Linear Solvers for Singularly Perturbed Problems Numerical Analysis - - PowerPoint PPT Presentation

Linear Solvers for Singularly Perturbed Problems

Numerical Analysis for Singularly Perturbed Problems

(dedicated to the 60th birthday of Martin Stynes) 16th–18th Nov, 2011

Niall Madden, NUI Galway

Joint work with Scott MacLachlan, Tufts University, MA.

5 10 15 5 10 15 −2 2 4 6 5 10 15 5 10 15 −1 1 2 3 5 10 15 5 10 15 0.5 1 1.5 5 10 15 5 10 15 0.2 0.4 0.6 0.8 5 10 15 5 10 15 0.1 0.2 0.3 0.4 5 10 15 5 10 15 0.05 0.1 0.15 0.2 5 10 15 5 10 15 0.05 0.1 5 10 15 5 10 15 0.02 0.04 0.06 5 10 15 5 10 15 0.01 0.02 0.03 5 10 15 5 10 15 0.005 0.01 0.015 5 10 15 5 10 15 2 4 6 8 x 10

−3

5 10 15 5 10 15 1 2 3 4 x 10

−3

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 1/25

The abstract

The abstract for this talk was: There has been great interest in the design and analysis of numerical schemes that yield robust solutions to singularly perturbed problems. However, there have been only a small number of studies concerning the solution to the associated linear systems. In this talk we discuss some difficulties that arise when applying standard direct solvers to finite difference discretizations of reaction-diffusion equations. Furthermore, we will present some recent developments of multigrid approaches for these problems. Particular emphasis will be given to coupled systems of reaction-diffusion problems... That there are been relatively few studies on linear solvers for SPPs, does not suggest there have been none, e.g.., [Roos, 1996, Ansari and Hegarty, 2003] and by the Zaragoza group. The focus here is on reaction-diffusion problems in one and two

• dimensions. In particular addressing some issues that have an effect on

direct solvers. Systems are less of an issue for this talk than was planned when the abstract was written.

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 2/25

Overview of the talk

This work began (in earnest) Oct-Dec of last year at the Institute for Mathematics and Its Applications, the University of Minnesota, Minneapolis. It is joint work with Scott MacLachlan, Tufts University. Our collaboration has been supported by the NSF under grant DMS-0811022.

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 3/25

Overview of the talk

In this talk I will describe some work on solvers for discretizations of singularly perturbed reaction-diffusion equations. Particularly: Some observations on the use of direct solvers; Towards a multigrid preconditioner. Moreover...

• 1. I’ll explain why it is Martin Stynes’ fault that I’m interested in this problem.
• 2. We reconsider some difficulties associated with applying a direct solver to a

standard discretization, and offer an explanation for its poor performance.

• 3. Demonstrate that an Algebraic Multigrid (AMG) solver can be applied with

some success.

• 4. Describe some progress in the development of a Geometric Multigrid (GMG)

solver for the problem.

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 4/25

Background

My interest in these differential equations stems from my time as a graduate student in University College Cork, working under Martin’s supervision, as part

• f a project led by Gareth Thomas.

The main thesis topic was supposed to be the development of methods for problems in wave-current interactions. The equations we had to solve included a k-ǫ turbulence model: a coupled system of two nonlinear reaction-diffusion problems with leading coefficients with different orders of magnitude.

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 5/25

Background

My interest in these differential equations stems from my time as a graduate student in University College Cork, working under Martin’s supervision, as part

• f a project led by Gareth Thomas.

The main thesis topic was supposed to be the development of methods for problems in wave-current interactions. The equations we had to solve included a k-ǫ turbulence model: a coupled system of two nonlinear reaction-diffusion problems with leading coefficients with different orders of magnitude. Earlier numerical analyses of fitted meshes for coupled systems, particularly [Shishkin, 1995] and [Matthews et al., 2002] led us apply piecewise uniform meshes to this applied problem. An analysis of the approach for the linear case was given in [Madden and Stynes, 2003].

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 5/25

Background

Some years later we returned to the topic of analysing fitted mesh methods coupled systems. In [Clavero et al., 2005] an analysis is given for a Shishkin mesh for an (uncoupled) reaction-diffusion problem in two dimensions. This was extended to a system of M coupled equations in [Kellogg et al., 2008]. −ε2      ∆ . . . ∆ . . . . . . . . . ... . . . ∆      u + Au = f. When generating the numerical results for that paper, we noticed that the time required by the linear solver (“backslash” in Matlab) did not appear to be uniform in the perturbation parameter.

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 6/25

Background

More recently, I was fortunate to be involved in a project with Martin on the development of a two-scale sparse grid finite element method for (uncoupled and coupled) reaction-diffusion problems in two dimensions [Liu et al., 2009]. The algorithm reduces the number of unknowns from O(N2) to O(N3/2) without compromising the accuracy of the underlying scheme. The price paid for this includes increased bandwidth in the linear system, and an increase in the number of non-zero entries. Justifying the assertion that the two-scale approach is more efficient than the standard Galerkin approach required a systematic study of the CPU time taken by the linear solver.

Galerkin Mesh Sparse Grid Mesh(es)

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 7/25

Background

More recently, I was fortunate to be involved in a project with Martin on the development of a two-scale sparse grid finite element method for (uncoupled and coupled) reaction-diffusion problems in two dimensions [Liu et al., 2009]. The algorithm reduces the number of unknowns from O(N2) to O(N3/2) without compromising the accuracy of the underlying scheme. The price paid for this includes increased bandwidth in the linear system, and an increase in the number of non-zero entries. Justifying the assertion that the two-scale approach is more efficient than the standard Galerkin approach required a systematic study of the CPU time taken by the linear solver.

Galerkin sparsity pattern Sparse Grid pattern

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 7/25

Background

More recently, I was fortunate to be involved in a project with Martin on the development of a two-scale sparse grid finite element method for (uncoupled and coupled) reaction-diffusion problems in two dimensions [Liu et al., 2009]. The algorithm reduces the number of unknowns from O(N2) to O(N3/2) without compromising the accuracy of the underlying scheme. The price paid for this includes increased bandwidth in the linear system, and an increase in the number of non-zero entries. Justifying the assertion that the two-scale approach is more efficient than the standard Galerkin approach required a systematic study of the CPU time taken by the linear solver.

