The Experts below are selected from a list of 258 Experts worldwide ranked by ideXlab platform

Carsten Witt - One of the best experts on this subject based on the ideXlab platform.

  • GECCO - Rigorous Runtime Analysis of a (µ+1) ES for the Sphere Function
    2020
    Co-Authors: Jens Jägersküpper, Carsten Witt
    Abstract:

    Evolutionary algorithms (EAs) are general, randomized search heuristics applied successfully to optimization prob- lems both in discrete and in continuous search spaces. In recent years, substantial progress has been made in theoreti- cal runtime analysis of EAs, in particular for pseudo-Boolean fitness Functions f : {0,1} n → R. Compared to this, little is known about the runtime of simple and, in particular, more complex EAs for continuous Functions f : R n → R. In this paper, a first rigorous runtime analysis of a popu- lation-based EA in continuous search spaces is presented. A simple (� +1) evolution strategy ((� +1)ES) that uses Gaus- sian mutations adapted by the 1/5-rule as its search operator is studied on the well-known Sphere Function and the influ- ence ofand n on its runtime is examined. By generalizing the proof technique of randomized family trees, developed before w.r.t. discrete search spaces, asymptotically upper and lower bounds on the time for the population to make a predefined progress are derived. Furthermore, the util- ity of the 1/5-rule in population-based evolution strategies is shown. Finally, the behavior of the (� +1)ES on multi- modal Functions is discussed.

  • rigorous runtime analysis of a μ 1 es for the Sphere Function
    Genetic and Evolutionary Computation Conference, 2005
    Co-Authors: Jens Jägersküpper, Carsten Witt
    Abstract:

    Evolutionary algorithms (EAs) are general, randomized search heuristics applied successfully to optimization problems both in discrete and in continuous search spaces. In recent years, substantial progress has been made in theoretical runtime analysis of EAs, in particular for pseudo-Boolean fitness Functions f:(0,1)n → R. Compared to this, little is known about the runtime of simple and, in particular, more complex EAs for continuous Functions f: Rn → R.In this paper, a first rigorous runtime analysis of a population-based EA in continuous search spaces is presented. A simple (μ+1) evolution strategy ((μ+1)ES) that uses Gaussian mutations adapted by the 1/5-rule as its search operator is studied on the well-known Sphere Functionand the influence of μ and n on its runtime is examined. By generalizing the proof technique of randomized family trees, developed before w.r.t. discrete search spaces, asymptotically upper and lower bounds on the time for the population to make a predefined progress are derived. Furthermore, the utility of the 1/5-rule in population-based evolution strategies is shown. Finally, the behavior of the (μ+1)ES on multimodal Functions is discussed.

  • Rigorous runtime analysis of a (μ+1)ES for the Sphere Function
    Proceedings of the 2005 conference on Genetic and evolutionary computation - GECCO '05, 2005
    Co-Authors: Jens Jägersküpper, Carsten Witt
    Abstract:

    Evolutionary algorithms (EAs) are general, randomized search heuristics applied successfully to optimization problems both in discrete and in continuous search spaces. In recent years, substantial progress has been made in theoretical runtime analysis of EAs, in particular for pseudo-Boolean fitness Functions f:(0,1)n → R. Compared to this, little is known about the runtime of simple and, in particular, more complex EAs for continuous Functions f: Rn → R.In this paper, a first rigorous runtime analysis of a population-based EA in continuous search spaces is presented. A simple (μ+1) evolution strategy ((μ+1)ES) that uses Gaussian mutations adapted by the 1/5-rule as its search operator is studied on the well-known Sphere Functionand the influence of μ and n on its runtime is examined. By generalizing the proof technique of randomized family trees, developed before w.r.t. discrete search spaces, asymptotically upper and lower bounds on the time for the population to make a predefined progress are derived. Furthermore, the utility of the 1/5-rule in population-based evolution strategies is shown. Finally, the behavior of the (μ+1)ES on multimodal Functions is discussed.

Jens Jägersküpper - One of the best experts on this subject based on the ideXlab platform.

