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Numerical Analysis Seminar
Seminar information archive ～04/24｜Next seminar｜Future seminars 04/25～
| Date, time & place | Tuesday 16:30 - 18:00 002Room #002 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | Norikazu Saito, Takahito Kashiwabara |
Seminar information archive
2017/10/10
16:50-18:20 Room #002 (Graduate School of Math. Sci. Bldg.)
Yumiharu Nakano (Tokyo Institute of Technology)
Meshfree collocation methods for linear and fully nonlinear parabolic equations
Yumiharu Nakano (Tokyo Institute of Technology)
Meshfree collocation methods for linear and fully nonlinear parabolic equations
2017/07/04
16:50-18:20 Room #002 (Graduate School of Math. Sci. Bldg.)
Ming-Cheng Shiue (National Chiao Tung University)
Boundary conditions for Limited-Area Models (English)
Ming-Cheng Shiue (National Chiao Tung University)
Boundary conditions for Limited-Area Models (English)
[ Abstract ]
The problem of boundary conditions in a limited domain is recognized an important problem in geophysical fluid dynamics. This is due to that boundary conditions are proposed to have high resolution over a region of interest. The challenges for proposing later boundary conditions are of two types: on the computational side, if the proposed boundary conditions are not appropriate, it is well-known that the error from the lateral boundary can propagate into the computational domain and make a major effect on the numerical solution; on the mathematical side, the negative result of Oliger and Sundstrom that these equations including the inviscid primitive equations and shallow water equations in the multilayer case are not well-posed for any set of local boundary conditions.
In this talk, three-dimensional inviscid primitive equations and (one-layer and two-layer) shallow water equations which have been used in the limited-area numerical weather prediction modelings are considered. Our goals of this work are two folds: one is to propose boundary conditions which are physically suitable. That is, they let waves move freely out of the domain without producing spurious waves; the other is to numerically implement these boundary conditions by proposing suitable numerical methods. Numerical experiments are presented to demonstrate that these proposed boundary conditions and numerical schemes are suitable.
The problem of boundary conditions in a limited domain is recognized an important problem in geophysical fluid dynamics. This is due to that boundary conditions are proposed to have high resolution over a region of interest. The challenges for proposing later boundary conditions are of two types: on the computational side, if the proposed boundary conditions are not appropriate, it is well-known that the error from the lateral boundary can propagate into the computational domain and make a major effect on the numerical solution; on the mathematical side, the negative result of Oliger and Sundstrom that these equations including the inviscid primitive equations and shallow water equations in the multilayer case are not well-posed for any set of local boundary conditions.
In this talk, three-dimensional inviscid primitive equations and (one-layer and two-layer) shallow water equations which have been used in the limited-area numerical weather prediction modelings are considered. Our goals of this work are two folds: one is to propose boundary conditions which are physically suitable. That is, they let waves move freely out of the domain without producing spurious waves; the other is to numerically implement these boundary conditions by proposing suitable numerical methods. Numerical experiments are presented to demonstrate that these proposed boundary conditions and numerical schemes are suitable.
2017/06/13
16:50-18:20 Room #002 (Graduate School of Math. Sci. Bldg.)
Hirofumi Notsu (Kanazawa University)
Numerical analysis of viscoelastic fluid models (Japanese)
Hirofumi Notsu (Kanazawa University)
Numerical analysis of viscoelastic fluid models (Japanese)
[ Abstract ]
Numerical methods for viscoelastic fluid models are studied. In viscoelastic fluid models the stress tensor is often written as a sum of the viscous stress tensor depending linearly on the strain rate tensor and the extra stress tensor for the viscoelastic contribution. In order to describe the viscoelastic contribution another equation for the extra stress tensor is required. In the talk we mainly deal with the Oldroyd-B and the Peterlin models among several proposed viscoelastic fluid models, and present error estimates of finite element schemes based on the method of characteristics. The key issue in the estimates is the treatment of the divergence of the extra stress tensor appearing in the equation for the velocity and the pressure.
Numerical methods for viscoelastic fluid models are studied. In viscoelastic fluid models the stress tensor is often written as a sum of the viscous stress tensor depending linearly on the strain rate tensor and the extra stress tensor for the viscoelastic contribution. In order to describe the viscoelastic contribution another equation for the extra stress tensor is required. In the talk we mainly deal with the Oldroyd-B and the Peterlin models among several proposed viscoelastic fluid models, and present error estimates of finite element schemes based on the method of characteristics. The key issue in the estimates is the treatment of the divergence of the extra stress tensor appearing in the equation for the velocity and the pressure.