Again it was noticed that the standard Matlab solver did not behave in a uniform manner for our test problem...

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 7/25

Background Galerkin sparsity pattern Sparse Grid pattern

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 8/25

Example

A two-dimensional model problem

−ε2∆u + bu = f

• n (0, 1) × (0, 1) + BCs.

0.5 1 0.5 1 1 2 3

The solution exhibits boundary and interior layers, in this example resolved using a piecewise uniform mesh.

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 9/25

Example

We’ll solve this problem using a finite difference scheme on a tensor-product

• grid. Let. ¯

ΩN

x and ¯

ΩN

y be arbitrary meshes with N intervals on [0, 1]. Set

ΩN = {(xi, yj)}N

i,j=0 to be the tensor product of ¯

ΩN

x and ¯

ΩN

y .

Set hi = xi − xi−1, kj = yj − yj−1, and ¯ hi = (xi+1 − xi−1)/2, ¯ kj = (yj+1 − yj−1)/2. Define the scaled 5-point second-order central difference operator: ∆N :=         ¯ hi kj+1 ¯ kj hi −¯ kj 1 hi + 1 hi+1

• − ¯

hi 1 kj + 1 kj+1

• ¯

kj hi+1 ¯ hi kj         . Then (neglecting boundary conditions), the numerical scheme is:

• − ε2∆N + ¯

h¯ kB

• U = ¯

h¯ kF. The system matrix is symmetric positive definite, and the system of equations should be easy to solve.

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 10/25

Example

If we take (for example) the now-standard piecewise-uniform Shishkin mesh with transition point Set τε = min 1 4, 2 ε β ln N

• ,

then we can generate a uniformly convergent numerical solution.

1 − τε τε τε 1 − τε

Theorem ([Clavero et al., 2005])

Let u be the solution to DE and U the solution to the discrete problem on the Shishkin mesh above then there is C independent of ε and N such that u − U∞, ¯

ΩN C(N−1 ln N)2.

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 11/25

Example

Maximum (pointwise) errors in the domain.

ε2 N = 64 N = 128 N = 256 N = 512 N = 1024 N = 2048 1 4.570e-04 1.144e-04 2.861e-05 7.152e-06 1.788e-06 4.470e-07 1e-02 1.620e-02 4.112e-03 1.031e-03 2.579e-04 6.449e-05 1.612e-05 1e-04 1.640e-01 6.109e-02 2.059e-02 6.606e-03 2.045e-03 6.199e-04 1e-06 1.646e-01 6.123e-02 2.064e-02 6.622e-03 2.050e-03 6.214e-04 1e-08 1.647e-01 6.125e-02 2.065e-02 6.623e-03 2.050e-03 6.215e-04 1e-10 1.647e-01 6.125e-02 2.065e-02 6.624e-03 2.050e-03 6.215e-04 1e-10 1.647e-01 6.125e-02 2.065e-02 6.624e-03 2.050e-03 6.215e-04

As expected, the computed solution is robust with respect to ε and gives almost second-order accuracy. The understanding of course is that the cost of implementing the scheme does not depend on ε. And indeed it should be if we are using a direct solver since the linear systems in the classical case and singularly perturbed case have the same structure.

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 12/25

Applying a direct solver

We implemented the scheme described in C, compiled with the standard gcc compiler, using -03 optimisations. The linear solver is HSL’s MA57 for sparse symmetric linear systems (not necessarily definite) [Duff, 2004]. The following are the wall-clock solve-times on a single core of an Intel Xeon (E5410) 2.33GHz processor. Solve time in seconds for Shishkin mesh

ε2 N = 64 N = 128 N = 256 N = 512 N = 1024 N = 2048 1 0.0100 0.0800 0.6500 3.4300 19.7000 117.9900 1e-02 0.0200 0.0800 0.7100 3.4400 19.6600 117.8300 1e-04 0.0200 0.1000 0.7800 3.4600 19.6900 118.0000 1e-06 0.0400 0.0900 1.0300 9.7000 102.5300 882.9000 1e-08 0.0200 0.1300 1.5000 11.2200 72.6700 461.8600 1e-10 0.0300 0.1700 1.3600 7.8900 49.2900 295.6100 1e-12 0.0300 0.1800 1.2600 6.8900 41.0400 233.3700

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 13/25

Applying a direct solver

We implemented the scheme described in C, compiled with the standard gcc compiler, using -03 optimisations. The linear solver is HSL’s MA57 for sparse symmetric linear systems (not necessarily definite) [Duff, 2004]. The following are the wall-clock solve-times on a single core of an Intel Xeon (E5410) 2.33GHz processor. We observe inappropriate scaling in ε. Why?? Solve time in seconds for Shishkin mesh

ε2 N = 64 N = 128 N = 256 N = 512 N = 1024 N = 2048 1 0.0100 0.0800 0.6500 3.4300 19.7000 117.9900 1e-02 0.0200 0.0800 0.7100 3.4400 19.6600 117.8300 1e-04 0.0200 0.1000 0.7800 3.4600 19.6900 118.0000 1e-06 0.0400 0.0900 1.0300 9.7000 102.5300 882.9000 1e-08 0.0200 0.1300 1.5000 11.2200 72.6700 461.8600 1e-10 0.0300 0.1700 1.3600 7.8900 49.2900 295.6100 1e-12 0.0300 0.1800 1.2600 6.8900 41.0400 233.3700

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 13/25

Applying a direct solver

We implemented the scheme described in C, compiled with the standard gcc compiler, using -03 optimisations. The linear solver is HSL’s MA57 for sparse symmetric linear systems (not necessarily definite) [Duff, 2004]. The following are the wall-clock solve-times on a single core of an Intel Xeon (E5410) 2.33GHz processor. Solve time in seconds for Bakhvalov mesh