  • GECCO - Rigorous Runtime Analysis of a (µ+1) ES for the Sphere Function
    2020
    Co-Authors: Jens Jägersküpper, Carsten Witt
    Abstract:

    Evolutionary algorithms (EAs) are general, randomized search heuristics applied successfully to optimization prob- lems both in discrete and in continuous search spaces. In recent years, substantial progress has been made in theoreti- cal runtime analysis of EAs, in particular for pseudo-Boolean fitness Functions f : {0,1} n → R. Compared to this, little is known about the runtime of simple and, in particular, more complex EAs for continuous Functions f : R n → R. In this paper, a first rigorous runtime analysis of a popu- lation-based EA in continuous search spaces is presented. A simple (� +1) evolution strategy ((� +1)ES) that uses Gaus- sian mutations adapted by the 1/5-rule as its search operator is studied on the well-known Sphere Function and the influ- ence ofand n on its runtime is examined. By generalizing the proof technique of randomized family trees, developed before w.r.t. discrete search spaces, asymptotically upper and lower bounds on the time for the population to make a predefined progress are derived. Furthermore, the util- ity of the 1/5-rule in population-based evolution strategies is shown. Finally, the behavior of the (� +1)ES on multi- modal Functions is discussed.

  • GECCO - Aiming for a theoretically tractable CSA variant by means of empirical investigations
    Proceedings of the 10th annual conference on Genetic and evolutionary computation - GECCO '08, 2008
    Co-Authors: Jens Jägersküpper, Mike Preuss
    Abstract:

    Evolution Strategies (ES) for black-box optimization of a Function f:Rn->R are investigated. Namely, we consider the cumulative step-size adaptation (CSA) for the variance of multivariate zero-mean normal distributions, which are commonly used to sample new candidate solutions within Evolution Strategies (ES). Four simplifications of CSA are proposed and investigated empirically and evaluated statistically. The background for these four new CSA-derivatives, however, is NOT performance tuning, but our aim to accomplish a probabilistic/theoretical runtime analysis of an ES using some kind of a CSA in the near future, and a better understanding of this step-size control mechanisms. Therefore, we consider two test problems, namely the Sphere Function without and with Gaussian noise.

  • rigorous runtime analysis of a μ 1 es for the Sphere Function
    Genetic and Evolutionary Computation Conference, 2005
    Co-Authors: Jens Jägersküpper, Carsten Witt
    Abstract:

    Evolutionary algorithms (EAs) are general, randomized search heuristics applied successfully to optimization problems both in discrete and in continuous search spaces. In recent years, substantial progress has been made in theoretical runtime analysis of EAs, in particular for pseudo-Boolean fitness Functions f:(0,1)n → R. Compared to this, little is known about the runtime of simple and, in particular, more complex EAs for continuous Functions f: Rn → R.In this paper, a first rigorous runtime analysis of a population-based EA in continuous search spaces is presented. A simple (μ+1) evolution strategy ((μ+1)ES) that uses Gaussian mutations adapted by the 1/5-rule as its search operator is studied on the well-known Sphere Functionand the influence of μ and n on its runtime is examined. By generalizing the proof technique of randomized family trees, developed before w.r.t. discrete search spaces, asymptotically upper and lower bounds on the time for the population to make a predefined progress are derived. Furthermore, the utility of the 1/5-rule in population-based evolution strategies is shown. Finally, the behavior of the (μ+1)ES on multimodal Functions is discussed.

  • Rigorous runtime analysis of a (μ+1)ES for the Sphere Function
    Proceedings of the 2005 conference on Genetic and evolutionary computation - GECCO '05, 2005
    Co-Authors: Jens Jägersküpper, Carsten Witt
    Abstract:

    Evolutionary algorithms (EAs) are general, randomized search heuristics applied successfully to optimization problems both in discrete and in continuous search spaces. In recent years, substantial progress has been made in theoretical runtime analysis of EAs, in particular for pseudo-Boolean fitness Functions f:(0,1)n → R. Compared to this, little is known about the runtime of simple and, in particular, more complex EAs for continuous Functions f: Rn → R.In this paper, a first rigorous runtime analysis of a population-based EA in continuous search spaces is presented. A simple (μ+1) evolution strategy ((μ+1)ES) that uses Gaussian mutations adapted by the 1/5-rule as its search operator is studied on the well-known Sphere Functionand the influence of μ and n on its runtime is examined. By generalizing the proof technique of randomized family trees, developed before w.r.t. discrete search spaces, asymptotically upper and lower bounds on the time for the population to make a predefined progress are derived. Furthermore, the utility of the 1/5-rule in population-based evolution strategies is shown. Finally, the behavior of the (μ+1)ES on multimodal Functions is discussed.

