Elegant Mathematics Ltd

Do the same with EMBoltzmann solver and complete this 3D simulation in one minute on modern PC!

Our last achievements in CFD allow to simulate sub-, super- and hypersonic aerodynamics and aeroacoustics solving deterministic Boltzmann equation with the complexity of modern Navier-Stocks solver.

NVIDIA Corporation choose Elegant Mathematics Ltd. as the official CUDA consultants

Our last achievements in streaming processor software development are mentioned now at NVIDIA CUDA Zone:

NMR data processing with CUDA
Dense and hierarchical CUDA enabled linear system solvers
Iterative linear system solver on CUDA
CFD with Boltzmann equations on CUDA

Past EM activities

Our Applications

Our Solvers

Fluid Dynamics

Elegant Mathematics Ltd researches and develops the software for numerical simulation of the most physical effects connected with transport of matter. The spectrum of our mathematical solutions ranges from construction of solid state models, which are simulated with the Lame equation (software package EMLibHMatrix), covering the majority of convection-diffusion models, up to the solution of under- and supersonic computational fluid dynamics models, which we successfully solve using the Boltzmann equation approach (software package EMBoltzmann). All our models are based on the advanced scientific development in the field of finite elements and adaptive grids.

Elegant Solvers are exceptionally robust and are able to solve highly ill-conditioned problems, even those, which can not be solved by any other reasonable cost method (software packages EMLibSparse and EMLibIter).

Our new algorithm for the deterministic solution of the Boltzmann equation is based on the multilinear decomposition of velocity space. It enables us to compute the results with the same computational efforts like with the Navier-Stokes approach. Moreover, the simulation of the Boltzmann equation enhances considerably the accuracy of received results.

Solving problems of numerical subsonic gas dynamics for the given shape of a plane, stream speed and an attack angle, you can compute turbulence and all forces acting on the plane. Based on these results, you learn a behavior of a plane on various sites of the flight. If you provide a set of possible geometries of a plane and run the simulations for them, it is possible to foretell flight characteristics for each of them and to find the most appropriate one.

Moreover, if optimum values of force distribution, acting on a plane, are given, you can optimize a form of the wing, (software package EMLibMinimize), without constructing a large set of different wings, and carrying out tests with them.

Our algorithms are used at all stages of development of marine power plant turbines and helicopter propellers.

It is often difficult – and sometimes impossible – to foresee the right form of turbine blades. At the same time, designing and checking on the efficiency of different shapes of blades turn out to be unprofitable, since this method demands great expense of time and resources.

Based on the blade's geometry, temperature, liquid pressure, fluid stream velocity, and other provided physical characteristics, our algorithms will compute both the efficiency of accumulation, in case of power plant turbines, and the efficiency of feedback, in case of marine turbines. Thus, for each possible form of the blade you can learn its characteristics without the actual construction of expensive test experiments to find the optimum parameters.

Using our adaptive grid algorithms with moving boundaries, (software package EMLibGrid), you will obtain not only a full picture of fluid motion, but also of all forces and energy transfer that act to the boundaries. We also suggest to automate the process of search for the optimal shape of blades. Thus, it is enough to specify the criterion function (for example, the efficiency of a turbine), possible restrictions (for example, a condition for a subsonic fluid flow), and varied parameters (for example, some points of spline interpolation for all surface of a turbine blade – software package EMLibSmooth). In this case the search for the optimum design of your turbine is carried out automatically with the help of our EMLibMinimize package.

Suppose that you have the best shape of a turbine blade, it is powerful and optimal in respect to computational fluid dynamics (CFD). Then you need to have it forged with the maximal durability. Here we will help you again, solving Lame equation for non-elastic deformation, (software package EMLibHMatrix). It will allow you to choose the most effective strategy of forging the blade, reducing probability of occurrence of shifts and material defects, and to construct the turbine with the maximal durability and reliability.

Our new project deals with numerical simulations of hypersonic flows with deterministic and stochastic models for the Boltzmann equation. We solve the 3-dimensional, real-time Boltzmann equation with an adaptive grid and billion particles for the stochastic approach, and with million values of average velocities on each physical space cell for deterministic approaches. It gives a chance to carefully predict hypersonic gas flows at the speed close to space flights and strongly turbulent flows, (software packages EMBoltzmann, EMParBoltzmann and EMGPUBoltzmann). Thus, these programs make it possible to compute:

atmospheric entry heating of lander surfaces,
supersonic flow in propulsion compressors,
flow separation phenomenon,

and many other important physical phenomena. The mathematical model, included in the Boltzmann equations, allows simulating most of shock waves and does not distort results even in high Mach numbers.

We suggest to optimize your Ram-, and Scramjets by means of simulating combustion process based on the three-dimensional Boltzmann equation.

For the accurate simulation we offer you a wide range of mesh generators and proper simulation algorithms, as follows:

tensor uniform grids with Toplitz matrices and fast Fourier transformation,
tensor nonuniform grids with Kronecker matrices,

for velocity space simulations (software package EMLibMDD);

adaptive grids with tetrahedral finite elements of first and higher orders,
mesh-free and dual mesh approaches based on the Delaunay tessellation,

for the discretization of physical space (software package EMLibGrid);

3D high-order finite elements

for the best approximation of boundary elements (software package EMLibGrid).

