Propagating systems of dense linear integer constraints

In interval propagation approaches to solving nonlinear constraints over reals it is common to build stronger propagators from systems of linear equations. This, as far as we are aware, is not pursued for integer finite domain propagation. In this paper we show how we use interval Gauss–Jordan elimination to build stronger propagators for an integer propagation solver. In a similar fashion we present an interval Fourier elimination preconditioning technique to generate redundant linear constraints from a system of linear inequalities. We show how to convert the resulting interval propagators into integer propagators. This allows us to use existing integer solvers. We give experiments that show how these preconditioning techniques can improve propagation performance on dense systems.     

Preconditioning for a Class of Spectral Differentiation Matrices

We propose an efficient preconditioning technique for the numerical solution of first-order partial differential equations (PDEs). This study has been motivated by the computation of an invariant torus of a system of ordinary differential equations. We find the torus by discretizing a nonlinear first-order PDE with a full two-dimensional Fourier spectral method and by applying Newton’s method. This leads to large nonsymmetric linear algebraic systems. The sparsity pattern of these systems makes the use of direct solvers prohibitively expensive. Commonly used iterative methods, e.g., GMRes, BiCGStab and CGNR (Conjugate Gradient applied to the normal equations), are quite slow to converge. Our preconditioner is derived from the solution of a PDE with constant coefficients; it has a fast implementation based on the Fast Fourier Transform (FFT). It effectively increases the clustering of the spectrum, and speeds up convergence significantly. We demonstrate the performance of the preconditioner in a number of linear PDEs and the nonlinear PDE arising from the Van der Pol oscillator     

Theoretically supported scalable BETI method for variational inequalities

Summary The Boundary Element Tearing and Interconnecting (BETI) methods were recently introduced as boundary element counterparts of the well established Finite Element Tearing and Interconnecting (FETI) methods. Here we combine the BETI method preconditioned by the projector to the “natural coarse grid” with recently proposed optimal algorithms for the solution of bound and equality constrained quadratic programming problems in order to develop a theoretically supported scalable solver for elliptic multidomain boundary variational inequalities such as those describing the equilibrium of a system of bodies in mutual contact. The key observation is that the “natural coarse grid” defines a subspace that contains the solution, so that the preconditioning affects also the non-linear steps. The results are validated by numerical experiments.      

Efficient monolithic simulation techniques for the stationary Lattice Boltzmann equation on general meshes

In this paper, we present special discretization and solution techniques for the numerical simulation of the Lattice Boltzmann equation (LBE). In Hübner and Turek (Computing, 81:281–296, 2007), the concept of the generalized mean intensity had been proposed for radiative transfer equations which we adapt here to the LBE, treating it as an analogous (semi-discretized) integro-differential equation with constant characteristics. Thus, we combine an efficient finite difference-like discretization based on short-characteristic upwinding techniques on unstructured, locally adapted grids with fast iterative solvers. The fully implicit treatment of the LBE leads to nonlinear systems which can be efficiently solved with the Newton method, even for a direct solution of the stationary LBE. With special exact preconditioning by the transport part due to the short-characteristic upwinding, we obtain an efficient linear solver for transport dominated configurations (macroscopic Stokes regime), while collision dominated cases (Navier-Stokes regime for larger Re numbers) are treated with a special block-diagonal preconditioning. Due to the new generalized equilibrium formulation (GEF) we can combine the advantages of both preconditioners, i.e. independence of the number of unknowns for convection-dominated cases with robustness for stiff configurations. We further improve the GEF approach by using hierarchical multigrid algorithms to obtain grid-independent convergence rates for a wide range of problem parameters, and provide representative results for various benchmark problems. Finally, we present quantitative comparisons between a highly optimized CFD-solver based on the Navier-Stokes equation (FeatFlow) and our new LBE solver (FeatLBE).     

Real-time reliability prediction for dynamic systems with both deteriorating and unreliable components

As an important technology for predictive maintenance, failure prognosis has attracted more and more attentions in recent years. Real-time reliability prediction is one effective solution to failure prognosis. Considering a dynamic system that is composed of normal, deteriorating and unreliable components, this paper proposes an integrated approach to perform real-time reliability prediction for such a class of systems. For a deteriorating component, the degradation is modeled by a time-varying fault process which is a linear or approximately linear function of time. The behavior of an unreliable component is described by a random variable which has two possible values corresponding to the operating and malfunction conditions of this component. The whole proposed approach contains three algorithms. A modified interacting multiple model particle filter is adopted to estimate the dynamic system’s state variables and the unmeasurable time-varying fault. An exponential smoothing algorithm named the Holt’s method is used to predict the fault process. In the end, the system’s reliability is predicted in real time by use of the Monte Carlo strategy. The proposed approach can effectively predict the impending failure of a dynamic system, which is verified by computer simulations based on a three-vessel water tank system.     