2017/04/25
16:50-18:20 Room #002 (Graduate School of Math. Sci. Bldg.)
Koya Sakakibara (University of Tokyo)
Theory and application of the method of fundamental solutions (日本語)
Koya Sakakibara (University of Tokyo)
Theory and application of the method of fundamental solutions (日本語)
2017/04/11
16:50-18:20 Room #002 (Graduate School of Math. Sci. Bldg.)
Shinya Uchiumi (Waseda University)
Some issues in the Lagrange-Galerkin method and solutions: computability, dependence on the viscosity and inflow boundary conditions (日本語)
Shinya Uchiumi (Waseda University)
Some issues in the Lagrange-Galerkin method and solutions: computability, dependence on the viscosity and inflow boundary conditions (日本語)
2017/01/16
16:50-18:20 Room #117 (Graduate School of Math. Sci. Bldg.)
Hideo Kawarada (AMSOK and Chiba University)
Mathematical model for the generation of calcium carbonate scale (日本語)
Hideo Kawarada (AMSOK and Chiba University)
Mathematical model for the generation of calcium carbonate scale (日本語)
2016/11/21
16:50-18:20 Room #002 (Graduate School of Math. Sci. Bldg.)
Sotirios E. Notaris (National and Kapodistrian University of Athens)
Gauss-Kronrod quadrature formulae (English)
Sotirios E. Notaris (National and Kapodistrian University of Athens)
Gauss-Kronrod quadrature formulae (English)
[ Abstract ]
In 1964, the Russian mathematician A.S. Kronrod, in an attempt to estimate practically the error term of the well-known Gauss quadrature formula, presented a new quadrature rule, which since then bears his name. It turns out that the new rule was related to some polynomials that Stieltjes developed some 70 years earlier, through his work on continued fractions and the moment problem. We give an overview of the Gauss-Kronrod quadrature formulae, which are interesting from both the mathematical and the applicable point of view.
The talk will be expository without requiring any previous knowledge of numerical integration.
In 1964, the Russian mathematician A.S. Kronrod, in an attempt to estimate practically the error term of the well-known Gauss quadrature formula, presented a new quadrature rule, which since then bears his name. It turns out that the new rule was related to some polynomials that Stieltjes developed some 70 years earlier, through his work on continued fractions and the moment problem. We give an overview of the Gauss-Kronrod quadrature formulae, which are interesting from both the mathematical and the applicable point of view.
The talk will be expository without requiring any previous knowledge of numerical integration.
2016/10/31
16:50-18:20 Room #002 (Graduate School of Math. Sci. Bldg.)
Shohiro Sho (Osaka University)
Numerical methods of a quantum hydrodynamic model for semiconductor devices (日本語)
Shohiro Sho (Osaka University)
Numerical methods of a quantum hydrodynamic model for semiconductor devices (日本語)
2016/07/11
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Hiroshi Fujiwara (Kyoto University)
Towards fast and reliable numerical computations of the stationary radiative transport equation (日本語)
Hiroshi Fujiwara (Kyoto University)
Towards fast and reliable numerical computations of the stationary radiative transport equation (日本語)
[ Abstract ]
The radiative transport equation (RTE) is a mathematical model of near-infrared light propagation in human tissue, and its analysis is required to develop a new noninvasive monitoring method of our body or brain activities. Since stationary RTE describes light intensity depending on a position and a direction, a discretization model of 3D-RTE is essentially a five dimensional problem. Therefore to establish a reliable and practical numerical method, both theoretical numerical analysis and computing techniques are required.
We firstly introduce huge-scale computation examples of RTE with bio-optical data. A high-accurate numerical cubature on the unit sphere and a hybrid parallel computing technique using GPGPU realize fast computation. Secondly we propose a semi-discrete upwind finite volume method to RTE. We also show its error estimate in two dimensions.
This talk is based on joint works with Prof. Y.Iso, Prof. N.Higashimori, and Prof. N.Oishi (Kyoto University).
The radiative transport equation (RTE) is a mathematical model of near-infrared light propagation in human tissue, and its analysis is required to develop a new noninvasive monitoring method of our body or brain activities. Since stationary RTE describes light intensity depending on a position and a direction, a discretization model of 3D-RTE is essentially a five dimensional problem. Therefore to establish a reliable and practical numerical method, both theoretical numerical analysis and computing techniques are required.