Multiphysics

To find a successful solution of a real problem, it is often necessary to simulate several physical phenomena at a time, which are described by different mathematical models.

For example, you are going to construct an injector for a fuel engine and it is required to optimize its functioning in real time on different operating points, you need to solve the CFD task for fuel propagation together with magnet wave propagation. While constructing a sea jet engine it is important to consider not only hydrodynamics of liquids, but also gas dynamics equations, moving boundary problems, and gas-liquid equilibrium.

A wide range of physical phenomena successfully simulated by our company, and broad experience in their numerical implementation are at your disposal. We are ready to research your problem and find the accurate solution of it.

Wave Simulation

Direct Wave Simulation Problems:

I. Numerical Simulation of Magnetostatics for Prediction of Electro- and Magnetic Field Around Solenoids

This modeling allows to predict parameters of a magnetic field of any form of the magnetic coil or superconducting solenoid without gathering the test stand.

In case you make the challenge with different model parameters, for example, proportions of dimensions of the solenoid, number of coils, etc., – you can find optimum values of these parameters concerning the efficiency or other required characteristics. The test stand assembly is often rather labor-consuming and financially unprofitable. In some cases such a stand is impossible to be constructed because the customer can afford to construct only one device with expected properties. For such cases numerical modeling and optimization of parameters of the device are carried out before this device is designed.

We also provide you with self-acting searching algorithm for the best device.s Thus, it is enough to specify the objective function (for example, the efficiency), varied parameters (for example, thickness of a wire, number of coils in the solenoid), and ranges of their values. In this case, the search for the optimum design of your device will be made automatically for you by our package EMLibMinimize.

We use modern BEM-FEM coupling algorithms for simulation. The boundary element method (BEM) is used for the simulation of magnetic parts, and the finite element method (FEM) for isolators. As we need to move magnetic parts, we do not recalculate the BEM matrix, thus, it reduces numerical errors that occur in FEM during remeshing. It allows us to solve most effectively and precisely the systems with moving parts, for example, rotation of the core of an electromagnet, movement of an electromagnetic valve, and many other similar problems.

II. Numerical Modeling of Radars and Radar Invisibility

For the successful simulation of stealth properties, or optimization of radar antennas, one need to discretize large volumes of physical space with a very fine grid, so the grid step size would be considerably smaller than the wave length. It leads to huge systems of linear equations. The system matrix in these equations is often so large, that it does not fit the main memory of the modern computer. Therefore we have developed for you parallel implementations of these algorithms (software packages with labels MPP and GPU), as well as the algorithms with effective disk memory usage (software packages with labels Out-of-Core).

Here we are using the last achievements in the numerical linear algebra developed by our company. These methods allow to solve sparse unstructured linear equation systems (software package EMLibSparse), and structured linear equation systems (software package EMLibMDD). So, systems of linear equations with several millions of unknowns are solved in a few seconds on a notebook; and systems of linear equations with several billions of unknowns are solved within a reasonable time in a small Linux cluster.

The software package LRA_CDENSE of the predecessor company Elegant Mathematics Inc was used by Lockheed Martin Concern for radar invisibility of stealth aircrafts. The relative wave number in these problems exceeded 500. Now you can make this simulation using our new improved packages EMLibIter, EMLibSparse, EMLibHMatrix on workstations, and with the EMParLibIter, EMParLibSparse packages on massively-parallel computers.

The experience obtained over the last 17 years in software development for the solution of ill-conditioned linear systems allows us to take part in the radar antennas development for the most aerospace applications.

3D Inverse Maxwell Solutions

I. Ultrasound Non-Destructive Diagnostics, Tomography, Acoustic Geological Exploration and Upstreaming

We will help you to find a three-dimensional structure of the mineral distribution if only a few emitters and targets of sound waves on the surface are available, and the studied object has considerably larger dimensions than the wave length.

The current problems can be divided into two basic classes:

The studied object is between sources and detectors, and wave reflection from heterogeneity of the object is not considered. Common applications here are those of medical tomography and, partially, ultrasonic non-destructive diagnostics of materials.
Sources and receivers are situated in such a way, that waves from sources are reflected on heterogeneity of the studied object and registered by receivers. Common applications are problems of seismic prospecting of oil- and gas-fields, and some applications of ultrasonic not-destructive diagnostics of materials.

As this method does not describe wave interference, its use is limited by the condition, that the studied object should have much greater sizes than the characteristic wave length of radiation (software package EMInverse).

The basic complexity of the solution lies in the correct discretization and solution of huge ill-conditioned linear system. The discretization is carried out by finite elements of first order, so the size of each element corresponds approximately to the average size of heterogeneity in the position where the corresponding finite element is situated.

In case we have no a priory information about heterogeneity distribution, we can start calculations with a regular finite element grid, and later, try to improve the grid according to computed properties using the Voronoi-Delaunay approach (software package EMLibGrid). It allows to improve the accuracy of computation, to reduce the total amount of unknown parameters, to save computational time and reduce memory requirements (software packages EMLibIter and EMLibSparse).

Our company has 17 years experience of development of iterative methods for the solution of linear systems of equations that allows us to find the most suitable and steady method of the solution of linear systems and, if necessary, to apply a correct regularization to a singular matrix (software package EMLibIter).

