A Non-Monotone Conjugate Subgradient Type Method for Minimization of Convex Functions

We suggest a conjugate subgradient type method without any line-search for minimization of convex non differentiable functions. Unlike the custom methods of this class, it does not require monotone decrease of the goal function and reduces the implementation cost of each iteration essentially. At the same time, its step-size procedure takes into account behavior of the method along the iteration points. Preliminary results of computational experiments confirm efficiency of the proposed modification.Comment: 11 page

Bridging Proper Orthogonal Decomposition methods and augmented Newton-Krylov algorithms: an adaptive model order reduction for highly nonlinear mechanical problems

This article describes a bridge between POD-based model order reduction techniques and the classical Newton/Krylov solvers. This bridge is used to derive an efficient algorithm to correct, "on-the-fly", the reduced order modelling of highly nonlinear problems undergoing strong topological changes. Damage initiation problems are addressed and tackle via a corrected hyperreduction method. It is shown that the relevancy of reduced order model can be significantly improved with reasonable additional costs when using this algorithm, even when strong topological changes are involved

Small steps and giant leaps: Minimal Newton solvers for Deep Learning

We propose a fast second-order method that can be used as a drop-in replacement for current deep learning solvers. Compared to stochastic gradient descent (SGD), it only requires two additional forward-mode automatic differentiation operations per iteration, which has a computational cost comparable to two standard forward passes and is easy to implement. Our method addresses long-standing issues with current second-order solvers, which invert an approximate Hessian matrix every iteration exactly or by conjugate-gradient methods, a procedure that is both costly and sensitive to noise. Instead, we propose to keep a single estimate of the gradient projected by the inverse Hessian matrix, and update it once per iteration. This estimate has the same size and is similar to the momentum variable that is commonly used in SGD. No estimate of the Hessian is maintained. We first validate our method, called CurveBall, on small problems with known closed-form solutions (noisy Rosenbrock function and degenerate 2-layer linear networks), where current deep learning solvers seem to struggle. We then train several large models on CIFAR and ImageNet, including ResNet and VGG-f networks, where we demonstrate faster convergence with no hyperparameter tuning. Code is available

Different Techniques and Algorithms for Biomedical Signal Processing

This paper is intended to give a broad overview of the complex area of biomedical and their use in signal processing. It contains sufficient theoretical materials to provide some understanding of the techniques involved for the researcher in the field. This paper consists of two parts: feature extraction and pattern recognition. The first part provides a basic understanding as to how the time domain signal of patient are converted to the frequency domain for analysis. The second part provides basic for understanding the theoretical and practical approaches to the development of neural network models and their implementation in modeling biological syste

Analysis of Iterative Methods for the Steady and Unsteady Stokes Problem: Application to Spectral Element Discretizations

A new and detailed analysis of the basic Uzawa algorithm for decoupling of the pressure and the velocity in the steady and unsteady Stokes operator is presented. The paper focuses on the following new aspects: explicit construction of the Uzawa pressure-operator spectrum for a semiperiodic model problem; general relationship of the convergence rate of the Uzawa procedure to classical inf-sup discretization analysis; and application of the method to high-order variational discretization

Comparison of two methods for describing the strain profiles in quantum dots

The electronic structure of interfaces between lattice-mismatched semiconductor is sensitive to the strain. We compare two approaches for calculating such inhomogeneous strain -- continuum elasticity (CE, treated as a finite difference problem) and atomistic elasticity (AE). While for small strain the two methods must agree, for the large strains that exist between lattice-mismatched III-V semiconductors (e.g. 7% for InAs/GaAs outside the linearity regime of CE) there are discrepancies. We compare the strain profile obtained by both approaches (including the approximation of the correct C_2 symmetry by the C_4 symmetry in the CE method), when applied to C_2-symmetric InAs pyramidal dots capped by GaAs.Comment: To appear in J. Appl. Physic

Coupling problem in thermal systems simulations

Building energy simulation is playing a key role in building design in order to reduce the energy consumption and, consequently, the CO2 emissions. An object-oriented tool called NEST is used to simulate all the phenomena that appear in a building. In the case of energy and momentum conservation and species transport, the current solver behaves well, but in the case of mass conservation it takes a lot of time to reach a solution. For this reason, in this work, instead of solving the continuity equations explicitly, an implicit method based on the Trust Region algorithm is proposed. Previously, a study of the properties of the model used by NEST-Building software has been done in order to simplify the requirements of the solver. For a building with only 9 rooms the new solver is a thousand times faster than the current method