We firstly introduce huge-scale computation examples of RTE with bio-optical data. A high-accurate numerical cubature on the unit sphere and a hybrid parallel computing technique using GPGPU realize fast computation. Secondly we propose a semi-discrete upwind finite volume method to RTE. We also show its error estimate in two dimensions.
This talk is based on joint works with Prof. Y.Iso, Prof. N.Higashimori, and Prof. N.Oishi (Kyoto University).
2016/06/13
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Atsushi Suzuki (Osaka University)
Dissection : A direct solver with kernel detection for finite element matrices
(日本語)
Atsushi Suzuki (Osaka University)
Dissection : A direct solver with kernel detection for finite element matrices
(日本語)
[ Abstract ]
Large-scale sparse matrices are solved in finite element analyses of elasticity and/or flow problems. In some cases, the matrix may be singular, e.g. due to pressure ambiguity of the Navier-Stokes equations, or due to rigid body movements of sub-domain elasticity problems by a domain decomposition method. Therefore, it is better the linear solver understands rank-deficiency of the matrix.
By assuming the matrix is factorized into LDU form with a symmetric partial permutation, and by introducing a threshold to postpone factorization for pseudo null pivots, solvability of the last Schur complement matrix will be examined. Usual procedure for rank-deficiency problem is based on computation of eigenvalues or singular values and an introduced threshold determines the null space. However, developed new algorithm in DOI:10.1002/nme.4729 is based on computation of residuals combined with orthogonal projections onto supposed image spaces and there is no necessary to introduce a threshold for understanding zero value in floating point. The algorithm uses higher precision arithmetic, e.g. quadruple precision, to distinguish numerical round-off errors that occurred during factorization of the whole sparse matrix from ones during the kernel detection procedure itself.
This is joint work with François-Xavier Roux (LJLL, UPMC/ONERA).
Large-scale sparse matrices are solved in finite element analyses of elasticity and/or flow problems. In some cases, the matrix may be singular, e.g. due to pressure ambiguity of the Navier-Stokes equations, or due to rigid body movements of sub-domain elasticity problems by a domain decomposition method. Therefore, it is better the linear solver understands rank-deficiency of the matrix.
By assuming the matrix is factorized into LDU form with a symmetric partial permutation, and by introducing a threshold to postpone factorization for pseudo null pivots, solvability of the last Schur complement matrix will be examined. Usual procedure for rank-deficiency problem is based on computation of eigenvalues or singular values and an introduced threshold determines the null space. However, developed new algorithm in DOI:10.1002/nme.4729 is based on computation of residuals combined with orthogonal projections onto supposed image spaces and there is no necessary to introduce a threshold for understanding zero value in floating point. The algorithm uses higher precision arithmetic, e.g. quadruple precision, to distinguish numerical round-off errors that occurred during factorization of the whole sparse matrix from ones during the kernel detection procedure itself.
This is joint work with François-Xavier Roux (LJLL, UPMC/ONERA).
2016/05/23
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Keiichi Morikuni (University of Tsukuba)
Inner-iteration preconditioning for least squares problems and its applications (日本語)
Keiichi Morikuni (University of Tsukuba)
Inner-iteration preconditioning for least squares problems and its applications (日本語)
[ Abstract ]
We discuss inner-iteration preconditioning for Krylov subspace methods for solving large-scale linear least squares problems. The preconditioning uses several steps of stationary iterative methods, and is efficient when the successive overrelaxation (SOR) method for the normal equations is employed. The SOR inner-iteration left/right-preconditioned generalized minimal residual (BA/AB-GMRES) methods determine a least squares solution/the minimum-norm solution of linear systems of equations without breakdown even in the rank-deficient case. The inner-iteration preconditioning requires less memory than incomplete matrix factorization-type one, and is effective for ill-conditioned and/or rank-deficient least squares problems.
We present applications of inner-iteration preconditioning to solutions of (1) general least squares problems, which is to find a least squares solution whose Euclidean norm is minimum; (2) linear systems of equations which arise in an interior-point method for solving linear programming problems. In (1), we focus on a two-step procedure for the solution of general least squares problems; the first step is to determine a least squares solution and the second step is to determine the minimum-norm solution to a linear system of equation. The solution of each step can be done by using the inner-iteration preconditioned GMRES methods. Numerical experiments show that the SOR inner-iteration preconditioned GMRES methods are more efficient than previous methods for some problems. In (2), the linear systems of equations at each interior-point step become ill-conditioned in the late phase of the interior-point iterations. To solve the linear systems of equation robustly, the inner-iteration preconditioning applies. We present efficient techniques to apply the inner-iteration preconditioning to the linear systems of equations. Numerical experiments on benchmark problems show that the inner-iteration preconditioning is robust compared to previous methods. (2) is joint work with Yiran Cui (University College London), Takashi Tsuchiya (National Graduate Institute for Policy Studies) , and Ken Hayami (National Institute of Informatics and SOKENDAI).