L M Yang - One of the best experts on this subject based on the ideXlab platform.

  • an implicit simplified Sphere Function based gas kinetic scheme for simulation of 3d incompressible isothermal flows
    Computers & Fluids, 2018
    Co-Authors: L M Yang, Wenming Yang, Yong Wang
    Abstract:

    Abstract In this work, an implicit simplified Sphere Function-based gas kinetic scheme (SGKS) is presented for simulation of 3D incompressible isothermal flows. At first, the numerical fluxes of governing equations are reconstructed by the local solution of Boltzmann equation with Sphere Function distribution. Due to incompressible limit, the Sphere at cell interface can be approximately considered to be symmetric as shown in the work. Besides that, the energy equation is usually not needed for simulation of incompressible isothermal flows. With all these simplifications, the formulations of the simplified SGKS can be expressed concisely and explicitly. Secondly, the commonly-used implicit Lower-Upper Symmetric Gauss-Seidel (LU-SGS) method is adopted to further improve the computational efficiency and numerical stability of present scheme. In LU-SGS method, only a forward and a backward sweep are needed for marching the conservative variables in time. As a result, the simplified SGKS with the LU-SGS method can be implemented easily. Numerical experiments, including the 3D lid-driven cavity flow and flow over a backward-facing step, showed that the incompressible isothermal flows can be well simulated by the developed scheme and its computational efficiency is significantly higher than that of the original SGKS and the lattice Boltzmann flux solver (LBFS). In addition, it was found that the present scheme with the LU-SGS method is more efficient than that with the explicit Euler method, and the speedup ratio is about 2 to 5.

  • an immersed boundary simplified Sphere Function based gas kinetic scheme for simulation of 3d incompressible flows
    Physics of Fluids, 2017
    Co-Authors: L M Yang, Wenming Yang, Yong Wang, J Wu
    Abstract:

    In this work, an immersed boundary-simplified Sphere Function-based gas kinetic scheme (SGKS) is presented for the simulation of 3D incompressible flows with curved and moving boundaries. At first, the SGKS [Yang et al., “A three-dimensional explicit Sphere Function-based gas-kinetic flux solver for simulation of inviscid compressible flows,” J. Comput. Phys. 295, 322 (2015) and Yang et al., “Development of discrete gas kinetic scheme for simulation of 3D viscous incompressible and compressible flows,” J. Comput. Phys. 319, 129 (2016)], which is often applied for the simulation of compressible flows, is simplified to improve the computational efficiency for the simulation of incompressible flows. In the original SGKS, the integral domain along the spherical surface for computing conservative variables and numerical fluxes is usually not symmetric at the cell interface. This leads the expression of numerical fluxes at the cell interface to be relatively complicated. For incompressible flows, the Sphere at ...

  • Comparative study of 1D, 2D and 3D simplified gas kinetic schemes for simulation of inviscid compressible flows
    Applied Mathematical Modelling, 2017
    Co-Authors: L M Yang, J Wu, Yong Wang
    Abstract:

    Abstract With assumption that all the particles in the phase velocity space are concentrated on a circle and on a Sphere, the circular Function-based gas kinetic scheme and Sphere Function-based gas kinetic scheme have been developed by Shu and his coworkers [21] , [22] , [23] . These schemes are simpler than the Maxwellian Function-based gas kinetic schemes. The simplicity is due to the fact that the integral domain of phase velocity of circular Function and Sphere Function is a finite region while the integral domain of Maxwellian distribution Function is infinite. In this work, the 1D delta Function-based gas kinetic scheme is also developed to form a complete set of the simplified gas kinetic schemes. The 1D, 2D and 3D simplified gas kinetic schemes can be viewed as the truly 1D, 2D and 3D flux solvers since they are based on the multi-dimensional Boltzmann equation. On the other hand, to solve the 3D flow problem, the tangential velocities are needed to be approximated by some ways for the 1D and 2D simplified gas kinetic schemes, and to solve the 1D flow problem, the tangential velocities should be taken as zero for the 2D and 3D simplified gas kinetic schemes. The performances of these three schemes for simulation of inviscid compressible flows are investigated in this work by their application to solve the test problems from 1D to 3D cases. Numerical results showed that the efficiency of the delta Function-based gas kinetic scheme is slightly superior to that of the circular Function- and Sphere Function-based gas kinetic schemes, while its stability is inferior significantly to the latter. For simulation of the 3D hypersonic flows, the Sphere Function-based gas kinetic scheme could be the best choice.