ε2 N = 64 N = 128 N = 256 N = 512 N = 1024 N = 2048 1 0.0100 0.0600 0.6400 3.4400 19.7000 117.7200 1e-02 0.0300 0.0900 0.7000 3.4400 19.6700 117.8800 1e-04 0.0300 0.0900 0.7400 3.4700 19.7100 118.0000 1e-06 0.0200 0.1100 1.0200 10.2000 107.9900 921.0700 1e-08 0.0200 0.1500 1.5500 10.8900 71.6400 459.1800 1e-10 0.0300 0.1800 1.3700 7.9900 48.2400 286.0500 1e-12 0.0400 0.1900 1.2600 6.8900 41.0400 229.0800

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 13/25

Applying a direct solver

We implemented the scheme described in C, compiled with the standard gcc compiler, using -03 optimisations. The linear solver is HSL’s MA57 for sparse symmetric linear systems (not necessarily definite) [Duff, 2004]. The following are the wall-clock solve-times on a single core of an Intel Xeon (E5410) 2.33GHz processor. Solve time in seconds for Uniform mesh

ε2 N = 64 N = 128 N = 256 N = 512 N = 1024 N = 2048 1 0.0200 0.0800 0.6500 3.4200 19.6900 117.7500 1e-02 0.0200 0.1000 0.7800 3.4300 19.6500 117.7800 1e-04 0.0300 0.1000 0.7400 3.4400 19.7100 117.8900 1e-06 0.0400 0.0900 0.9100 6.6800 55.6700 422.0100 1e-08 0.0400 0.1600 1.6700 12.7800 89.6900 583.8500 1e-10 0.0400 0.2200 1.4000 8.6400 47.8300 271.5800 1e-12 0.0400 0.2000 1.3300 7.1300 37.2400 197.3900

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 13/25

Applying a direct solver

Further experiments have been conducted using “CHOLMOD” [Chen et al., 2009], (supernodal sparse Cholesky factorization and update/downdate). The results are comparable: a similar scaling with ε is

• bserved. (Solve times are roughly half that of MA57 since it uses that the

system matrix is positive definite) Experiments with Matlab are (of course) consistent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 14/25

Applying a direct solver

Further experiments have been conducted using “CHOLMOD” [Chen et al., 2009], (supernodal sparse Cholesky factorization and update/downdate). The results are comparable: a similar scaling with ε is

• bserved. (Solve times are roughly half that of MA57 since it uses that the

system matrix is positive definite) Experiments with Matlab are (of course) consistent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . After some investigation (and with thanks for Iain Duff) we have identified that the problem is related to performance problems with denormalized numbers. Denormalised numbers are outside the range of “normal numbers” as defined by IEEE standards, and so not handled in the same manner by the hardware (often involve software trap). Arithmetic operations involving such numbers are typically much more expensive. Suppose we are using a Cholesky-based algorithm to solve Ax = f. When we factorise A = LLT, we expect the factors to fill in the band...

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 14/25

Applying a direct solver

In the figure on the left below we show the sparsity pattern for the system matrix in the case ε2 = 1, highlighting any non-zero |lij| < eps in red (there aren’t any).

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 15/25

Applying a direct solver

Next we show the same images, but for ε2 = 10−6, again highlighting any non-zero |lij| < eps in red. Roughly, the entries lij are scaled like ε2(i−j) for i − j < N − 1.

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 16/25

Applying a direct solver

How can we remedy this?

Our choices include

• 1. The issue can be avoided (at least with gcc) by using
• -funsafe-math-optimizations.

However, this compromises the integrity of the code. Furthermore, there is no comparable option (as far as I know) when calling the solver from Matlab.

• 2. One can reimplement the solver for this special case.
• 3. Don’t use a direct solver...

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 17/25

Multigrid

Recall Geometric multigrid (GMG): Construct a sequence of meshes Ω1, Ω2, Ω2, ΩN/2, ΩN. These have an associated sequence of spaces of piecewise bilinear functions: V1

0, V2 0, . . . , VN/2

, VN

0 .

Define projection operators Ik

k−1 : Vk−1

→ Vk

0 ,

and restriction operators Ik−1

k

: Vk

0 → Vk−1

. Denote a single iteration of the iterative solver as S(b, x). Take an initial guess x[0]

N ,

Improve: x[0] = S(b, x[0]), Compute the residual: r[0] = Ax[0] − b, Let d[0] = x[0] − x, and solve PTAPd[0] = PTr[0], where P = IN/2

N

. Set x[1] = x[0] − Pd[0] Repeat.

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 18/25

Multigrid

Algebraic multigrid

As a first attempt to check the viability of a iterative solver, we tried employing an Algebraic Multigrid Approach (AMG). This is a variant on GMG that aims to be a black-box solver. The algorithm infers the geometry of the problem based on the structure of the linear system. Can be quite competitive, though unlikely to be as efficient as a well-designed GMG approach. However, it is very useful for proof of concept...

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 19/25

Multigrid

Algebraic multigrid

Recall again the solve times for the direct solver: Solve time in seconds for Shishkin mesh

ε2 N = 64 N = 128 N = 256 N = 512 N = 1024 N = 2048 1 0.0100 0.0800 0.6500 3.4300 19.7000 117.9900 1e-02 0.0200 0.0800 0.7100 3.4400 19.6600 117.8300 1e-04 0.0200 0.1000 0.7800 3.4600 19.6900 118.0000 1e-06 0.0400 0.0900 1.0300 9.7000 102.5300 882.9000 1e-08 0.0200 0.1300 1.5000 11.2200 72.6700 461.8600 1e-10 0.0300 0.1700 1.3600 7.8900 49.2900 295.6100 1e-12 0.0300 0.1800 1.2600 6.8900 41.0400 233.3700

Significantly better results are obtained using AMG (irrespective of compiler

• ptimizations).