We discuss inner-iteration preconditioning for Krylov subspace methods for solving large-scale linear least squares problems. The preconditioning uses several steps of stationary iterative methods, and is efficient when the successive overrelaxation (SOR) method for the normal equations is employed. The SOR inner-iteration left/right-preconditioned generalized minimal residual (BA/AB-GMRES) methods determine a least squares solution/the minimum-norm solution of linear systems of equations without breakdown even in the rank-deficient case. The inner-iteration preconditioning requires less memory than incomplete matrix factorization-type one, and is effective for ill-conditioned and/or rank-deficient least squares problems.
We present applications of inner-iteration preconditioning to solutions of (1) general least squares problems, which is to find a least squares solution whose Euclidean norm is minimum; (2) linear systems of equations which arise in an interior-point method for solving linear programming problems. In (1), we focus on a two-step procedure for the solution of general least squares problems; the first step is to determine a least squares solution and the second step is to determine the minimum-norm solution to a linear system of equation. The solution of each step can be done by using the inner-iteration preconditioned GMRES methods. Numerical experiments show that the SOR inner-iteration preconditioned GMRES methods are more efficient than previous methods for some problems. In (2), the linear systems of equations at each interior-point step become ill-conditioned in the late phase of the interior-point iterations. To solve the linear systems of equation robustly, the inner-iteration preconditioning applies. We present efficient techniques to apply the inner-iteration preconditioning to the linear systems of equations. Numerical experiments on benchmark problems show that the inner-iteration preconditioning is robust compared to previous methods. (2) is joint work with Yiran Cui (University College London), Takashi Tsuchiya (National Graduate Institute for Policy Studies) , and Ken Hayami (National Institute of Informatics and SOKENDAI).
2016/05/09
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Ken'ichiro Tanaka (Musashino University)
Potential theoretic approach to design of formulas for function approximation and numerical integration in weighted Hardy spaces
(日本語)
Ken'ichiro Tanaka (Musashino University)
Potential theoretic approach to design of formulas for function approximation and numerical integration in weighted Hardy spaces
(日本語)
2016/04/18
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Takahito Kashiwabara (University of Tokyo)
Error estimate for the finite element method in a smooth domain (日本語)
Takahito Kashiwabara (University of Tokyo)
Error estimate for the finite element method in a smooth domain (日本語)
2016/04/04
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Eric Chung (Chinese University of Hong Kong)
Staggered discontinuous Galerkin methods for the incompressible Navier-Stokes equations (English)
Eric Chung (Chinese University of Hong Kong)
Staggered discontinuous Galerkin methods for the incompressible Navier-Stokes equations (English)
[ Abstract ]
In this talk, we present a staggered discontinuous Galerkin method for the approximation of the incompressible Navier-Stokes equations. Our new method combines the advantages of discontinuous Galerkin methods and staggered meshes, and results in many good properties, namely local and global conservations, optimal convergence and superconvergence through the use of a local postprocessing technique. Another key feature is that our method provides a skew-symmetric discretization of the convection term, with the aim of giving a better conservation property compared with existing discretizations. We also analyze the stability and convergence of the method. In addition, we will present some numerical results to show the performance of the proposed method.
In this talk, we present a staggered discontinuous Galerkin method for the approximation of the incompressible Navier-Stokes equations. Our new method combines the advantages of discontinuous Galerkin methods and staggered meshes, and results in many good properties, namely local and global conservations, optimal convergence and superconvergence through the use of a local postprocessing technique. Another key feature is that our method provides a skew-symmetric discretization of the convection term, with the aim of giving a better conservation property compared with existing discretizations. We also analyze the stability and convergence of the method. In addition, we will present some numerical results to show the performance of the proposed method.
2015/10/26
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Fredrik Lindgren (Osaka University)
Numerical approximation of spinodal decomposition in the presence of noise (English)
Fredrik Lindgren (Osaka University)
Numerical approximation of spinodal decomposition in the presence of noise (English)
[ Abstract ]
Numerical approximations of stochastic partial differential equations (SPDE) has evolved to a vivid subfield of computational mathematics in the last decades. It poses new challenges both for numerical analysis and the theory of SPDE.