II. Ground Penetrating Radars and Upstreaming

We will help you to find the structure of a three-dimensional object using interference, non-stationary time distribution of sound and electromagnetic waves.

Solving this mathematical problem, you can predict a distribution of petroliferous stratum on the basis of seismic prospecting, and simultaneously conduct correction of drilling. Our algorithms were successfully applied for numerical modeling of distribution of petroliferous stratum, and for correct direction of oil drilling. Sources and receivers of electromagnetic signals were placed on the drill and allowed "to see" the end of a petroliferous stratum already 5-7 meters before, without drilling a dead rock.

For the solution of this problem we use a new method with two different discretization grids. The first grid is used for distribution of electromagnetic waves, it is structured and fine, to make better approximation for physical phenomena of the Maxwell equations. The second grid is used for dielectric permeability approximation, and this grid is based on the 3D adaptive Voronoi-Delaunay tessellation. This dual grid approach allows describing magnetic and electric fields with minimum complexity. At the same time, adaptive grid, on which dielectric permeability is calculated, does not overload the system of equations with excessive unknown variables.

The software packages of Elegant Mathematics run on supercomputers in the High-Performance Supercomputing Center in Maui of Hawaii (USA) on behalf of Mobil Oil Corporation for the solution of ill-conditioned sparse problems with billion unknowns (software package A_SPARSE_T3D – the predecessor of our EMParLibIter and EMParLibSparse software packages).

Nowadays, our algorithms, (software package EMMaxwell), solve 3D inverse ground penetrating radar problems.

Signal Processing

Elegant Mathematics Ltd has several successful approaches for image and video stream compression based the modern Multilinear Adaptive Cross Approximation method. It can increase the compression ratio for video streams, and it is very effective for noise reduction. Using the new numerical method together with the ultra-broadband lenses, we can construct a 3D surface of a pictured object and give the information about materials over its surface.

If there are some pictures of the object, which were made from different points, a new picture of this object at an arbitrary point can be attained, provided that this point is not far from an effective central distance. This method allows substantial simplification of air and space admission: while flying past the object, it is sufficient to make a photo of the desired object only from a few points. Then, using our algorithm based on the EMLibMDD software package, you can restore the picture of the object from any other desired point, and even its 3D surface.

These algorithms can be used also for recognition of flying objects by fragmentary picturing from the ground and comparing them with the picture in the data base. In this case, we do not need to have all possible pictures of the flying object from all possible points in the data base.

We have a wide experience of work with various computer architectures from Embedded Systems up to TOP 1 supercomputers. If necessary, we are ready to transfer our algorithms on your built-in microprocessors, or to offer you a complete solution using the microprocessors of our partners. Successful results in this area are shown with our software package EMGPULibMDD, which was ported to NVIDIA 2xx and Tesla GPU processors.

Nanotechnology

Elegant Mathematics Ltd successfully develops and introduces numerical algorithms for biology, biochemistry and nanoscience.

Our new solution methods as to the Schroedinger equation and its Hartree-Fock approximation and electron density theory are widely used for semiconductor design.

Similar algorithms are developed for exact calculation of electronic density and prediction of a chemical activity of molecules.

Carrying out scientific researches at the level of microcosm of molecules, we try to describe our macrocosm more precisely. So, for example, computing a volume integral of collision energy of molecules, we are able to find out the precise form of the collision kernel in the Boltzmann equation. The last is used for the solution of computational fluid dynamic problems (software package EMBoltzmann).

Already for more than ten years, our multilinear decomposition solver (software package EMLibMDD) has been computing pure fluorescence spectra and relative concentrations of separate substances on the basis of several fluorescence spectra of different mixes without reference spectra. The similar method can be applied in a high-performance liquid chromatography.

Our multilinear decomposition algorithms are applied for increasing accuracy of protein analysis based on nuclear magnetic resonance.

Joint scientific research of Elegant Mathematics Ltd and Scandinavian National Center (Goteborg, Sweden) in the field of signal processing of nuclear magnetic resonance data has no analogues, its results were honored with being published in Nature.

Engine Optimization

Modeling of the fuel engine and optimization of its functioning in real time with different operating points. Due to the data evolutions per minute, quantity, temperature and pressure of inlet air and fuel, detonation time and properties of mixtures in the exhaust, one can build a mathematical model that makes regulation of the fuel force engine possible. It allows the optimization of several parameters of the engine, for example, to tune for fuel consumption or increasing of engine's power.

EMLibIter

The software library is developed for iterative solutions of linear systems and eigenvalue problems. This package contains CG, GMRES, GMRESF, NGMRES, BiCGStab algorithms for the linear system solutions, and Lanczos, Arnoldy and Davidson methods for eigenvalue problems. All these implementations allow you to apply any external preconditioner and Newton acceleration technique.

Free CUDA CG! Take advantage from our full featured 150GFlop/s Conjugated Gradient CUDA and CPU solvers for float, double and quad precision for free.

Optimized for Multi-Core, Vector-Pipeline, Out-of-Core, GPU and MPP; complex numbers support.

Multi-Core – symmetric multi-core multi-processing architectures, for example, Xeon Quad Core;
Vector-Pipeline – vector-pipelined processors and instructions, for example, processors with SSE2 instruction sets;
MPP – massively-parallel distributed memory computer systems, for example, Linux Clusters;
GPU, Cell – co-processors and powerful graphic cards of NVIDIA and Cell IBM;
Out-of-Core – special mathematical algorithms, which allow to use a hard disk memory as a main memory without large slowdown of computations.