  • Development of discrete gas kinetic scheme for simulation of 3D viscous incompressible and compressible flows
    Journal of Computational Physics, 2016
    Co-Authors: L M Yang, Yan Wang
    Abstract:

    The Sphere Function-based gas kinetic scheme (GKS), which was presented by Shu and his coworkers 23 for simulation of inviscid compressible flows, is extended to simulate 3D viscous incompressible and compressible flows in this work. Firstly, we use certain discrete points to represent the spherical surface in the phase velocity space. Then, integrals along the spherical surface for conservation forms of moments, which are needed to recover 3D Navier-Stokes equations, are approximated by integral quadrature. The basic requirement is that these conservation forms of moments can be exactly satisfied by weighted summation of distribution Functions at discrete points. It was found that the integral quadrature by eight discrete points on the spherical surface, which forms the D3Q8 discrete velocity model, can exactly match the integral. In this way, the conservative variables and numerical fluxes can be computed by weighted summation of distribution Functions at eight discrete points. That is, the application of complicated formulations resultant from integrals can be replaced by a simple solution process. Several numerical examples including laminar flat plate boundary layer, 3D lid-driven cavity flow, steady flow through a 90° bending square duct, transonic flow around DPW-W1 wing and supersonic flow around NACA0012 airfoil are chosen to validate the proposed scheme. Numerical results demonstrate that the present scheme can provide reasonable numerical results for 3D viscous flows. It is the first time to extend the Sphere Function-based GKS for simulation of 3D viscous flows.D3Q8 model is firstly presented.The complicated Sphere Function-based GKS is replaced by a simple solution process.3D incompressible and compressible viscous flows can be accurately simulated by present scheme.

  • a three dimensional explicit Sphere Function based gas kinetic flux solver for simulation of inviscid compressible flows
    Journal of Computational Physics, 2015
    Co-Authors: L M Yang, Jie Wu
    Abstract:

    Abstract In this work, a truly three-dimensional (3D) flux solver is presented for simulation of inviscid compressible flows. Like the conventional multi-dimensional gas-kinetic scheme, in the present work, the local solution of 3D Boltzmann equation at the cell interface is used to evaluate the flux. On the other hand, different from most of the existing gas-kinetic schemes, which are constructed from Maxwellian distribution Function, the present flux solver is derived from a simple distribution Function defined on the spherical surface in the phase velocity space. As a result, the explicit expression of flux vector at the cell interface can be simply given. Since the simple distribution Function is defined on the spherical surface, for simplicity, it is termed as Sphere Function hereafter. In addition, to simulate fluid flow problems with strong shock waves, the non-equilibrium part of the distribution Function is regarded as numerical dissipation and involved in evaluating the inviscid flux at the cell interface. The weight of the non-equilibrium part is controlled by introducing a switch Function which ranges from 0 to 1. In the smooth region, the switch Function takes a value close to zero, while around the strong shock wave, it tends to one. To validate the proposed flux solver, several transonic, supersonic and hypersonic inviscid flows are simulated. Numerical results showed that the present solver can provide accurate numerical results for three-dimensional inviscid flows with strong shock waves.

Anne Auger - One of the best experts on this subject based on the ideXlab platform.