Solve time in seconds for AMG solver on a uniform mesh

ε2 N = 64 N = 128 N = 256 N = 512 N = 1024 N = 2048 1 0.0600 0.3400 1.0300 5.1000 22.3900 107.5400 1e-02 0.0500 0.1900 0.8600 3.8200 15.8600 67.1900 1e-04 0.0400 0.2000 0.8200 4.3700 15.9500 65.5300 1e-06 0.0300 0.1600 0.7400 3.2300 14.2000 66.7900 1e-08 0.0300 0.1400 0.6200 2.9000 12.1200 51.9600 1e-10 0.0300 0.1400 0.6200 2.9900 13.3200 56.9000 1e-12 0.0300 0.1100 0.5500 2.3800 10.6800 46.2200

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 20/25

Multigrid

Geometric Multigrid

We are motivated by the apparent success of the AMG solver to consider a constructing a Geometric Multigrid solver. For that, and the remainder of this talk we consider a one-dimensional

• problems. The simplest possible example is

−ε2u′′ + u = f on Ω = (0, 1). The numerical scheme is the standard finite different method, and in this example is applied on a Shishkin mesh. 1D problem, GMG-PCG V(1,1) cycles, CPU time N 215 216 217 218 219 220 ε = 10−4 0.136 0.238 0.494 1.03 2.11 4.73 ε = 10−5 0.146 0.261 0.638 1.13 2.51 4.32 ε = 10−6 0.150 0.284 0.640 1.33 2.73 5.58 The results look promising, but don’t exploit the singularly perturbed nature of the problem.

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 21/25

Multigrid

A boundary layer preconditioner

To improve performance in the case where ε ≪ N, we can design a preconditioner specific to this problem. Reordering the degrees of freedom in the system matrix A so that those going with the interior nodes come first, then those associated with the boundary layer come second, including the transition points: A = AII AIB ABI ABB

• .

Since A is symmetric, we know that AIB = ABI

T.

We can then show

Let hI denote the interior mesh-spacing, so that the off-diagonal entries in AII, and the entries in AIB and ABI are all −ε2/hI. Let δ = ε2/h2

I.

Then, defining AD =

• hII

ABB

• , (1 − 2δ)AD A (1 + 5δ)AD.

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 22/25

Multigrid

A boundary layer preconditioner

This suggests the algorithm:

1 Apply one MG V-cycle to the left boundary layer, including the transition

point as an endpoint in the finest grid of the multigrid.

2 Apply one MG V-cycle to the right boundary layer, including the transition

point as an endpoint in the finest grid of the multigrid.

3 Apply a diagonal scaling to the interior points, multiplying by h−1 I I.

1D problem, GMG-PCG V(1,1) cycles, CPU time N 215 216 217 218 219 220 ε = 10−4 0.136 0.238 0.494 1.03 2.11 4.73 ε = 10−5 0.146 0.261 0.638 1.13 2.51 4.32 ε = 10−6 0.150 0.284 0.640 1.33 2.73 5.58 1D problem, BLMG-PCG V(1,1) cycles, CPU time N 215 216 217 218 219 220 ε = 10−4 0.365 1.38 5.47

• ε = 10−5

0.082 0.208 0.678 2.51 9.52 37.39 ε = 10−6 0.083 0.164 0.336 0.693 1.55 4.87

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 23/25

Summary

It is not clear that, in practical situations, it can be assumed that direct solvers are “robust” when solving linear systems arising from some popular methods for singular perturbed reaction-diffusion problems; AMG seems to work well “out of the box”, but there are difficulties in extending to, say, non-symmetric problems. Furthermore, rigorous analysis isn’t easy. ... For any iterative scheme, determining the appropriate stopping criteria isn’t entirely obvious. GMG is likely to be more efficient than AMG, and more amenable to analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 24/25

Summary

It is not clear that, in practical situations, it can be assumed that direct solvers are “robust” when solving linear systems arising from some popular methods for singular perturbed reaction-diffusion problems; AMG seems to work well “out of the box”, but there are difficulties in extending to, say, non-symmetric problems. Furthermore, rigorous analysis isn’t easy. ... For any iterative scheme, determining the appropriate stopping criteria isn’t entirely obvious. GMG is likely to be more efficient than AMG, and more amenable to analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Thank you for your attention, but especially, thank you Martin!

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 24/25

Summary

Thanks, and happy birthday Martin!

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 25/25

Summary

Thanks, and happy birthday Martin!

Martin Joseph Stynes 1977 Oregon State University Wen Guo 1993 National University of Ireland, Cork Guangfu Sun 1993 National University of Ireland, Cork Torsten Olaf Linss 1998 National University of Ireland, Cork Niall Madden 2000 National University of Ireland, Cork Maria Caridad Natividad 2001 University of the Philippines Aidan Naughton 2006 National University of Ireland, Cork Martin Viscor 2011 National University of Ireland, Cork Sebastian Franz 2008 Technische Universität Dresden Meghan Stephens 2009 National University of Ireland, Galway Philip Marshall Anselone 1957 Oregon State University Arvid Turner Lonseth 1939 University of California, Berkeley Hans Lewy 1925 Georg-August-Universität Göttingen Richard Courant 1910 Georg-August-Universität Göttingen David Hilbert 1885 Universität Königsberg

• C. L. Ferdinand (Carl Louis) Lindemann

• C. Felix (Christian) Klein

Academic Genealogy of Martin Stynes The Mathematics Genealogy Project is a service of North Dakota State University and the American Mathematical Society http://www.genealogy.math.ndsu.nodak.edu

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 25/25

Summary

Thanks, and happy birthday Martin!