EMLibSparse

The software library is developed for solutions of large sparse linear systems and eigenvalue problems. This package contains all iterative methods of the EMLibIter package. The wide range of preconditioners allows to considerably reduce computational costs. The package contains several incomplete LU and Cholesky preconditioners with optimal thresholds for neglecting preconditioner entries. To reduce computational expenses such algorithms can be implemented as of maximum degree, approximate maximum degree, nested dissection, minimal Schur complement norm and optimal permutation.

Free CUDA CG! Take advantage from our full featured 150GFlop/s Conjugated Gradient CUDA and CPU solvers for float, double and quad precision for free.

Optimized for Multi-Core, Vector-Pipeline, Out-of-Core and partially for MPP; complex numbers support.

Multi-Core – symmetric multi-core multi-processing architectures, for example, Xeon Quad Core;
Vector-Pipeline – vector-pipelined processors and instructions, for example, processors with SSE2 instruction sets;
MPP – massively-parallel distributed memory computer systems, for example, Linux Clusters;
GPU, Cell – co-processors and powerful graphic cards of NVIDIA and Cell IBM;
Out-of-Core – special mathematical algorithms, which allow to use a hard disk memory as a main memory without large slowdown of computations.

EMLibHMatrix

The software library is developed for solutions of dense linear systems and eigenvalue problems. This package contains all iterative methods of EMLibIter package together with the complete set of H-Matrix arithmetic. It allows to create, multiply, add up, solve systems with hierarchical matrixes, using direct and iterative methods with hierarchical preconditioners.

Free CUDA CG! Take advantage from our full featured 150GFlop/s Conjugated Gradient CUDA and CPU solvers for float, double and quad precision for free.

Optimized for Multi-Core, Vector-Pipeline, Out-of-Core and partially for MPP; complex numbers support.

Multi-Core – symmetric multi-core multi-processing architectures, for example, Xeon Quad Core;
Vector-Pipeline – vector-pipelined processors and instructions, for example, processors with SSE2 instruction sets;
MPP – massively-parallel distributed memory computer systems, for example, Linux Clusters;
GPU, Cell – co-processors and powerful graphic cards of NVIDIA and Cell IBM;
Out-of-Core – special mathematical algorithms, which allow to use a hard disk memory as a main memory without large slowdown of computations.

EMLibGrid

The software library is developed for construction and recalculation of the adaptive Voronoi-Delaunay grids in three-dimensional space, and for generation of piece-wise constant and linear finite elements on these grids.

Optimized for Multi-Core and Vector-Pipeline.

Multi-Core – symmetric multi-core multi-processing architectures, for example, Xeon Quad Core;
Vector-Pipeline – vector-pipelined processors and instructions, for example, processors with SSE2 instruction sets;
MPP – massively-parallel distributed memory computer systems, for example, Linux Clusters;
GPU, Cell – co-processors and powerful graphic cards of NVIDIA and Cell IBM;
Out-of-Core – special mathematical algorithms, which allow to use a hard disk memory as a main memory without large slowdown of computations.

EMLibSmooth

The software library is developed for construction of spline approximations of any multivariate partially defined functions, or data sets. The package also includes smoothing and noise filtration algorithms.

Optimized for Multi-Core and Vector-Pipeline, complex numbers support.

Multi-Core – symmetric multi-core multi-processing architectures, for example, Xeon Quad Core;
Vector-Pipeline – vector-pipelined processors and instructions, for example, processors with SSE2 instruction sets;
MPP – massively-parallel distributed memory computer systems, for example, Linux Clusters;
GPU, Cell – co-processors and powerful graphic cards of NVIDIA and Cell IBM;
Out-of-Core – special mathematical algorithms, which allow to use a hard disk memory as a main memory without large slowdown of computations.

EMLibMinimize

The software library is developed for unconditional minimization of piecewise smooth function, or for the solution of nonlinear system of equations. This software package automatically tunes to minimization problem choosing between conjugated directions, Broyden, BFGS, and Quasi-Newton methods. The gradient can be very efficiently computed using a very powerful method of Baur-Strassen.

Optimized for Multi-Core and Vector-Pipeline.

Multi-Core – symmetric multi-core multi-processing architectures, for example, Xeon Quad Core;
Vector-Pipeline – vector-pipelined processors and instructions, for example, processors with SSE2 instruction sets;
MPP – massively-parallel distributed memory computer systems, for example, Linux Clusters;
GPU, Cell – co-processors and powerful graphic cards of NVIDIA and Cell IBM;
Out-of-Core – special mathematical algorithms, which allow to use a hard disk memory as a main memory without large slowdown of computations.

EMLibMDD

The software library is developed for multilinear decompositions of all sorts: Parallel Factor (PARAFAC), Alternate Least Squares (ALS), Parallel Decomposition (PARDEC), Multilinear Adaptive Cross Approximation (MACA). Using this library you can treat both fully populated and partially defined data sets.

Optimized for Multi-Core, Vector-Pipeline, partially for Out-of-Core, MPP and GPU, complex numbers support.