Usability levels for sparse linear algebra components

Sparse matrix computations are ubiquitous in high‐performance computing applications and often are their most computationally intensive part. In particular, efficient solution of large‐scale linear systems may drastically improve the overall application performance. Thus, the choice and implementation of the linear system solver are of paramount importance. It is difficult, however, to navigate through a multitude of available solver packages and to tune their performance to the problem at hand, mainly because of the plethora of interfaces, each requiring application adaptations to match the specifics of solver packages. For example, different ways of setting parameters and a variety of sparse matrix formats hinder smooth interactions of sparse matrix computations with user applications. In this paper, interfaces designed for components that encapsulate sparse matrix computations are discussed in the light of their matching with application usability requirements. Consequently, we distinguish three levels of interfaces, high, medium, and low, corresponding to the degree of user involvement in the linear system solution process and in sparse matrix manipulations. We demonstrate when each interface design choice is applicable and how it may be used to further users' scientific goals. Component computational overheads caused by various design choices are also examined, ranging from low level, for matrix manipulation components, to high level, in which a single component contains the entire linear system solver. Published in 2007 by John Wiley & Sons, Ltd.     

Parallel iterative solvers for sparse linear systems in circuit simulation

For the solution of sparse linear systems from circuit simulation whose coefficient matrices include a few dense rows and columns, a parallel iterative algorithm with distributed Schur complement preconditioning is presented. The parallel efficiency of the solver is increased by transforming the equation system into a problem without dense rows and columns as well as by exploitation of parallel graph partitioning methods. The costs of local, incomplete LU decompositions are decreased by fill-in reducing reordering methods of the matrix and a threshold strategy for the factorization. The efficiency of the parallel solver is demonstrated with real circuit simulation problems on PC clusters.     

Adaptation of robot’s perception of fuzzy linguistic information by evaluating vocal cues for controlling a robot manipulator

This article proposes a method for adapting a robot’s perception of fuzzy linguistic information by evaluating vocal cues. The robot’s perception of fuzzy linguistic information such as “very little” depends on the environmental arrangements and the user’s expectations. Therefore, the robot’s perception of the corresponding environment is modified by acquiring the user’s perception through vocal cues. Fuzzy linguistic information related to primitive movements is evaluated by a behavior evaluation network (BEN). A vocal cue evaluation system (VCES) is used to evaluate the vocal cues for modifying the BEN. The user’s satisfactory level for the robot’s movements and the user’s willingness to change the robot’s perception are identified based on a series of vocal cues to improve the adaptation process. A situation of cooperative rearrangement of the user’s working space is used to illustrate the proposed system by a PA-10 robot manipulator.     

Parallel methods for optimality criteria-based topology optimization

Topology optimization problems require the repeated solution of finite element problems that are often extremely ill-conditioned due to highly heterogeneous material distributions. This makes the use of iterative linear solvers inefficient unless appropriate preconditioning is used. Even then, the solution time for topology optimization problems is typically very high. These problems are addressed by considering the use of non-overlapping domain decomposition-based parallel methods for the solution of topology optimization problems. The parallel algorithms presented here are based on the solid isotropic material with penalization (SIMP) formulation of the topology optimization problem and use the optimality criteria method for iterative optimization. We consider three parallel linear solvers to solve the equilibrium problem at each step of the iterative optimization procedure. These include two preconditioned conjugate gradient (PCG) methods: one using a diagonal preconditioner and one using an incomplete LU factorization preconditioner with a drop tolerance. A third substructuring solver that employs a hybrid of direct and iterative (PCG) techniques is also studied. This solver is found to be the most effective of the three solvers studied, both in terms of parallel efficiency and in terms of its ability to mitigate the effects of ill-conditioning. In addition to examining parallel linear solvers, we consider the parallelization of the iterative optimality criteria method. To tackle checkerboarding and mesh dependence, we propose a multi-pass filtering technique that limits the number of “ghost” elements that need to be exchanged across interprocessor boundaries.     

Algebraic multigrid methods based on element preconditioning

This paper presents a new algebraic multigrid (AMG) solution strategy for large linear systems with a sparse matrix arising from a finite element discretization of some self-adjoint, second order, scalar, elliptic partial differential equation. The AMG solver is based on Ruge/Stübens method. Ruge/Stübens algorithm is robust for M-matrices, but unfortunately the “region of robustness“ between symmetric positive definite M-matrices and general symmetric positive definite matrices is very fuzzy.

For this reason the so-called element preconditioning technique is introduced in this paper. This technique aims at the construction of an M-matrix that is spectrally equivalent to the original stiffness matrix. This is done by solving small restricted optimization problems. AMG applied to the spectrally equivalent M-matrix instead of the original stiffness matrix is then used as a preconditioner in the conjugate gradient method for solving the original problem.