In this talk we will discuss the strength and weaknesses of the \emph{semigroup approach} to SPDE when it is combined with the idea of viewing a single-step method in time as a \emph{rational approximation of a semigroup}. We shall apply this framework to the stochastic Allen-Cahn equation, a parabolic semi-linear SPDE where the non-linearity is non-globally Lipschitz continuous, but has a \emph{one-sided Lipschitz condition}, and the deterministic equation has a Lyapunov functional.
We focus on semi-discretisation in time, the first step in Rothe's method, and show how the semigroup approach allows for convergence proofs under the assumption that the numerical solution admits moment bounds. However, this assumption turns out to be difficult to verify in the semi-group framework, and the rates achieved are not sharp. This is due to the fact that the one-sided Lipschitz condition, being a variational inequality, can't be utilised. We thus turn to variational methods to solve this issue.
If time admits we shall also comment on the stochastic Cahn-Hilliard equation where the non-linearity has a one-sided Lipschitz condition in a lower norm, only. However, the fact of convergence can still be proved.
This is joint work with Daisuke Furihata (Osaka University), Mih\'aly Kov\'acs (University of Otago, New Zealand), Stig Larsson (Chalmers University of Technology, Sweden) and Shuji Yoshikawa (Ehime University).
Numerical approximations of stochastic partial differential equations (SPDE) has evolved to a vivid subfield of computational mathematics in the last decades. It poses new challenges both for numerical analysis and the theory of SPDE.
In this talk we will discuss the strength and weaknesses of the \emph{semigroup approach} to SPDE when it is combined with the idea of viewing a single-step method in time as a \emph{rational approximation of a semigroup}. We shall apply this framework to the stochastic Allen-Cahn equation, a parabolic semi-linear SPDE where the non-linearity is non-globally Lipschitz continuous, but has a \emph{one-sided Lipschitz condition}, and the deterministic equation has a Lyapunov functional.
We focus on semi-discretisation in time, the first step in Rothe's method, and show how the semigroup approach allows for convergence proofs under the assumption that the numerical solution admits moment bounds. However, this assumption turns out to be difficult to verify in the semi-group framework, and the rates achieved are not sharp. This is due to the fact that the one-sided Lipschitz condition, being a variational inequality, can't be utilised. We thus turn to variational methods to solve this issue.
If time admits we shall also comment on the stochastic Cahn-Hilliard equation where the non-linearity has a one-sided Lipschitz condition in a lower norm, only. However, the fact of convergence can still be proved.
This is joint work with Daisuke Furihata (Osaka University), Mih\'aly Kov\'acs (University of Otago, New Zealand), Stig Larsson (Chalmers University of Technology, Sweden) and Shuji Yoshikawa (Ehime University).
2015/06/29
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Yoshio Komori (Kyushu Institute of Technology)
Stabilized Runge-Kutta methods for the weak approximation of solutions of stochastic differential equations (日本語)
Yoshio Komori (Kyushu Institute of Technology)
Stabilized Runge-Kutta methods for the weak approximation of solutions of stochastic differential equations (日本語)
[ Abstract ]
We are concerned with numerical methods which give weak approximations for stiff It\^{o} stochastic differential equations (SDEs). Implicit methods are one of good candidates to deal with such SDEs. In fact, a well-designed implicit method has been recently proposed by Abdulle and his colleagues [Abdulle et al. 2013a]. On the other hand, it is well known that the numerical solution of stiff SDEs leads to a stepsize reduction when explicit methods are used. However, there are some classes of explicit methods that are well suited to solving some types of stiff SDEs. One such class is the class of stochastic orthogonal Runge-Kutta Chebyshev (SROCK) methods [Abdulle et al. 2013b]. SROCK methods reduce to Runge-Kutta Chebyshev methods when applied to ordinary differential equations (ODEs). Another promising class of methods is the class of explicit methods that reduce to explicit exponential Runge-Kutta (RK) methods [Hochbruck et al. 2005, 2010] when applied to semilinear ODEs.
In this talk, we will propose new exponential RK methods which achieve weak order two for multi-dimensional, non-commutative SDEs with a semilinear drift term. We will analytically investigate their stability properties in mean square, and will check their performance in numerical experiments.
(This is a joint work with D. Cohen and K. Burrage.)