Multi-Core – symmetric multi-core multi-processing architectures, for example, Xeon Quad Core;
Vector-Pipeline – vector-pipelined processors and instructions, for example, processors with SSE2 instruction sets;
MPP – massively-parallel distributed memory computer systems, for example, Linux Clusters;
GPU, Cell – co-processors and powerful graphic cards of NVIDIA and Cell IBM;
Out-of-Core – special mathematical algorithms, which allow to use a hard disk memory as a main memory without large slowdown of computations.

EMBoltzmann, EMParBoltzmann, EMGPUBoltzmann

Packages for CFD calculations of sub-, super- and hypersonic aerodynamics and aeroacoustics, turbine compressors and combustion chambers.

The main difference of these packages from EMPartBoltzmann series of packages is based on the grid discretized deterministic approach. This method is heavier for computations, however it guaranties the quadratic convergence of physical variables on the amount of discretized velocity space finite elements.

The software package is designed for the solution of computational fluid dynamics problems with a special emphasis on the solution of super- and hypersonic challenges. Our recent scientific results allow to solve Boltzmann equations with the complexity comparable to the Navier-Stokes algorithms. It vanishes a lot of unstable empirical models and gives us a chance to simulate ram- and scramjets, atmospheric entry heating of lander surfaces, supersonic turbulence, boundary layer problems, flow-separation phenomenon, and many others. Velocity subspace is discretized with an adaptive tensor grid and piece-wise linear finite elements, and physical subspace is discretized with adaptive Voronoi-Delaunay grids.

EMPartBoltzmann

Packages for CFD calculations of sub-, super- and hypersonic aerodynamics and aeroacoustics, turbine compressors and combustion chambers.

These software packages solve the Boltzmann equation with the mesh free particle method for arbitrary gas or gas mixture with possible chemical reactions.

The key idea of these packages is based on the mesh free approach, i.e. you need to provide only a 2D grid with boundary conditions, and this grid can evolve according to your conditions during the computational time. It gives a possibility to make simple and fast estimations of your properties for a computing task, positioning of shock waves and their forms, separation phenomena, and others.

EMBoltzmannModules

This is a supplement package to the EMBoltzmann software package that contains modules for the super- and hypersonic combustion and moving boundaries. It is well tuned for simulation of turbofan, turbojet, ram- and scramjet engines and computing properties of helicopter blades.

EMHydroBoltzmann

This program package solves the equation similar to the Boltzmann approach and allows us to have a multi-velocity model for incompressible fluid dynamics. It is applied to power plant turbines and sea propellers.

Together with the gas dynamical approach it can be used for complicated multi-phase problems for optimizing sea jet turbines, simulating a cavitation, and optimizing multi-phase high-speed flow for water missiles and submarines.

EMMaxwell

This software package is designed for direct 3D Maxwell problem solutions. It allows to simulate eddy current and other electromagnetic effects. Several modern approaches are implemented in this package to solve frequencies from Hz up to GHz range. Is allows us to simulate properties of low frequency electro-motors and high frequency radar cells.

EMInverseTomo, EMInverseMaxwell

These packages are designed for the solution of inverse aeroacoustics of geo- and oil-radar improvements, antenna design and fully featured stealth optimization. The software package EMInverseTomo solves simplification of the Maxwell problem to the case of no diffraction useful for tomography and similar tasks. The software package EMInverseMaxwell solves complete 3D vector Maxwell problems.

EMLame

This program package is designed to optimize forging of turbine compressor blades and other high-strength parts, computation of their static and dynamic loading.

GPU Iterative Linear System Solvers

An average performance of CG, BiCGStab, Lanczos with real arithmetic equals to 7 GFlop/s on a single NVIDIA 260. This performance is achieved from 100,000 unknowns. A complex version of these iterative methods increases twice, 14 GFlop/s. It is almost 50 times faster than on the Quad-Core Xeon 2.6 GHz processor with 666 FSB, which can deliver only 100 MFlop/s.

The main reason of our achievements lies in the comprehensive usage of fast GPU memory.

Block versions of CG [1,2], BiCGStab, GMRES/FGMRES/NGMRES [3], Arnoldy and Davidson algorithms provide even faster performance. In particular zGMRES with 10 simultaneous right hand sides achieves 70 GFlop/s on NVIDIA 260; so it is almost the peak performance of double-precision arithmetic. The similar algorithm on the Quad Core Xeon 2.6 GHz processor with 666 FSB produces only 3 GFlop/s.

Our Kronecker Preconditioner and Kronecker sparse matrix multiplication algorithm [4,5] show the incredible 250 GFlop/s on one NVIDIA GPU 260!

Free CUDA CG! Take advantage from our full featured 150GFlop/s Conjugated Gradient CUDA and CPU solvers for float, double and quad precision for free: EM-Free-CG.zip.