Martin Joseph Stynes 1977 Oregon State University

Wen Guo 1993 National University of Ireland, Cork Guangfu Sun 1993 National University of Ireland, Cork Torsten Olaf Linss 1998 National University of Ireland, Cork Niall Madden 2000 National University of Ireland, Cork Maria Caridad Natividad 2001 University of the Philippines Aidan Naughton 2006 National University of Ireland, Cork Martin Viscor 2011 National University of Ireland, Cork Sebastian Franz 2008 Technische Universität Dresden Meghan Stephens 2009 National University of Ireland, Galway Philip Marshall Anselone 1957 Oregon State University Arvid Turner Lonseth 1939 University of California, Berkeley Hans Lewy 1925 Georg-August-Universität Göttingen Richard Courant 1910 Georg-August-Universität Göttingen David Hilbert 1885 Universität Königsberg

• C. L. Ferdinand (Carl Louis) Lindemann

1873 Friedrich-Alexander-Universität Erlangen-Nürnberg

• C. Felix (Christian) Klein

1868 Rheinische Friedrich-Wilhelms-Universität Bonn Julius Plücker 1823 Philipps-Universität Marburg Rudolf Otto Sigismund Lipschitz 1853 Universität Berlin Christian Ludwig Gerling 1812 Georg-August-Universität Göttingen Gustav Peter Lejeune Dirichlet 1827 Rheinische Friedrich-Wilhelms-Universität Bonn Martin Ohm 1811 Friedrich-Alexander-Universität Erlangen-Nürnberg Carl Friedrich Gauß 1799 Universität Helmstedt Simeon Denis Poisson 1800 École Polytechnique Jean-Baptiste Joseph Fourier École Normale Supérieure Karl Christian von Langsdorf 1781 Georg-August-Universität Göttingen / Justus-Liebig-Universität Gießen / Universität Erfurt Johann Friedrich Pfaff 1786 Georg-August-Universität Göttingen Joseph Louis Lagrange 1754 Università di Torino Pierre-Simon Laplace Abraham Gotthelf Kästner 1739 Universität Leipzig Johann Elert Bode Handelsakademie Hamburg Leonhard Euler 1726 Universität Basel Giovanni Battista (Giambattista) Beccaria Jean Le Rond d'Alembert Christian August Hausen 1713 Martin-Luther-Universität Halle-Wittenberg Johann Georg Büsch 1752 Georg-August-Universität Göttingen Johann Bernoulli 1690, 1694 Universität Basel Johann Christoph Wichmannshausen 1685 Universität Leipzig Johann Andreas Planer 1686, 1709 Martin-Luther-Universität Halle-Wittenberg / Eberhard-Karls-Universität Tübingen Johann Andreas Segner 1726, 1734 Friedrich-Schiller-Universität Jena Jacob Bernoulli 1676 Universität Basel 1684 Universität Basel Nikolaus Eglinger 1660, 1661 Universität Basel Otto Mencke 1665 Universität Leipzig Johann Pasch 1683 Martin-Luther-Universität Halle-Wittenberg Rudolf Jakob Camerarius 1684, 1686 Eberhard-Karls-Universität Tübingen Georg Erhard Hamberger 1721 Friedrich-Schiller-Universität Jena Simon Paul Hilscher 1704 Friedrich-Schiller-Universität Jena Nicolas Malebranche 1672 Peter Werenfels 1649 Universität Basel Emmanuel Stupanus 1613 Universität Basel Johann Caspar Bauhin 1649 Universität Basel Georg Balthasar Metzger 1644, 1650 Friedrich-Schiller-Universität Jena / Universität Basel Franciscus de le Boë Sylvius 1634, 1637 Universiteit Leiden / Universität Basel Jakob Thomasius 1643 Universität Leipzig Gottfried Wilhelm Leibniz 1666 Universität Leipzig 1667 Universität Altdorf 1676 Académie royale des sciences de Paris Michael Walther, Jr. 1661, 1687 Martin-Luther-Universität Halle-Wittenberg Elias Rudolph Camerarius, Sr. 1663 Eberhard-Karls-Universität Tübingen Johann Adolph Wedel 1694 Friedrich-Schiller-Universität Jena Rudolf Wilhelm Krause 1671 Universiteit Leiden Theodor Zwinger, Jr. 1630 Universität Basel Petrus Ryff 1584 Universität Basel Friedrich Leibniz Aegidius Strauch 1651, 1657 Martin-Luther-Universität Halle-Wittenberg Johann Andreas Quenstedt 1643, 1644 Universität Helmstedt / Martin-Luther-Universität Halle-Wittenberg Johann Georg Macasius 1638, 1640 Friedrich-Schiller-Universität Jena Georg Wolffgang Wedel 1667 Friedrich-Schiller-Universität Jena / Universiteit Leiden Erhard Weigel 1650 Universität Leipzig Christiaan Huygens 1647, 1655 Universiteit Leiden / Université d'Angers Bartholomäus Leonhard Schwendendörffer Sebastian Beck 1610 Universität Basel Theodor Zwinger 1553 Collège de France 1559 Università di Padova John Craig 1580 Universität Basel Abraham Klein (Calovius) 1632 Universität Rostock Andreas Kunad Christoph Notnagel 1630 Martin-Luther-Universität Halle-Wittenberg Georg Calixt 1607 Universität Helmstedt Balthasar Widmarcter 1640 Friedrich-Schiller-Universität Jena Johannes Musaeus 1634 Universität Erfurt Werner Rolfinck 1625 Martin-Luther-Universität Halle-Wittenberg / Università di Padova Adolph Vorstius 1619, 1622 Universiteit Leiden / Università di Padova Philipp Müller 1604 Universität Leipzig Jan Jansz Stampioen, Jr. Frans van Schooten, Jr. 1635 Universiteit Leiden Johann Jacob Grynaeus 1564 Eberhard-Karls-Universität Tübingen Petrus (Pierre de La Ramée) Ramus 1536 Collège de Navarre Vittore Trincavelli Università di Padova Bassiano Landi 1542 Università di Padova Ambrosius Rhodius 1600, 1610 Martin-Luther-Universität Halle-Wittenberg Paul Röber 1613 Martin-Luther-Universität Halle-Wittenberg 1617 Martin-Luther-Universität Halle-Wittenberg Johannes Caselius 1560, 1566 Martin-Luther-Universität Halle-Wittenberg / Universität Leipzig / Università di Pisa Cornelius Martini 1592 Universität Helmstedt Jacobus Martini 1596 Universität Helmstedt Georg Großhain 1629 Martin-Luther-Universität Halle-Wittenberg Adriaan van den Spieghel 1603 Université Catholique de Louvain / Università di Padova Daniel Sennert 1594, 1599 Martin-Luther-Universität Halle-Wittenberg Gilbert Jacchaeus 1601/1603/1611 University of St. Andrews / Universität Helmstedt / Universiteit Leiden Christoph Meurer 1582 Universität Leipzig Jacobus Golius 1612, 1621 Universiteit Leiden Marin Mersenne 1611 Université Paris IV-Sorbonne Jacob Andreae 1553 