  • Investigating the impact of sequential selection in the (1,2)-CMA-ES on the noisy BBOB-2010 testbed
    Proceedings of the 12th annual conference comp on Genetic and evolutionary computation - GECCO '10, 2020
    Co-Authors: Anne Auger, Dimo Brockhoff, Nikolaus Hansen
    Abstract:

    International audienceSequential selection was introduced for Evolution Strategies (ESs) with the aim of accelerating their convergence---performing the evaluations of the different offspring sequentially and concluding an iteration immediately if one offspring is better than the parent. This paper investigates the impact of the application of sequential selection to the (1,2)-CMA-ES on the BBOB-2010 noisy benchmark testbed. The performance of the (1,2$^s$)-CMA-ES, where sequential selection is implemented, is compared to the baseline algorithm (1,2)-CMA-ES. Independent restarts for the two algorithms are conducted up to a maximum number of $10^{4} D$ Function evaluations, where $D$ is the dimension of the search space. The results show a slight improvement of the (1,2$^s$)-CMA-ES over the baseline (1,2)-CMA-ES on the Sphere Function with Cauchy noise and a stronger decline on the Sphere Function with moderate uniform noise. Overall, the (1,2$^s$)-CMA-ES seems slighly less reliable and we conclude that for the (1,2)-CMA-ES, sequential selection is no improvement on noisy Functions

  • GECCO - Drift theory in continuous search spaces: expected hitting time of the (1 + 1)-ES with 1/5 success rule
    Proceedings of the Genetic and Evolutionary Computation Conference, 2018
    Co-Authors: Youhei Akimoto, Anne Auger, Tobias Glasmachers
    Abstract:

    This paper explores the use of the standard approach for proving runtime bounds in discrete domains---often referred to as drift analysis---in the context of optimization on a continuous domain. Using this framework we analyze the (1+1) Evolution Strategy with one-fifth success rule on the Sphere Function. To deal with potential Functions that are not lower-bounded, we formulate novel drift theorems. We then use the theorems to prove bounds on the expected hitting time to reach a certain target fitness in finite dimension d. The bounds are akin to linear convergence. We then study the dependency of the different terms on d proving a convergence rate dependency of Θ(1/d). Our results constitute the first non-asymptotic analysis for the algorithm considered as well as the first explicit application of drift analysis to a randomized search heuristic with continuous domain.

  • Drift Theory in Continuous Search Spaces: Expected Hitting Time of the (1+1)-ES with 1/5 Success Rule
    arXiv: Neural and Evolutionary Computing, 2018
    Co-Authors: Youhei Akimoto, Anne Auger, Tobias Glasmachers
    Abstract:

    This paper explores the use of the standard approach for proving runtime bounds in discrete domains---often referred to as drift analysis---in the context of optimization on a continuous domain. Using this framework we analyze the (1+1) Evolution Strategy with one-fifth success rule on the Sphere Function. To deal with potential Functions that are not lower-bounded, we formulate novel drift theorems. We then use the theorems to prove bounds on the expected hitting time to reach a certain target fitness in finite dimension $d$. The bounds are akin to linear convergence. We then study the dependency of the different terms on $d$ proving a convergence rate dependency of $\Theta(1/d)$. Our results constitute the first non-asymptotic analysis for the algorithm considered as well as the first explicit application of drift analysis to a randomized search heuristic with continuous domain.

  • GECCO - Analysis of Linear Convergence of a (1 + 1)-ES with Augmented Lagrangian Constraint Handling
    Proceedings of the 2016 on Genetic and Evolutionary Computation Conference - GECCO '16, 2016
    Co-Authors: Asma Atamna, Anne Auger, Nikolaus Hansen
    Abstract:

    We address the question of linear convergence of evolution strategies on constrained optimization problems. In particular, we analyze a (1+1)-ES with an augmented Lagrangian constraint handling approach on Functions defined on a continuous domain, subject to a single linear inequality constraint. We identify a class of Functions for which it is possible to construct a homogeneous Markov chain whose stability implies linear convergence. This class includes all Functions such that the augmented Lagrangian of the problem, centered with respect to its value at the optimum and the corresponding Lagrange multiplier, is positive homogeneous of degree 2 (thus including convex quadratic Functions as a particular case). The stability of the constructed Markov chain is empirically investigated on the Sphere Function and on a moderately ill-conditioned ellipsoid Function.