The numerical experiments show the efficiency and the robustness of the new preconditioning method for a wide class of problems including problems with anisotropic elements.     

Parallel simulation of anisotropic diffusion with human brain DT-MRI Data

We conduct simulations for the 3D unsteady state anisotropic diffusion process with DT-MRI data in the human brain by discretizing the governing diffusion equation on Cartesian grid and adopting a high performance differential-algebraic equation (DAE) solver, the parallel version of implicit differential-algebraic (IDA) solver, to tackle the resulting large scale system of DAEs. Parallel preconditioning techniques including sparse approximate inverse and banded-block-diagonal preconditioners are used with the GMRES method to accelerate the convergence rate of the iterative solution. We then investigate and compare the efficiency and effectiveness of the two parallel preconditioners. The experimental results of the diffusion simulations on a parallel supercomputer show that the sparse approximate inverse preconditioning strategy, which is robust and efficient with good scalability, gives a much better overall performance than the banded-block-diagonal preconditioner.     

Don’t care in SMT: building flexible yet efficient abstraction/refinement solvers

This paper describes a method for combining “off-the-shelf” SAT and constraint solvers for building an efficient Satisfiability Modulo Theories (SMT) solver for a wide range of theories. Our method follows the abstraction/refinement approach to simplify the implementation of custom SMT solvers. The expected performance penalty by not using an interweaved combination of SAT and theory solvers is reduced by generalising a Boolean solution of an SMT problem first via assigning don’t care to as many variables as possible. We then use the generalised solution to determine a thereby smaller constraint set to be handed over to the constraint solver for a background theory. We show that for many benchmarks and real-world problems, this optimisation results in considerably smaller and less complex constraint problems. The presented approach is particularly useful for assembling a practically viable SMT solver quickly, when neither a suitable SMT solver nor a corresponding incremental theory solver is available. We have implemented our approach in the ABsolver framework and applied the resulting solver successfully to an industrial case-study: the verification problems arising in verifying an electronic car steering control system impose non-linear arithmetic constraints, which do not fall into the domain of any other available solver.     

Selection of modeling coordinates for forward dynamic multibody simulations

Modeling mechanical systems in a manner that allows the models to be simulated quickly is vital in many fields, such as real-time simulation and control. Modeling these systems using their symbolic equations, rather than the more widely-used numerical methods, generally produces faster solution times. However, the number, complexity, and computational efficiency of these equations is highly dependent upon which coordinate set was used to model the system. Most coordinate selection methods established thus far are based on the assumption that minimizing the number of modeling coordinates will produce models with faster simulation times. This paper will show that this technique is not always valid and proposes a new technique of selecting a system’s coordinates based on a series of heuristics. A large part of these heuristics will be established by closely analyzing a specific technique used to formulate a system’s equations, and the effect each step of this formulation process will have on the complexity of the final system equations.     

Denoising by BV-duality

In this paper we apply Meyer’s G-norm for image processing problems. We use a definition of the G-norm as norm of linear functionals on BV, which seems to be more feasible for numerical computation. We establish the equivalence between Meyer’s original definition and ours and show that computing the norm can be expressed as an interface problem. This allows us to define an algorithm based on the level set method for its solution. Alternatively we propose a fixed point method based on mean curvature type equations. A computation of the G-norm according to our definition additionally gives functions which can be used for denoising of simple structures in images under a high level of noise. We present some numerical computations of this denoising method which support this claim.Dedicated to David Gottlieb on his 60th birthday.Stefan Kindermann on leave from University Linz, Austria.     

Adaptive multigrid for finite element computations in plasticity

The solution of the system of equilibrium equations is the most time-consuming part in large-scale finite element computations of plasticity problems. The development of efficient solution methods are therefore of utmost importance to the field of computational plasticity. Traditionally, direct solvers have most frequently been used. However, recent developments of iterative solvers and preconditioners may impose a change. In particular, preconditioning by the multigrid technique is especially favorable in FE applications.The multigrid preconditioner uses a number of nested grid levels to improve the convergence of the iterative solver. Prolongation of fine-grid residual forces is done to coarser grids and computed corrections are interpolated to the fine grid such that the fine-grid solution successively is improved. By this technique, large 3D problems, invincible for solvers based on direct methods, can be solved in acceptable time at low memory requirements. By means of a posteriori error estimates the computational grid could successively be refined (adapted) until the solution fulfils a predefined accuracy level. In contrast to procedures where the preceding grids are erased, the previously generated grids are used in the multigrid algorithm to speed up the solution process.The paper presents results using the adaptive multigrid procedure to plasticity problems. In particular, different error indicators are tested.     