Eberhard-Karls-Universität Tübingen Johannes (Johann Sturm) Sturmius 1527 Université Catholique de Louvain Jacques Toussain 1521 Université de Paris Adrien Turnèbe 1532 Collège de France Pietro Pomponazzi Università di Padova Giovanni Battista della Monte Università degli Studi di Ferrara Università di Padova Andreas (Andries van Wesel) Vesalius 1537 Université Catholique de Louvain / Università di Padova Melchior Jöstel 1583, 1600 Martin-Luther-Universität Halle-Wittenberg Ernestus Hettenbach 1576, 1591 Martin-Luther-Universität Halle-Wittenberg Philipp Melanchthon 1511, 1514 Ruprecht-Karls-Universität Heidelberg / Eberhard-Karls-Universität Tübingen Johannes Hommel 1543 Martin-Luther-Universität Halle-Wittenberg Duncan Liddel 1582,1596 Universität Viadrina Frankfurt an der Oder / Universität Breslau / Universität Helmstedt Hieronymus (Girolamo Fabrici d'Acquapendente) Fabricius 1559 Università di Padova Jan Jessenius 1588, 1591 Universität Leipzig / Università di Padova Salomon Alberti 1564, 1574 Martin-Luther-Universität Halle-Wittenberg / Università di Padova Jacobus (Jacob Harmensz.) Arminius 1582 Philipps-Universität Marburg / Universiteit Leiden Moritz Valentin Steinmetz 1550, 1567 Universität Leipzig Willebrord (Snel van Royen) Snellius 1607 Universiteit Leiden Thomas Erpenius 1608 Universiteit Leiden Jakob Beuerlin 1551 Eberhard-Karls-Universität Tübingen Johannes Winter von Andernach 1527, 1532 Université Catholique de Louvain / Collège de Tréguier Nicolas (Nicolaes Cleynaerts) Clénard 1515, 1521 Université Catholique de Louvain Guillaume Budé 1486, 1491 Université d'Orléans / Université de Paris Pietro Roccabonella Università di Padova Niccolò Leoniceno 1453 Scuola Pubblica di Vicenza 1464 Università di Padova Nicoletto Vernia Università di Padova Antonio Musa Brasavola 1520 Università degli Studi di Ferrara Marco Musuro 1486 Università di Firenze Valentin (Valentinus Otho) Otto 1570 Martin-Luther-Universität Halle-Wittenberg Andreas Schato 1562, 1578 Martin-Luther-Universität Halle-Wittenberg Johann (Johannes Kapnion) Reuchlin 1477, 1481 Universität Basel / Université de Poitiers Jan (Johannes Campensis) van Campen 1519 Université Catholique de Louvain / Universität Ingolstadt Johannes Stöffler 1476 Universität Ingolstadt Paul Wittich 1566 Universität Leipzig / Martin-Luther-Universität Halle-Wittenberg Wolfgang Franz 1590 Martin-Luther-Universität Halle-Wittenberg Gabriele Falloppio 1547 Università di Padova / Università degli Studi di Ferrara Rudolph (Snel van Royen) Snellius 1572 Universität zu Köln / Ruprecht-Karls-Universität Heidelberg Georg Joachim von Leuchen Rheticus 1535 Martin-Luther-Universität Halle-Wittenberg Sebastian (Theodoricus) Dietrich 1544 Martin-Luther-Universität Halle-Wittenberg Caspar Peucer 1545 Martin-Luther-Universität Halle-Wittenberg Johann Hoffmann Ludolph van Ceulen Joseph Justus Scaliger 1563 Collège de France Balthasar Kaeuffelin 1521 Eberhard-Karls-Universität Tübingen Jacobus (Jacques Dubois) Sylvius 1530 Université de Paris / Université de Montpellier Rutger Rescius 1513 Université de Paris Jacobus (Jacques Masson) Latomus 1502 Collège de Montaigu 1519 Katholieke Universiteit Leuven Georgius Hermonymus Janus Lascaris 1472 Università di Padova Gaetano da Thiene Alessandro Sermoneta Pelope Ognibene (Omnibonus Leonicenus) Bonisoli da Lonigo Università di Mantova Jacob ben Jehiel Loans Johannes Argyropoulos 1444 Università di Padova Marsilio Ficino 1462 Università di Firenze Valentin Thau 1555 Universität Leipzig Heinrich Maius Matteo Realdo (Renaldus Columbus) Colombo 1544 Università di Padova Immanuel Tremellius 1549 University of Cambridge 1561 Ruprecht-Karls-Universität Heidelberg Valentine Naibod Martin-Luther-Universität Halle-Wittenberg / Universität Erfurt Nicolaus (Mikołaj Kopernik) Copernicus 1499 Uniwersytet Jagielloński / Università di Bologna / Università degli Studi di Ferrara / Università di Padova Johannes Volmar 1515 Martin-Luther-Universität Halle-Wittenberg Jean Tagault François Dubois 1516 Université de Paris Girolamo (Hieronymus Aleander) Aleandro 1499, 1508 Università di Padova Jan Standonck 1474, 1490 Collège Sainte-Barbe / Collège de Montaigu Desiderius Erasmus 1497 /1506 Collège de Montaigu / University of Turin Matthaeus Adrianus Basilios Bessarion 1436 Mystras Johannes Müller Regiomontanus 1457 Universität Leipzig / Universität Wien Demetrios Chalcocondyles 1452 Mystras / Accademia Romana Vittorino da Feltre 1416 Università di Padova Theodoros Gazes 1433 Constantinople / Università di Mantova Erasmus Reinhold 1535 Martin-Luther-Universität Halle-Wittenberg Thomas Cranmer 1515 University of Cambridge Domenico Maria Novara da Ferrara 1483 Università di Firenze Leonhard (Leonard Vitreatoris z Dobczyc) von Dobschütz 1489 Uniwersytet Jagielloński Bonifazius Erasmi 1509 Martin-Luther-Universität Halle-Wittenberg Moses Perez Scipione Fortiguerra 1493 Università di Firenze Georgios Plethon Gemistos 1380, 1393 Rudolf Agricola 1478 Università degli Studi di Ferrara Guarino da Verona 1408 Jakob Milich 1520, 1524 Albert-Ludwigs-Universität Freiburg im Breisgau / Universität Wien Gemma (Jemme Reinerszoon) Frisius 1529, 1536 Université Catholique de Louvain Luca Pacioli Angelo Poliziano 1477 Università di Firenze Demetrios Kydones Manuel Chrysoloras Elissaeus Judaeus Ulrich Zasius 1501 Albert-Ludwigs-Universität Freiburg im Breisgau Petrus (Pieter de Corte) Curtius 1513, 1530 Université Catholique de Louvain Georg von Peuerbach 1440 Universität Wien Cristoforo Landino Nilos Kabasilas Alexander Hegius 1474 Maarten (Martinus Dorpius) van Dorp 1504, 1515 Université Catholique de Louvain Johannes von Gmunden 1406 Universität Wien Thomas à Kempis Leo Outers 1485 Université Catholique de Louvain Heinrich von Langenstein 1363, 1375 Université de Paris