  • PPSN - How to Assess Step-Size Adaptation Mechanisms in Randomised Search
    Parallel Problem Solving from Nature – PPSN XIII, 2014
    Co-Authors: Nikolaus Hansen, Asma Atamna, Anne Auger
    Abstract:

    Step-size adaptation for randomised search algorithms like evolution strategies is a crucial feature for their performance. The adaptation must, depending on the situation, sustain a large diversity or entertain fast convergence to the desired optimum. The assessment of step-size adaptation mechanisms is therefore non-trivial and often done in too restricted scenarios, possibly only on the Sphere Function. This paper introduces a (minimal) methodology combined with a practical procedure to conduct a more thorough assessment of the overall population diversity of a randomised search algorithm in different scenarios. We illustrate the methodology on evolution strategies with σ-self-adaptation, cumulative step-size adaptation and two-point adaptation. For the latter, we introduce a variant that abstains from additional samples by constructing two particular individuals within the given population to decide on the step-size change. We find that results on the Sphere Function alone can be rather misleading to assess mechanisms to control overall population diversity. The most striking flaws we observe for self-adaptation: on the linear Function, the step-size increments are rather small, and on a moderately conditioned ellipsoid Function, the adapted step-size is 20 times smaller than optimal.

J Wu - One of the best experts on this subject based on the ideXlab platform.

  • an immersed boundary simplified Sphere Function based gas kinetic scheme for simulation of 3d incompressible flows
    Physics of Fluids, 2017
    Co-Authors: L M Yang, Wenming Yang, Yong Wang, J Wu
    Abstract:

    In this work, an immersed boundary-simplified Sphere Function-based gas kinetic scheme (SGKS) is presented for the simulation of 3D incompressible flows with curved and moving boundaries. At first, the SGKS [Yang et al., “A three-dimensional explicit Sphere Function-based gas-kinetic flux solver for simulation of inviscid compressible flows,” J. Comput. Phys. 295, 322 (2015) and Yang et al., “Development of discrete gas kinetic scheme for simulation of 3D viscous incompressible and compressible flows,” J. Comput. Phys. 319, 129 (2016)], which is often applied for the simulation of compressible flows, is simplified to improve the computational efficiency for the simulation of incompressible flows. In the original SGKS, the integral domain along the spherical surface for computing conservative variables and numerical fluxes is usually not symmetric at the cell interface. This leads the expression of numerical fluxes at the cell interface to be relatively complicated. For incompressible flows, the Sphere at ...

  • Comparative study of 1D, 2D and 3D simplified gas kinetic schemes for simulation of inviscid compressible flows
    Applied Mathematical Modelling, 2017
    Co-Authors: L M Yang, J Wu, Yong Wang
    Abstract:

    Abstract With assumption that all the particles in the phase velocity space are concentrated on a circle and on a Sphere, the circular Function-based gas kinetic scheme and Sphere Function-based gas kinetic scheme have been developed by Shu and his coworkers [21] , [22] , [23] . These schemes are simpler than the Maxwellian Function-based gas kinetic schemes. The simplicity is due to the fact that the integral domain of phase velocity of circular Function and Sphere Function is a finite region while the integral domain of Maxwellian distribution Function is infinite. In this work, the 1D delta Function-based gas kinetic scheme is also developed to form a complete set of the simplified gas kinetic schemes. The 1D, 2D and 3D simplified gas kinetic schemes can be viewed as the truly 1D, 2D and 3D flux solvers since they are based on the multi-dimensional Boltzmann equation. On the other hand, to solve the 3D flow problem, the tangential velocities are needed to be approximated by some ways for the 1D and 2D simplified gas kinetic schemes, and to solve the 1D flow problem, the tangential velocities should be taken as zero for the 2D and 3D simplified gas kinetic schemes. The performances of these three schemes for simulation of inviscid compressible flows are investigated in this work by their application to solve the test problems from 1D to 3D cases. Numerical results showed that the efficiency of the delta Function-based gas kinetic scheme is slightly superior to that of the circular Function- and Sphere Function-based gas kinetic schemes, while its stability is inferior significantly to the latter. For simulation of the 3D hypersonic flows, the Sphere Function-based gas kinetic scheme could be the best choice.