## 数値解析セミナー

開催情報 火曜日　16:30～18:00　数理科学研究科棟(駒場) 056号室 齊藤宣一 http://www.infsup.jp/utnas/

### 2016年04月18日(月)

16:30-18:00   数理科学研究科棟(駒場) 056号室

[ 講演概要 ]

### 2016年04月04日(月)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Eric Chung 氏 (Chinese University of Hong Kong)
Staggered discontinuous Galerkin methods for the incompressible Navier-Stokes equations (English)
[ 講演概要 ]
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年10月26日(月)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Fredrik Lindgren 氏 (大阪大学)
Numerical approximation of spinodal decomposition in the presence of noise (English)
[ 講演概要 ]
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年06月29日(月)

16:30-18:00   数理科学研究科棟(駒場) 056号室

Stabilized Runge-Kutta methods for the weak approximation of solutions of stochastic differential equations (日本語)
[ 講演概要 ]
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年06月15日(月)

16:30-18:00   数理科学研究科棟(駒場) 056号室

ハミルトン系に対する並列エネルギー保存解法 (日本語)
[ 講演概要 ]

### 2015年05月18日(月)

16:30-18:00   数理科学研究科棟(駒場) 056号室

エラーフリー変換を用いた行列積の高精度計算
(日本語)
[ 講演概要 ]
すべての成分が浮動小数点数である行列の積に関して，数値計算を用いて高信頼な結果を得る手法について研究を行っている．本講演では，HPCの技術者がチューニングをした高速なライブラリを直に使用する高精度行列積アルゴリズムについて紹介したい．アルゴリズムの概要，長所と短所，エラーフリーであることの証明から解説し，区間演算への応用や最近開発できた事後保証型のアルゴリズムについても紹介したい．

### 2015年04月27日(月)

16:30-18:00   数理科学研究科棟(駒場) 056号室

[ 講演概要 ]

### 2015年03月20日(金)

13:30-15:00   数理科学研究科棟(駒場) 122号室
Gadi Fibich 氏 (Tel Aviv University)
Asymmetric Auctions (English)
[ 講演概要 ]
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年02月18日(水)

14:30-16:00   数理科学研究科棟(駒場) 002号室

Harnack inequalities for supersolutions of fully nonlinear elliptic difference and differential equations (日本語)
[ 講演概要 ]

### 2015年02月18日(水)

16:30-18:00   数理科学研究科棟(駒場) 002号室

Precise and fast computation of elliptic integrals and elliptic functions (日本語)
[ 講演概要 ]
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年01月19日(月)

16:30-18:00   数理科学研究科棟(駒場) 056号室

[ 講演概要 ]
コンピュータによる数値計算においては、浮動小数点演算による丸め誤差、級数展開・ 反復の打ち切り誤差、連続問題の離散化誤差など、レベルの異なる誤差が発生すること があります。本講演では、「計算の品質」の観点から、何らかの方法で得られた近似解 に対し、真の解との誤差限界を数学的に厳密に見積もる数値計算手法について、特に残 差の果たす役割に注目しながら、分かりやすく紹介できればと思います。

### 2014年12月01日(月)

16:30-18:00   数理科学研究科棟(駒場) 056号室

Poisson方程式に対する一般化粒子法の誤差評価 (日本語)
[ 講演概要 ]
SPH法やMPS法に代表される粒子法は, 津波のような移動境界流れに対する数値計算手法の一つとして, 現在幅広く利用されている. 一方で, 近似解の誤差評価といった粒子法の数学的正当化は, 我々の知る限り十分に行われているとは言えない.
そこで我々は, 誤差評価の第一ステップとして, Poisson方程式に対するある一般化粒子法を導入し, その誤差評価を行った. 提案する粒子法は, SPH法やMPS法を含む, より広いクラスの粒子法を記述することが可能である. 本講演では, 粒子分布の正則性と接続性を導入し, これらの性質を持った粒子分布の下で, 近似解の誤差が重み関数の影響半径に関して2次収束することを示す. 我々の誤差評価では, 従来は工学的な経験則に基づいていた 参照関数の選択や粒子数と影響半径の組合わせの選択などに, 数学的に正当化されたある十分条件を与えていることが重要である.
[ 参考URL ]
http://www.infsup.jp/utnas/

### 2014年10月20日(月)

16:30-18:00   数理科学研究科棟(駒場) 056号室

Finite element method with various types of penalty on domain/boundary (ENGLISH)
[ 講演概要 ]
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.