Nikishin A., Yeremin A. Variable block CG algorithms for solving large sparse symmetric positive definite linear systems on parallel computers. I. General iterative scheme. SIAM J. Matrix Anal. Appl. 16(4), 1995, 1135-1153.
Nikishin A., Yeremin A. An automatic scheme for regulating the block size in the block conjugate gradient method for solving linear systems. Zap. Nauchn. Sem. POMI, 2000, J. Math. Sci., 114(6), 2003, 1844-1853.
Kharchenko S., Yeremin A. Multiplicative correction of a matrix on a sequence of subspaces. I. Basic algorithms and theory for the general non symmetric sign-indefinite case. Zap. Nauchn. Sem. POMI, 2002, J. Math. Sci., 121(4), 2004, 2546-2575.
Ibraghimov I. Application of the three-way decomposition for matrix compression. Numer. Lin. Alg. Appl. 2002; 9:551-565.
Ibraghimov I., Sublinear Complexity of Krylov Subspace Method for the Kronecker Product Matrices. In press in Numer. Lin. Alg. Appl, 2009.

GPU Dense Compressed/Hierarchical Linear System Solvers

The subject of discussion is preliminary benchmarks with the NVIDIA 260 GTX hardware for the solution of large dense linear systems with hierarchical structures.

A linear system with 81,920 unknowns was generated and solved in GPU with reasonable speedup in regards to Quad Core Xeon 2.66 HGz.

Matrix generation shows 15 times speedup and delivers the peak performance of 60 GFlop/s.

The iterative solver and compressed matrix multiplication algorithm produce up to 50 times speedup with the peak performance of 6 GFlop/s, equal to 45 GB/s of memory bandwidth.

Ibragimov. CUDA Benchmarks with Dense Hierarchical Linear System Solver. Elegant Math. Tech Report, EM-TR-11, 2009.

Free CUDA CG! Take advantage from our full featured 150GFlop/s Conjugated Gradient CUDA and CPU solvers for float, double and quad precision for free: EM-Free-CG.zip.

Particle Simulation Boltzmann Solver with GPU Kernel

This software package solves the Boltzmann equation with a particle method. It allows making mixtures of particles and particle transformation, i.e. chemical reactions. The collision part is almost 30 times faster than on the modern Quad Core Xeon CPUs. The free flow part is even faster – up to 120 times.

The "simple" problem with 15,000,000 particles can be solved within several minutes without out-of-core algorithms.

The true out-of-core GPU memory to the CPU memory allows to solve huge problems that fits only the main CPU memory. So it gives a possibility to solve real simulations with up to 1,000,000,000 particles during one computation week on one GPU together with the large main CPU memory of 64 Gb.

Deterministic Boltzmann Solver with GPU Acceleration

The main core part of the EMBoltzmann software package was tuned for NVIDIA GPU processors. This part is always about 90-95% of the total computation time of the EMBoltzmann package. The core is now almost 30 times faster than on CPU, so the total computation time was 10 times reduced.

I. Ibragimov, S. Rjasanow. Numerical solution of the Boltzmann equation on the uniform grid. Computing, 69(2):163-186, 2002.
I. Ibragimov, S. Rjasanow. Three Way Decomposition for the Boltzmann Equation. J. Comp. Math., 27(2-3):184-195, 2009.
I. Ibragimov. Fast numerical solution of Boltzmann equation. ZAMM, 7(1):1110101-1110102, 2007.
E. Ibragimova. Solution of industrial CFD problems with structured Boltzmann approach. ZAMM, 7(1):1110105-1110106, 2007.

Fast GPU 3D Inverse Maxwell Solver for Georadars

The recent results with the solution of 3D inverse Maxwell equation and their application to georadar signal processing move the solver from supercomputers to modern PCs. However, the solution time can take hours on a modern workstation. To speed up this computation the core part was tuned to the NVIDIA 2xx series of GPU. That made it possible to solve the most computation 50 times faster; so the total solution time on NVIDIA 280 proved to be 40 times shorter than on modern workstations.

Fast GPU Multilinear NMR Deconvolution

The multilinear decomposition has been recently approved as a new robust method for data processing of multidimensional Nuclear Magnetic Resonance. These results were published in "Nature". The application problem has so huge sizes, that modern workstations need several days to find a solution.

A new high performance implementation of this algorithm for the GPU NVIDIA 2xx stream processors is presented.

The algorithm is based on sparse implementation of parallel factor decomposition algorithm (PARAFAC) that performs alternate sparsely defined least squares minimization.

The nuclear magnetic resonance (NMR) data are usually huge and have a large amount of data entries. To handle them one needs to solve several (often hundreds) almost nonoverlapping regions with a considerably small rank, and then to make a final tune of the total large rank system.

This package allows to compute large regions independently using all power of NVIDIA GPU.

This part often shows significant slow down in the CPU architecture since it requires solving a lot of small problems where CPU cannot show all power.

Since these parts are multithreaded over NVIDIA multiprocessors, it gives us high performance improvement. The only bottleneck in this part is load balancing, however, it is not very important with usage of large data sets. Hence, with data sets published in the articles below, we reached 50-60 times speedup.

The final solution of the joined multidimensional problem with a large rank (often about 1,000 components) was again efficiently implemented, and showed us improvement 30-40 times as much as before compared with Quad Core Xeon workstations.

The core solver of mddnmr distributed by Goteborg University was originally developed by Dr. Ibragimov in 2003 when he was on the University of Saarbruecken position. Right now this package is not supported. Please, consider our fully supported EMLibMDD package instead.

Jaravine V., Ibraghimov I., Orekhov V. Removal of a Time Barrier for High-Resolution Multidimensional NMR Spectroscopy. Nature Methods 3(8):605-607, 2006.
Jaravine V., Zhuravleva A., Permi P., Ibraghimov I., Orekhov V. Hyperdimensional NMR Spectroscopy with Nonlinear Sampling. JACS 130(12):3927-3936, 2008.