A finite element based level set method for two-phase incompressible flows

We present a method that has been developed for the efficient numerical simulation of two-phase incompressible flows. For capturing the interface between the phases the level set technique is applied. The continuous model consists of the incompressible Navier–Stokes equations coupled with an advection equation for the level set function. The effect of surface tension is modeled by a localized force term at the interface (so-called continuum surface force approach). For spatial discretization of velocity, pressure and the level set function conforming finite elements on a hierarchy of nested tetrahedral grids are used. In the finite element setting we can apply a special technique to the localized force term, which is based on a partial integration rule for the Laplace–Beltrami operator. Due to this approach the second order derivatives coming from the curvature can be eliminated. For the time discretization we apply a variant of the fractional step θ-scheme. The discrete saddle point problems that occur in each time step are solved using an inexact Uzawa method combined with multigrid techniques. For reparametrization of the level set function a new variant of the fast marching method is introduced. A special feature of the solver is that it combines the level set method with finite element discretization, Laplace–Beltrami partial integration, multilevel local refinement and multigrid solution techniques. All these components of the solver are described. Results of numerical experiments are presented.     

Quantized stabilization for stochastic discrete‐time systems with multiplicative noises

This paper considers the problem of quadratic mean‐square stabilization of a class of stochastic linear systems using quantized state feedback. Different from the previous works where the system is restricted to be deterministic, we focus on stochastic systems with multiplicative noises in both the system matrix and the control input. A static quantizer is used in the feedback channel. It is shown that the coarsest quantization density that permits stabilization of a stochastic system with multiplicative noises in the sense of quadratic mean‐square stability is achieved with the use of a logarithmic quantizer, and the coarsest quantization density is determined by an algebraic Riccati equation, which is also the solution to a special stochastic linear control problem. Our work is then extended to exponential quadratic mean‐square stabilization of the same class of stochastic systems. Copyright © 2011 John Wiley & Sons, Ltd.     

Large-scale parallel topology optimization using a dual-primal substructuring solver

Parallel computing is an integral part of many scientific disciplines. In this paper, we discuss issues and difficulties arising when a state-of-the-art parallel linear solver is applied to topology optimization problems. Within the topology optimization framework, we cannot readjust domain decomposition to align with material decomposition, which leads to the deterioration of performance of the substructuring solver. We illustrate the difficulties with detailed condition number estimates and numerical studies. We also report the practical performances of finite element tearing and interconnection/dual–primal solver for topology optimization problems and our attempts to improve it by applying additional scaling and/or preconditioning strategies. The performance of the method is finally illustrated with large-scale topology optimization problems coming from different optimal design fields: compliance minimization, design of compliant mechanisms, and design of elastic surface wave-guides. The authors acknowledge the support of the Air Force Office of Scientific Research (AFOSR) under grant FA9550-05-1-0046. The computational facility was obtained under the grant AFOSR-DURIP FA9550-05-1-0291.     

Reinforcement learning technique using agent state occurrence frequency with analysis of knowledge sharing on the agent’s learning process in multiagent environments

Reinforcement learning techniques like the Q-Learning one as well as the Multiple-Lookahead-Levels one that we introduced in our prior work require the agent to complete an initial exploratory path followed by as many hypothetical and physical paths as necessary to find the optimal path to the goal. This paper introduces a reinforcement learning technique that uses a distance measure to the goal as a primary gauge for an autonomous agent’s action selection. In this paper, we take advantage of the first random walk to acquire initial information about the goal. Once the agent’s goal is reached, the agent’s first perceived internal model of the environment is updated to reflect and include said goal. This is done by the agent tracing back its steps to its origin starting point. We show in this paper, no exploratory or hypothetical paths are required after the goal is initially reached or detected, and the agent requires a maximum of two physical paths to find the optimal path to the goal. The agent’s state occurrence frequency is introduced as well and used to support the proposed Distance-Only technique. A computation speed performance analysis is carried out, and the Distance-and-Frequency technique is shown to require less computation time than the Q-Learning one. Furthermore, we present and demonstrate how multiple agents using the Distance-and-Frequency technique can share knowledge of the environment and study the effect of that knowledge sharing on the agents’ learning process.     

Dense Linear System: A Parallel Self-verified Solver

This article presents a parallel self-verified solver for dense linear systems of equations. This kind of solver is commonly used in many different kinds of real applications which deal with large matrices. Nevertheless, two key problems appear to limit the use of linear system solvers to a more extensive range of real applications: solution correctness and high computational cost. In order to solve the first one, verified computing would be an interesting choice. An algorithm that uses this concept is able to find a highly accurate and automatically verified result providing more reliability. However, the performance of these algorithms quickly becomes a drawback. Aiming at a better performance, parallel computing techniques were employed. Two main parts of this method were parallelized: the computation of the approximate inverse of matrix A and the preconditioning step. The results obtained show that these optimizations increase significantly the overall performance.