%) are purchased infrequently and account for 20% of the data while frequently purchased items account for 80% of the data. Therefore, the data is sparse, skewed and demonstrates a long tail. Attempts to incorporate the data from the long tail, so far have proved difficult and current Unilever recommendation systems do not incorporate the information about infrequently purchased items. At the same time, these items are more indicative of customers' preferences and Unilever would like to make recommendations from/about these items, i.e. give a rank ordering of available products in real time. Study Group suggested to use the approach of bipartite networks to construct a similarity matrix that would allow the recommendation scores for different products to be computed. Given a current basket and a customer ID, this approach gives recommendation scores for each available item and recommends the item with the highest score that is not already in the basket. The similarity matrix can be computed offline, while recommendation score calculations can be performed live. This report contains the summary of Study Group findings together with the insights into properties of the similarity matrix and other related issues, such as recommendation for the data collection

Exact stochastic simulations of intra-cellular transport by mechanically coupled molecular motors

Numerous processes in live cells depend on active, motor-driven transport of cargo and organelles along the filaments of the cytoskeleton. Understanding the resulting dynamics and the underlying biophysical and biochemical processes critically depends on computational models of intra-cellular transport. A number of motor{cargo models have hence been developed to reproduce experimentally observed transport dynamics on various levels of detail. Computer simulations of these models have so far exclusively relied on approximate time-discretization methods. Using a consensus motor{cargo model that unites several existing models from the literature we demonstrate that this simulation approach is not correct. The numerical errors do not vanish even for arbitrarily small time steps, rendering the algorithm inconsistent. We propose a novel exact simulation algorithm for intra-cellular transport models that is also computationally more efficient than the approximate one. Furthermore, we introduce a robust way of analyzing the different time scales in the model dynamics using velocity autocorrelation functions

On the Use of Conformal Models and Methods in Dosimetry for Nonuniform Field Exposure

Numerical artifacts affect the reliability of computational dosimetry of human exposure to low-frequency electromagnetic fields. In the guidelines of the International Commission of Non-Ionizing Radiation Protection (ICNIRP), a reduction factor of 3 was considered to take into account numerical uncertainties when determining the limit values for human exposure. However, the rationale for this value is unsure. The IEEE International Committee on Electromagnetic Safety has published a research agenda to resolve numerical uncertainties in low-frequency dosimetry. For this purpose, intercomparison of results computed using different methods by different research groups is important. In previous intercomparison studies for low-frequency exposures, only a few computational methods were used, and the computational scenario was limited to a uniform magnetic field exposure. This study presents an application of various numerical techniques used: different Finite Element Method (FEM) schemes, Method of Moments (MoM) and Boundary Element Method (BEM) variants and finally by using a hybrid FEM/BEM approach. As a computational example, the induced electric field in the brain by the coil used in transcranial magnetic stimulation is investigated. Intercomparison of the computational results are presented qualitatively. Some remarks are given for the effectiveness and limitations of application of the various computational methods

An asymptotic analysis of the buckling of a highly shear resistant vesicle

The static compression between two smooth plates of an axisymmetric capsule or vesicle is investigated by means of asymptotic analysis. The governing equations of the vesicle are derived from thin-shell theory and involve a bending stiffness B, a shear modulus H, the unstressed vesicle radius a and a constant surface-area constraint. The sixth-order free-boundary problem obtained by a balance-of-forces approach is addressed in the limit when the dimensionless parameter C = Ha2/B is large and the plate displacements are small. When the plate displacement is of order aC?1/2, the vesicle undergoes a sub-critical buckling instability which is captured by leading-order asymptotics. Asymptotic linear and quadratic force–displacement relations for the pre- and post-buckled solutions are determined. The leading-order post-buckled solution is described by a simple fourth-order problem, exhibiting stress-focusing with stretching and bending confined to a narrow boundary layer. In contrast, in the pre-buckled state, stretching occurs over a larger length scale than bending. The results are in good qualitative agreement with numerical simulations for finite values of

Fast neighbor lists for adaptive-resolution particle simulations

Particle methods provide a simple yet powerful framework for simulating both discrete and continuous systems either deterministically or stochastically. The inherent adaptivity of particle methods is particularly appealing when simulating multiscale models or systems that develop a wide spectrum of length scales. Evaluating particle–particle interactions using neighbor-finding algorithms such as cell lists or Verlet lists, however, quickly becomes inefficient in adaptive-resolution simulations where the interaction cutoff radius is a function of space. We present a novel adaptive-resolution cell list algorithm and the associated data structures that provide efficient access to the interaction partners of a particle, independent of the (potentially continuous) spectrum of cutoff radii present in a simulation. We characterize the computational cost of the proposed algorithm for a wide range of resolution spans and particle numbers, showing that the present algorithm outperforms conventional uniform-resolution cell lists in most adaptive-resolution settings

Discretization correction of general integral PSE Operators for particle methods

The general integral particle strength exchange (PSE) operators [J.D. Eldredge, A. Leonard, and T. Colonius, J. Comput. Phys. 180, 686–709 (2002)] approximate derivatives on scattered particle locations to any desired order of accuracy. Convergence is, however, limited to a certain range of resolutions. For high-resolution discretizations, the constant discretization error dominates and prevents further convergence. We discuss a consistent discretization correction framework for PSE operators that yields the desired rate of convergence for any resolution, both on uniform Cartesian and irregular particle distributions, as well as near boundaries. These discretization-corrected (DC) PSE operators also have no overlap condition, enabling the kernel width to become arbitrarily small for constant interparticle spacing. We show that, on uniform Cartesian particle distributions, this leads to a seamless transition between DC PSE operators and classical finite difference stencils. We further identify relationships between DC PSE operators and operators used in corrected smoothed particle hydrodynamics and reproducing kernel particle methods. We analyze the presented DC PSE operators with respect to accuracy, rate of convergence, computational efficiency, numerical dispersion, numerical diffusion, and stability

A self-organizing Lagrangian particle method for adaptive-resolution advection–diffusion simulations

We present a novel adaptive-resolution particle method for continuous parabolic problems. In this method, particles self-organize in order to adapt to local resolution requirements. This is achieved by pseudo forces that are designed so as to guarantee that the solution is always well sampled and that no holes or clusters develop in the particle distribution. The particle sizes are locally adapted to the length scale of the solution. Differential operators are consistently evaluated on the evolving set of irregularly distributed particles of varying sizes using discretization-corrected operators. The method does not rely on any global transforms or mapping functions. After presenting the method and its error analysis, we demonstrate its capabilities and limitations on a set of two- and three-dimensional benchmark problems. These include advection-diffusion, the Burgers equation, the Buckley-Leverett five-spot problem, and curvature-driven level-set surface refinement