Consulting

Our company develops software packages for the solution of industrial problems, thereto we often need to use a huge computer power: massively-parallel, vector-pipeline processors, and out-of-core solutions.

We work out the detailed task description for each customer, and then proceed to solve the challenges in the most effective and fast way. We help the customer to find the solution that best fits the available computer facilities. In case you are deciding what computer hardware to buy, we can help you to make this choice and optimize the price/performance ratio for these particular needs.

When our programs are used, we shall always accompany you and solve together your problems. You will never encounter problems with our software as we shall provide training and advise you on using our products.

We keep abreast of the quickly developing computer industry – many our algorithms are ported on such special hardware platforms like Tesla and 2xx GPU of NVIDIA and AMD-ATI. Based on broad experience on all possible massively-parallel and vector-pipelined platforms that have existed in the world since the end of the last century, we are ready to develop and optimize our algorithms for your actual computer resources.

Our algorithms allow solving problems by means of massively-parallel computers. If you have a computer with 1,000 processors, we solve your problem practically 1,000 times faster.

If your problem is so complex, that your computing facilities do not manage to solve it in a reasonable time, we will try to solve it on our facilities or to place an order for computational time in the leading supercomputer centers.

In case usage of the main memory is the most critical part of modeling, we offer our out-of-core solutions, which almost do not slow down our algorithms, allowing you to expand your main memory to the size of a hard disk.

Elegant Mathematics Ltd hopes for a successful co-operation with you!

Massively-Parallel and Multi-Core Consulting

NVIDIA CUDA Consulting

Last two years Elegant Mathematics develops many CUDA enabled applications that runs on many NVIDIA graphic cards and listed at NVIDIA CUDA Zone:

Hence our CUDA products solve problems in many industrial applications, i.e.:

signal processing for nuclear magnetic resonance, tomography, optical and fluorescence spectroscopy,
sub-, super- and hypersonic aerodynamics and aeroacoustics,
geological exploration problems for electro-magnetic and acoustic mineral investigations,
in addition, we have many basic and advanced numerical linear algebra solvers.

We would be happy to help you from the start choosing right NVIDIA hardware and estimate how many Tesla nodes do you need for solution of your project.

We can provide:

CUDA Consulting,
CUDA Training and teaching, preferably in central Europe,
CUDA outsourcing, and porting your algorithms to CUDA platforms.

Elegant Mathematics Ltd hopes for a successful co-operation with you!

Past Elegant Mathematics Activities

EM is pleasure to present GPL Open Source QBLAS – a quad precision BLAS package for robust calculations.
Conference talk "Sparse NMR Data Processing with Multi-Core and GPU Hardware" at Swiss NMR Symposium at 9 September 2009 in Geneva.
23.06.2009, scientific talk and company presentation at Technical University Braunschweig with the title Fast Numerical Solution of Boltzmann Equation.
Salon International de l'Aeronautique et de l'Espace Le Bourget 2009 - Paris Air Show, June 15-21, 2009
17.03.2009 Win a Sodern EADS competition for NVIDIA CUDA outsourcing
26.01.2009 Invited Talk "Numerical Experiments with Kinetic Boltzmann Solver" at Sankt-Peterbourg State University
16.01.2009 Invited Talk "Numerical Experiments with Kinetic Boltzmann Solver" at Heatphysics Institute, Novosibirsk, the invitation was sent by Akademition A. K. Rebrov
Publication in JACS: Jaravine V., Zhuravleva A., Permi P., Ibraghimov I., Orekhov V. Hyperdimensional NMR Spectroscopy with Nonlinear Sampling. JACS 130(12):3927-3936, 2008.
25.09.2008 Invited Talk "Fast Numerical Solution of Boltzmann Equation" at Heatphysics Institute, Novosibirsk, the invitation was sent by Akademition A. K. Godunov
25.09.2008 Conference talk and presentation at GPR 2008 conference: "GROT-12-3D Radars with the Real Time Solution of 3D Ground-Penetrating Inverse Problem"
Company presentation titeled "Numerical Simulations for Aircraft Industry" and Exibition meeting at Berlin Air Show in ILA Exhibition grounds, Conference Center, Hall 4.
19-23.08.2007 MAKS Airshow and Conference presentation titeled "New Trends of CFD Boltzmann Solutions for Industrial and Military Aviation".
6-th International Congress on Industrial and Applied Mathematics, Zurich, Switzerland, 16-20 July 2007
International Conference on Preconditioning Techniques, Toulouse, France, July 9-12, 2007
Salon International de l'Aeronautique et de l'Espace Le Bourget 2007 - Paris Air Show, June 18-24, 2007
12-15.03.2007 Brussel Von Karman Institute of Fluid Dynamics Converence "Advances on Propultion Technology for Hight-Speed Aircraft".
Publication in Nature M. Journal: Jaravine V., Ibraghimov I., Orekhov V. Removal of a Time Barrier for High-Resolution Multidimensional NMR Spectroscopy. Nature Methods 3(8):605-607, 2006.
TRICAP 2006 talk and presentation.

Into the High-Speed Century – with High-Performance Algorithms!