### 2014年07月28日(月)

16:30-18:00   数理科学研究科棟(駒場) 056号室

Computer assisted analysis of Craik’s and Pehlivan’s 3D dynamical systems (JAPANESE)
[ 講演概要 ]

[ 参考URL ]
http://www.infsup.jp/utnas/

### 2014年06月09日(月)

16:30-18:00   数理科学研究科棟(駒場) 056号室

[ 講演概要 ]
ハイブリッド型不連続Galerkin(HDG)法とは、要素内部の未知量に加え、要素間境界上の未知量を導入して定式化を行うという、新しいタイプの不連続Galerkin法である。本講演では、従来のHDG法の安定化項を弱めることによって得られる新手法(弱安定化HDG法)を紹介する。 弱安定化HDG法の理論誤差解析や、ガウス型数値積分公式による実装法、数値計算結果などについて示す。非適合有限要素法との関連性についても述べる。
[ 参考URL ]
http://www.infsup.jp/utnas/

### 2014年05月12日(月)

16:30-18:00   数理科学研究科棟(駒場) 002号室
Chien-Hong Cho 氏 (National Chung Cheng University)
On the finite difference approximation for blow-up solutions of the nonlinear wave equation (JAPANESE)
[ 講演概要 ]
We consider in this paper the 1-dim nonlinear wave equation $u_{tt}=u_{xx}+u^{1+\\alpha}$ $(\\alpha > 0)$ and its finite difference analogue. It is known that the solutions of the current equation becomes unbounded in finite time, a phenomenon which is often called blow-up. Numerical approaches on such kind of problems are widely investigated in the last decade. However, those results are mainly about parabolic blow-up problems. Compared with the parabolic ones, there is a remarkable property for the solution of the nonlinear wave equation -- the existence of the blow-up curve. That is, even though the solution has become unbounded at certain points, the solution continues to exist at other points and blows up at later times. We are concerned in this paper as to how a finite difference scheme can reproduce such a phenomenon.
[ 参考URL ]
http://www.infsup.jp/utnas/

### 2014年04月21日(月)

16:30-18:00   数理科学研究科棟(駒場) 056号室

[ 講演概要 ]
Stokes方程式やNavier-Stokes方程式の定常解に対する形状最適化問題は,これまで多く行われてきた.しかし, Navier-Stokes方程式の周期解に対しては十分に行われていない.本講演では,安定性理論を活用することで,Navier-Stokes方程式の周期解に対する形状最適化問題を人工血管の最適設計という現実の問題を通して考察する.
[ 参考URL ]
http://www.infsup.jp/utnas/

### 2014年02月13日(木)

16:00-17:30   数理科学研究科棟(駒場) 056号室
いつもと,開催曜日,開催時刻,教室が異なっております.ご注意ください.
Mitchell Luskin 氏 (University of Minnesota)
Numerical analysis of atomistic-to-continuum coupling methods (ENGLISH)
[ 講演概要 ]
The building blocks of micromechanics are the nucleation and movement of point, line, and surface defects and their long-range elastic interactions. Computational micromechanics has begun to extend the predictive scope of theoretical micromechanics, but mathematical theory able to assess the accuracy and efficiency of multiscale methods is needed for computational micromechanics to reach its full potential.

Many materials problems require the accuracy of atomistic modeling in small regions, such as the neighborhood of a crack tip. However, these localized defects typically interact through long range elastic fields with a much larger region that cannot be computed atomistically. Materials scientists have proposed many methods to compute solutions to these multiscale problems by coupling atomistic models near a localized defect with continuum models where the deformation is nearly uniform on the atomistic scale. During the past several years, a mathematical structure has been given to the description and formulation of atomistic-to-continuum coupling methods, and corresponding numerical analysis and benchmark computational experiments have clarified the relation between the various methods and their sources of error. Our numerical analysis has enabled the development of more accurate and efficient coupling methods.
[ 参考URL ]
http://www.infsup.jp/utnas/

### 2014年01月28日(火)

16:30-18:00   数理科学研究科棟(駒場) 118号室

[ 講演概要 ]

[ 参考URL ]
http://www.infsup.jp/utnas/

### 2013年11月12日(火)

16:30-18:00   数理科学研究科棟(駒場) 002号室
http://www.infsup.jp/utnas/

method of numerical integration''による摩擦型境界条件問題の数値解析について (JAPANESE)
[ 講演概要 ]

[ 参考URL ]
http://www.infsup.jp/utnas/

### 2013年10月29日(火)

16:30-18:00   数理科学研究科棟(駒場) 002号室

[ 講演概要 ]

[ 参考URL ]
http://www.infsup.jp/utnas/

### 2013年07月23日(火)

16:30-18:00   数理科学研究科棟(駒場) 002号室

[ 講演概要 ]

[ 参考URL ]
http://www.infsup.jp/utnas/

### 2013年07月16日(火)

16:30-18:00   数理科学研究科棟(駒場) 002号室

[ 講演概要 ]

[ 参考URL ]
http://www.infsup.jp/utnas/

### 2013年07月02日(火)

16:30-18:00   数理科学研究科棟(駒場) 002号室

[ 講演概要 ]

[ 参考URL ]
http://www.infsup.jp/utnas/

### 2013年06月25日(火)

16:30-18:00   数理科学研究科棟(駒場) 002号室

ウェーブレット変換と区間演算に基づく電子透かし法 (JAPANESE)
[ 講演概要 ]

この方法は、精度保証付き数値計算の分野で主に使われてきた区間演算が、電子透かしに利用された初めての例でもある。本講演では,今まで開発してきたデジタル画像の電子透かし法とこれを音声信号へ適用する方法、さらには改ざん検知への応用について述べ,実験結果により提案方法の有効性を示す.
[ 参考URL ]
http://www.infsup.jp/utnas/