Uniform Order Eleven of Eight-Step Hybrid Block Backward Differentiation Formulae for the Solution of Stiff Ordinary Differential Equations

In this research, we developed a uniform order eleven of eight step Second derivative hybrid block backward differentiation formula for integration of stiff systems in ordinary differential equations. The single continuous formulation developed is evaluated at some grid point of x=x_(n+j),j=0,1,2,3,4,5 and6 and its first derivative was also evaluated at off-grid point x=x_(n+j),j=15/2 and grid point x=x_(n+j),j=8. The method is suitable for the solution of stiff ordinary differential equations and the accuracy and stability properties of the newly constructed method are investigated and are shown to be A-stable. Our numerical results obtained are compared with the theoretical solutions as well as ODE23 solver.

Spatial Distribution of Mass and Speed on Movement of Two Shrapnel Discs of Variable Thickness in Explosive Load

Results of studies of the explosive expansion of a set of shrapnel discs of natural crushing are described in the article. Set consisted of two disks of different thickness, but of a fixed total mass. The studies were carried out by computer simulation of shock-wave processes in a continuous formulation using the ANSYS/LS-DYNA program. The program of computer design foresees development of three-dimensional certainly-element model including, in accordance with symmetry of the examined system fourth part of the examined explosive system of casting block, with imposed on its knots of the proper scopes terms concerted with taken mixed Lagrangian-Eulerian approach within the framework of the continuum model. The effect of the order of installing disks of different thickness on the distribution of their mass and its velocity in the middle of the meridional angle of expansion is established. The analysis of the computer simulation presented on the basis of numerical studies on the distribution of the mass of the disks and its velocity of motion suggests that to create a narrow high-speed uniformly filled fragmentary mass of the axial flow, it is necessary to change the geometric shape of the disc so that in the central angular zones of the disks. This allows the velocity of the fragment mass to be aligned along the radius of the discs and to fill the first angular zones with the required mass of fragments.

Gradient-Based Optimization for Bayesian Preference Elicitation

Effective techniques for eliciting user preferences have taken on added importance as recommender systems (RSs) become increasingly interactive and conversational. A common and conceptually appealing Bayesian criterion for selecting queries is expected value of information (EVOI). Unfortunately, it is computationally prohibitive to construct queries with maximum EVOI in RSs with large item spaces. We tackle this issue by introducing a continuous formulation of EVOI as a differentiable network that can be optimized using gradient methods available in modern machine learning computational frameworks (e.g., TensorFlow, PyTorch). We exploit this to develop a novel Monte Carlo method for EVOI optimization, which is much more scalable for large item spaces than methods requiring explicit enumeration of items. While we emphasize the use of this approach for pairwise (or k-wise) comparisons of items, we also demonstrate how our method can be adapted to queries involving subsets of item attributes or “partial items,” which are often more cognitively manageable for users. Experiments show that our gradient-based EVOI technique achieves state-of-the-art performance across several domains while scaling to large item spaces.