We are concerned with numerical methods which give weak approximations for stiff It\^{o} stochastic differential equations (SDEs). Implicit methods are one of good candidates to deal with such SDEs. In fact, a well-designed implicit method has been recently proposed by Abdulle and his colleagues [Abdulle et al. 2013a]. On the other hand, it is well known that the numerical solution of stiff SDEs leads to a stepsize reduction when explicit methods are used. However, there are some classes of explicit methods that are well suited to solving some types of stiff SDEs. One such class is the class of stochastic orthogonal Runge-Kutta Chebyshev (SROCK) methods [Abdulle et al. 2013b]. SROCK methods reduce to Runge-Kutta Chebyshev methods when applied to ordinary differential equations (ODEs). Another promising class of methods is the class of explicit methods that reduce to explicit exponential Runge-Kutta (RK) methods [Hochbruck et al. 2005, 2010] when applied to semilinear ODEs.
In this talk, we will propose new exponential RK methods which achieve weak order two for multi-dimensional, non-commutative SDEs with a semilinear drift term. We will analytically investigate their stability properties in mean square, and will check their performance in numerical experiments.
(This is a joint work with D. Cohen and K. Burrage.)
2015/06/15
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Yuto Miyatake (Nagoya University)
Parallel energy-preserving methods for Hamiltonian systems (日本語)
Yuto Miyatake (Nagoya University)
Parallel energy-preserving methods for Hamiltonian systems (日本語)
2015/05/18
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Katsuhisa Ozaki (Shibaura Institute of Technology)
Accurate matrix multiplication by error-free transformation (日本語)
Katsuhisa Ozaki (Shibaura Institute of Technology)
Accurate matrix multiplication by error-free transformation (日本語)
2015/04/27
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Akitoshi Takayasu (Waseda University)
A method of verified computations for solutions to semilinear parabolic equations using an analytic semigroup (日本語)
Akitoshi Takayasu (Waseda University)
A method of verified computations for solutions to semilinear parabolic equations using an analytic semigroup (日本語)
2015/03/20
13:30-15:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Gadi Fibich (Tel Aviv University)
Asymmetric Auctions (English)
Gadi Fibich (Tel Aviv University)
Asymmetric Auctions (English)
[ Abstract ]
Auctions are central to the modern economy, both on-line and off-line. A fundamental result in auction theory is that when bidders are symmetric (identical), then under quite general conditions, all auctions are revenue equivalent. While it is known that this result does not hold when bidders are asymmetric, the effect of bidders' asymmetry is poorly understood, since asymmetric auctions are much harder to analyze.
In this talk I will discuss the mathematical theory of asymmetric auctions. I will focus on asymmetric first-price auctions, where the mathematical model is given by a nonstandard system of $n$ nonlinear ordinary differential equations, with $2n$ boundary conditions and a free boundary. I will present various analytic and numerical approaches for this system. Then I will present some recent results on asymptotic revenue equivalence of asymmetric auctions.
Joint work with A. Gavious and N. Gavish.
Auctions are central to the modern economy, both on-line and off-line. A fundamental result in auction theory is that when bidders are symmetric (identical), then under quite general conditions, all auctions are revenue equivalent. While it is known that this result does not hold when bidders are asymmetric, the effect of bidders' asymmetry is poorly understood, since asymmetric auctions are much harder to analyze.
In this talk I will discuss the mathematical theory of asymmetric auctions. I will focus on asymmetric first-price auctions, where the mathematical model is given by a nonstandard system of $n$ nonlinear ordinary differential equations, with $2n$ boundary conditions and a free boundary. I will present various analytic and numerical approaches for this system. Then I will present some recent results on asymptotic revenue equivalence of asymmetric auctions.
Joint work with A. Gavious and N. Gavish.