In our hi-tech century many industrial branches are confronted by difficulties with production, which requires precisely defined properties. It is often necessary to simulate these characteristics to improve the quality and reduce production costs. Most often it occurs in industries with intensive power- and labor-consumption where the development overhead makes up the main part of the total production costs. Sometimes it is impossible to predict the properties of a product before it leaves the assembly line. In this case, the only solution is to make numerical simulations using algorithms, which are precisely developed for specific industrial problems.

Development of the high-performance software is the main duty of our team. The core of our software products is based on our high-performance matrix algorithms that are unique with respect to their accuracy and efficiency.

Based on the wide experience of work with industrial customers, we apply the newest scientific methods and the latest technical achievements for the solution of industrial tasks. We elaborate specific software products individualized for each of our customers, and we are always ready to work with highly complicated requirements. We shall satisfy needs of the most particular customer, and base our cooperation on mutual understanding and decency.

Elena Ibragimova, Managing Director

About Elegant Mathematics Ltd.

Elegant Mathematics was found in 1992 in the USA, (Washington State), for development and manufacture of linear systems and eigenvalue solvers for vector-pipelined and massively-parallel algorithms, that was requested by the industry of the nineties in the last century, for the solution of mathematical, physical, chemical, aerodynamic and other tasks.

Our experts worked on the newest computer facilities of that time: 32 processor vector-pipelined system Cray C90, massively-parallel computers Cray T3D-T3E with 1,024 processors installed in NASA, Cray Research Inc, and the University of Pittsburgh, on many massively-parallel clusters with IRIX, DEC, RS6000, HP, Convex processors in the Hawaiian High-Performance Center, and on a huge amount of Linux clusters worldwide.

In the beginning of the 21st century our company underwent significant changes. There came a new generation of employees, we adjusted to new industrial problems and entered the European market.

In 2006 Elegant Mathematics moved its activity to Germany where its head office is situated at the present. The staff of our company are highly qualified specialists, who received Master's and PhD degrees on graduating from the worldwide recognized universities. Our mathematicians, physicists, chemists and programmers search out new technologies and directions in the industry. The results of their research and development activities are regularly published in scientific journals, such as Nature, JACS, LAA, etc. Hence, you are assured to find the high standard of scientific knowledge at your disposal.

All our products result from the search of the optimal task solution for the customers and application of our know-how in software development. We work out the detailed task description for each customer, and then proceed to solve the challenges in the most effective and fast way. We help the customer to find the solution that best fits the available computer facilities, and to optimize the price/performance ratio for these particular needs.

Here you can download booklets about our company

24 pages, in English, 4Mb
24 Blatt, auf Deutsch, 4Mb
24 pages, en Francaise, 4Mb

To make our pricing easy to understand, please, check your requirements online.
Software Prices

Order amount of licences with agreement on years

120,00 Euro per 1 Licence and 1 Year + VAT

Prices include all updates, new releases and support during validity of licences.

Consulting/Software development

50,00 Euro per Hour + VAT

Prices don't include travel and accommodation expenses in case if your company is more than 300 km away from our main office.

Free CUDA CG! Take advantage from our full featured 150GFlop/s Conjugated Gradient CUDA and CPU solvers for float, double and quad precision for free: EM-Free-CG.zip.

Quad Precision BLAS

A software package with complete BLAS and ATLAS functionality that allow you to make basic linear algebra subroutine with quad precision.

You do not need a hardware support of quad precision, each quad precision value is implemented as sum of two doubles a+b, and a is bigger then b*eps, where eps is machine precision.

This package is allow you to construct iterative linear system solvers and other memory bounded algorithms with high precision and very few overhead in computational time. So, many modern x86 computers run our CG and other iterative linear system solvers based on this package only two times slower, that on double precision achieving 31 decimal precision digits on a solution!

This package is also useful for hardware with no support of double precision, like 8xx and 9xx series of NVIDIA GPU graphic cards, AMD streaming processors and IBM Cell, however, for specific hardware we are strictly recommend to ask us for corresponding version.
You may download:

qblas1.1-src.tar.gz sources for the most Linux and Windows (Cygwin) platform with development files and examples;
libqblas32.a precompiled 32 bit version of this library;
libqblas64.a precompiled 64 bit version of this library;
qblas.h C Header file;
qblas-demo.c C test example that runs CG on quad precision.

This is open source software copyrighted by Elegant Mathematics Ltd and distributed under GNU General Public License.

In case if you want to incorporate this library or its portions into your commercial projects, you can obtain this package or any other derivatives (for example, our iterative linear system solvers) with the commercial license that you can order from Elegant Mathematics Ltd.

Our Contacts

Main Office

Elegant Mathematics Ltd,
Hanauer Muehle 2,
66564, Ottweiler,
Germany

Office in England

Elegant Mathematics Ltd,
Omega 4 No. 116, 6 Roach Road,
London E3 2PA,
United Kingdom

Tel: +49 6821 207 11 74
Fax: +49 6821 920 63 93

Email: info@elegant-mathematics.com

Legal registration numbers:
Cardiff 05975337
HRB 16570
tax payer's account number 030/146/00565
EU VAT account number DE 257663693

Our technical support and information office is always available for you. You can contact us at any time from any point of the world by our contact phones, and receive competitive guidance and consulting about our products and services in English and German languages.

View Larger Map