2015/02/18
14:30-16:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Nao Hamamuki (Hokkaido University)
Harnack inequalities for supersolutions of fully nonlinear elliptic difference and differential equations (日本語)
Nao Hamamuki (Hokkaido University)
Harnack inequalities for supersolutions of fully nonlinear elliptic difference and differential equations (日本語)
2015/02/18
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Toshio Fukushima (National Astronomical Observatory)
Precise and fast computation of elliptic integrals and elliptic functions (日本語)
Toshio Fukushima (National Astronomical Observatory)
Precise and fast computation of elliptic integrals and elliptic functions (日本語)
[ Abstract ]
Summarized is the recent progress of the methods to compute (i) Legendre's normal form complete elliptic integrals of all three kinds, $K(m)$, $E(m)$, and $\Pi(n|m)$, (ii) Legendre's normal form incomplete elliptic integrals of all three kinds, $F(\phi|m)$, $E(\phi|m)$, and $\Pi(\phi,n|m)$, (iii) Jacobian elliptic functions, $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, $\mathrm{dn}(u|m)$, and $\mathrm{am}(u|m)$, (iv) the inverse functions of $K(m)$ and $E(m)$, $m_K(K)$ and $m_E(E)$, (v) the inverse of a general incomplete elliptic integral in Jacobi's form, $G(\mathrm{am}(u|m),n|m)$, with respect to $u$, and (vi) the partial derivatives of $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, $dn(u|m)$, $E(\mathrm{am}(u|m)|m)$, and $\Pi(\mathrm{am}(u|m),n|m)$ with respect to $u$ and those of $F(\phi|m)$, $E(\phi|m)$, and $\Pi(\phi,n|m)$ with respect to $\phi$. In order to avoid the information loss when $n\ll 1$ and/or $m \ll 1$, focused are the associate incomplete elliptc integrals defined as $B(\phi|m)=[E(\phi|m)-(1-m)F(\phi|m)]/m$, $D(\phi|m)=[F(\phi|m)-E(\phi|m)]/m$, and $J(\phi,n|m)=[\Pi(\phi,n|m)-F(\phi|m)]/n$, and their complete versions, $B(m)=[E(m)-(1-m)K(m)]/m$, $D(m)=[K(m)-E(m)]/m$, and $J(n|m)=[\Pi(n|m)-K(m)]/n$. The main techniques used are (i) the piecewise approximation for single variable functions as $K(m)$, and (ii) the combination of repeated usage of the half and double argument transformations and the truncated Maclaurin series expansions with respect to $u = F(\phi|m)$. The new methods are of the full double precision accuracy without any chance of cancellation against small input arguments. They run significantly faster than the existing methods: (i) 2.5 times faster than Cody's Chebyshev polynomial approximations for $K(m)$ and $E(m)$, (ii) 2.5 times faster than Bulirsch's cel for $\Pi(n|m)$, (iii) slightly faster than Bulirsch's el1 for $F(\phi|m)$, (iv) 3.5 times faster than Carlson's $R_D$ for $E(\phi|m)$, (v) 3.5 times faster than Carlson's $R_C$, $R_D$, $R_F$, and $R_J$ for $\Pi(\phi,n|m)$, and (vi) 1.5 times faster than Bulirsch's \texttt{sncndn} for $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, and $\mathrm{dn}(u|m)$.
Summarized is the recent progress of the methods to compute (i) Legendre's normal form complete elliptic integrals of all three kinds, $K(m)$, $E(m)$, and $\Pi(n|m)$, (ii) Legendre's normal form incomplete elliptic integrals of all three kinds, $F(\phi|m)$, $E(\phi|m)$, and $\Pi(\phi,n|m)$, (iii) Jacobian elliptic functions, $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, $\mathrm{dn}(u|m)$, and $\mathrm{am}(u|m)$, (iv) the inverse functions of $K(m)$ and $E(m)$, $m_K(K)$ and $m_E(E)$, (v) the inverse of a general incomplete elliptic integral in Jacobi's form, $G(\mathrm{am}(u|m),n|m)$, with respect to $u$, and (vi) the partial derivatives of $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, $dn(u|m)$, $E(\mathrm{am}(u|m)|m)$, and $\Pi(\mathrm{am}(u|m),n|m)$ with respect to $u$ and those of $F(\phi|m)$, $E(\phi|m)$, and $\Pi(\phi,n|m)$ with respect to $\phi$. In order to avoid the information loss when $n\ll 1$ and/or $m \ll 1$, focused are the associate incomplete elliptc integrals defined as $B(\phi|m)=[E(\phi|m)-(1-m)F(\phi|m)]/m$, $D(\phi|m)=[F(\phi|m)-E(\phi|m)]/m$, and $J(\phi,n|m)=[\Pi(\phi,n|m)-F(\phi|m)]/n$, and their complete versions, $B(m)=[E(m)-(1-m)K(m)]/m$, $D(m)=[K(m)-E(m)]/m$, and $J(n|m)=[\Pi(n|m)-K(m)]/n$. The main techniques used are (i) the piecewise approximation for single variable functions as $K(m)$, and (ii) the combination of repeated usage of the half and double argument transformations and the truncated Maclaurin series expansions with respect to $u = F(\phi|m)$. The new methods are of the full double precision accuracy without any chance of cancellation against small input arguments. They run significantly faster than the existing methods: (i) 2.5 times faster than Cody's Chebyshev polynomial approximations for $K(m)$ and $E(m)$, (ii) 2.5 times faster than Bulirsch's cel for $\Pi(n|m)$, (iii) slightly faster than Bulirsch's el1 for $F(\phi|m)$, (iv) 3.5 times faster than Carlson's $R_D$ for $E(\phi|m)$, (v) 3.5 times faster than Carlson's $R_C$, $R_D$, $R_F$, and $R_J$ for $\Pi(\phi,n|m)$, and (vi) 1.5 times faster than Bulirsch's \texttt{sncndn} for $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, and $\mathrm{dn}(u|m)$.
2015/01/19
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Yoshitaka Watanabe (Kyushu University)
Between error and residual in numerical computations (日本語)
Yoshitaka Watanabe (Kyushu University)
Between error and residual in numerical computations (日本語)
2014/12/01
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Yusuke Imoto (Kyushu University)
An error estimate of a generalized particle method for Poisson equations
(日本語)
[ Reference URL ]
http://www.infsup.jp/utnas/
Yusuke Imoto (Kyushu University)
An error estimate of a generalized particle method for Poisson equations
(日本語)
[ Reference URL ]
http://www.infsup.jp/utnas/
2014/10/20
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Guanyu Zhou (The University of Tokyo)
Finite element method with various types of penalty on domain/boundary (ENGLISH)
Guanyu Zhou (The University of Tokyo)
Finite element method with various types of penalty on domain/boundary (ENGLISH)
[ Abstract ]
We are concerned with several penalty methods (on domain/boundary)
combining with finite element method to solve some partial differential equations. The penalty methods are very useful and widely applied to various problems. For example, to solve the Navier-Stokes equations in moving boundary domain, the finite element method requires to construct the boundary fitted mesh at every times step, which is very time-consuming. The fictitious domain method is proposed to tackle this problem. It is to reformulate the equation to a larger fixed domain, called the fictitious domain, to which we can take a uniform mesh independent on the original moving boundary. The reformulation is based on a penalty method on do- main. Some penalty methods are proposed to approximate the boundary conditions which are not easy to handle with general FEM, such as the slip boundary condition to Stokes/Navier-Stokes equations, the unilateral boundary condition of Signorini’s type to Stokes equations, and so on. It is known that the variational crimes occurs if the finite element spaces or the implementation methods are not chosen properly for slip boundary condition. By introducing a penalty term to the normal component of velocity on slip boundary, we can solve the equations in FEM easily. For the boundary of Signorini’s type, the variational form is an inequality, to which the FEM is not easy to applied. However, we can approximate the variational inequality by a variation equation with penalty term, which can be solve by FEM directly. In above, we introduced several penalty methods with finite element approximation. In this work, we investigate the well-posedness of those penalty method, and obtain the error estimates of penalty; moreover, we consider the penalty methods combining with finite element approximation and show the error estimates.
We are concerned with several penalty methods (on domain/boundary)
combining with finite element method to solve some partial differential equations. The penalty methods are very useful and widely applied to various problems. For example, to solve the Navier-Stokes equations in moving boundary domain, the finite element method requires to construct the boundary fitted mesh at every times step, which is very time-consuming. The fictitious domain method is proposed to tackle this problem. It is to reformulate the equation to a larger fixed domain, called the fictitious domain, to which we can take a uniform mesh independent on the original moving boundary. The reformulation is based on a penalty method on do- main. Some penalty methods are proposed to approximate the boundary conditions which are not easy to handle with general FEM, such as the slip boundary condition to Stokes/Navier-Stokes equations, the unilateral boundary condition of Signorini’s type to Stokes equations, and so on. It is known that the variational crimes occurs if the finite element spaces or the implementation methods are not chosen properly for slip boundary condition. By introducing a penalty term to the normal component of velocity on slip boundary, we can solve the equations in FEM easily. For the boundary of Signorini’s type, the variational form is an inequality, to which the FEM is not easy to applied. However, we can approximate the variational inequality by a variation equation with penalty term, which can be solve by FEM directly. In above, we introduced several penalty methods with finite element approximation. In this work, we investigate the well-posedness of those penalty method, and obtain the error estimates of penalty; moreover, we consider the penalty methods combining with finite element approximation and show the error estimates.