Academic Genealogy of Martin Stynes The Mathematics Genealogy Project is a service of North Dakota State University and the American Mathematical Society http://www.genealogy.math.ndsu.nodak.edu

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 25/25

Summary

Thanks, and happy birthday Martin!

Martin Joseph Stynes 1977 Oregon State University

Wen Guo 1993 National University of Ireland, Cork Guangfu Sun 1993 National University of Ireland, Cork Torsten Olaf Linss 1998 National University of Ireland, Cork Niall Madden 2000 National University of Ireland, Cork Maria Caridad Natividad 2001 University of the Philippines Aidan Naughton 2006 National University of Ireland, Cork Martin Viscor 2011 National University of Ireland, Cork Sebastian Franz 2008 Technische Universität Dresden Meghan Stephens 2009 National University of Ireland, Galway Philip Marshall Anselone 1957 Oregon State University Arvid Turner Lonseth 1939 University of California, Berkeley Hans Lewy 1925 Georg-August-Universität Göttingen Richard Courant 1910 Georg-August-Universität Göttingen David Hilbert 1885 Universität Königsberg

• C. L. Ferdinand (Carl Louis) Lindemann

1873 Friedrich-Alexander-Universität Erlangen-Nürnberg

• C. Felix (Christian) Klein

1868 Rheinische Friedrich-Wilhelms-Universität Bonn Julius Plücker 1823 Philipps-Universität Marburg Rudolf Otto Sigismund Lipschitz 1853 Universität Berlin Christian Ludwig Gerling 1812 Georg-August-Universität Göttingen Gustav Peter Lejeune Dirichlet 1827 Rheinische Friedrich-Wilhelms-Universität Bonn Martin Ohm 1811 Friedrich-Alexander-Universität Erlangen-Nürnberg Carl Friedrich Gauß 1799 Universität Helmstedt Simeon Denis Poisson 1800 École Polytechnique Jean-Baptiste Joseph Fourier École Normale Supérieure Karl Christian von Langsdorf 1781 Georg-August-Universität Göttingen / Justus-Liebig-Universität Gießen / Universität Erfurt Johann Friedrich Pfaff 1786 Georg-August-Universität Göttingen Joseph Louis Lagrange 1754 Università di Torino Pierre-Simon Laplace Abraham Gotthelf Kästner 1739 Universität Leipzig Johann Elert Bode Handelsakademie Hamburg Leonhard Euler 1726 Universität Basel Giovanni Battista (Giambattista) Beccaria Jean Le Rond d'Alembert Christian August Hausen 1713 Martin-Luther-Universität Halle-Wittenberg Johann Georg Büsch 1752 Georg-August-Universität Göttingen Johann Bernoulli 1690, 1694 Universität Basel Johann Christoph Wichmannshausen 1685 Universität Leipzig Johann Andreas Planer 1686, 1709 Martin-Luther-Universität Halle-Wittenberg / Eberhard-Karls-Universität Tübingen Johann Andreas Segner 1726, 1734 Friedrich-Schiller-Universität Jena Jacob Bernoulli 1676 Universität Basel 1684 Universität Basel Nikolaus Eglinger 1660, 1661 Universität Basel Otto Mencke 1665 Universität Leipzig Johann Pasch 1683 Martin-Luther-Universität Halle-Wittenberg Rudolf Jakob Camerarius 1684, 1686 Eberhard-Karls-Universität Tübingen Georg Erhard Hamberger 1721 Friedrich-Schiller-Universität Jena Simon Paul Hilscher 1704 Friedrich-Schiller-Universität Jena Nicolas Malebranche 1672 Peter Werenfels 1649 Universität Basel Emmanuel Stupanus 1613 Universität Basel Johann Caspar Bauhin 1649 Universität Basel Georg Balthasar Metzger 1644, 1650 Friedrich-Schiller-Universität Jena / Universität Basel Franciscus de le Boë Sylvius 1634, 1637 Universiteit Leiden / Universität Basel Jakob Thomasius 1643 Universität Leipzig Gottfried Wilhelm Leibniz 1666 Universität Leipzig 1667 Universität Altdorf 1676 Académie royale des sciences de Paris Michael Walther, Jr. 1661, 1687 Martin-Luther-Universität Halle-Wittenberg Elias Rudolph Camerarius, Sr. 1663 Eberhard-Karls-Universität Tübingen Johann Adolph Wedel 1694 Friedrich-Schiller-Universität Jena Rudolf Wilhelm Krause 1671 Universiteit Leiden Theodor Zwinger, Jr. 1630 Universität Basel Petrus Ryff 1584 Universität Basel Friedrich Leibniz Aegidius Strauch 1651, 1657 Martin-Luther-Universität Halle-Wittenberg Johann Andreas Quenstedt 1643, 1644 Universität Helmstedt / Martin-Luther-Universität Halle-Wittenberg Johann Georg Macasius 1638, 1640 Friedrich-Schiller-Universität Jena Georg Wolffgang Wedel 1667 Friedrich-Schiller-Universität Jena / Universiteit Leiden Erhard Weigel 1650 Universität Leipzig Christiaan Huygens 1647, 1655 Universiteit Leiden / Université d'Angers Bartholomäus Leonhard Schwendendörffer Sebastian Beck 1610 Universität Basel Theodor Zwinger 1553 Collège de France 1559 Università di Padova John Craig 1580 Universität Basel Abraham Klein (Calovius) 1632 Universität Rostock Andreas Kunad Christoph Notnagel 1630 Martin-Luther-Universität Halle-Wittenberg Georg Calixt 1607 Universität Helmstedt Balthasar Widmarcter 1640 Friedrich-Schiller-Universität Jena Johannes Musaeus 1634 Universität Erfurt Werner Rolfinck 1625 Martin-Luther-Universität Halle-Wittenberg / Università di Padova Adolph Vorstius 1619, 1622 Universiteit Leiden / Università di Padova Philipp Müller 1604 Universität Leipzig Jan Jansz Stampioen, Jr. Frans van Schooten, Jr. 1635 Universiteit Leiden Johann Jacob Grynaeus 1564 Eberhard-Karls-Universität Tübingen Petrus (Pierre de La Ramée) Ramus 1536 Collège de Navarre Vittore Trincavelli Università di Padova Bassiano Landi 1542 Università di Padova Ambrosius Rhodius 1600, 1610 Martin-Luther-Universität Halle-Wittenberg Paul Röber 1613 Martin-Luther-Universität Halle-Wittenberg 1617 Martin-Luther-Universität Halle-Wittenberg Johannes Caselius 1560, 1566 Martin-Luther-Universität Halle-Wittenberg / Universität Leipzig / Università di Pisa Cornelius Martini 1592 Universität Helmstedt Jacobus Martini 1596 Universität Helmstedt Georg Großhain 1629 Martin-Luther-Universität Halle-Wittenberg Adriaan van den Spieghel 1603 Université Catholique de Louvain / Università di Padova Daniel Sennert 1594, 1599 Martin-Luther-Universität Halle-Wittenberg Gilbert Jacchaeus 1601/1603/1611 University of St. Andrews / Universität Helmstedt / Universiteit Leiden Christoph Meurer 1582 Universität Leipzig Jacobus Golius 1612, 1621 Universiteit Leiden Marin Mersenne 1611 Université Paris IV-Sorbonne Jacob Andreae 1553 Eberhard-Karls-Universität Tübingen Johannes (Johann Sturm) Sturmius 1527 Université Catholique de Louvain Jacques Toussain 1521 Université de Paris Adrien Turnèbe 1532 Collège de France Pietro Pomponazzi Università di Padova Giovanni Battista della Monte Università degli Studi di Ferrara Università di Padova Andreas (Andries van Wesel) Vesalius 1537 Université Catholique de Louvain / Università di Padova Melchior Jöstel 1583, 1600 Martin-Luther-Universität Halle-Wittenberg Ernestus Hettenbach 1576, 1591 Martin-Luther-Universität Halle-Wittenberg Philipp Melanchthon 1511, 1514 Ruprecht-Karls-Universität Heidelberg / Eberhard-Karls-Universität Tübingen Johannes Hommel 1543 Martin-Luther-Universität Halle-Wittenberg Duncan Liddel 1582,1596 Universität Viadrina Frankfurt an der Oder / Universität Breslau / Universität Helmstedt Hieronymus (Girolamo Fabrici d'Acquapendente) Fabricius 1559 Università di Padova Jan Jessenius 1588, 1591 Universität Leipzig / Università di Padova Salomon Alberti 1564, 1574 Martin-Luther-Universität Halle-Wittenberg / Università di Padova Jacobus (Jacob Harmensz.) Arminius 1582 Philipps-Universität Marburg / Universiteit Leiden Moritz Valentin Steinmetz 1550, 1567 Universität Leipzig Willebrord (Snel van Royen) Snellius 1607 Universiteit Leiden Thomas Erpenius 1608 Universiteit Leiden Jakob Beuerlin 1551 Eberhard-Karls-Universität Tübingen Johannes Winter von Andernach 1527, 1532 Université Catholique de Louvain / Collège de Tréguier Nicolas (Nicolaes Cleynaerts) Clénard 1515, 1521 Université Catholique de Louvain Guillaume Budé 1486, 1491 Université d'Orléans / Université de Paris Pietro Roccabonella Università di Padova Niccolò Leoniceno 1453 Scuola Pubblica di Vicenza 1464 Università di Padova Nicoletto Vernia Università di Padova Antonio Musa Brasavola 1520 Università degli Studi di Ferrara Marco Musuro 1486 Università di Firenze Valentin (Valentinus Otho) Otto 1570 Martin-Luther-Universität Halle-Wittenberg Andreas Schato 1562, 1578 Martin-Luther-Universität Halle-Wittenberg Johann (Johannes Kapnion) Reuchlin 1477, 1481 Universität Basel / Université de Poitiers Jan (Johannes Campensis) van Campen 1519 Université Catholique de Louvain / Universität Ingolstadt Johannes Stöffler 1476 Universität Ingolstadt Paul Wittich 1566 Universität Leipzig / Martin-Luther-Universität Halle-Wittenberg Wolfgang Franz 1590 Martin-Luther-Universität Halle-Wittenberg Gabriele Falloppio 1547 Università di Padova / Università degli Studi di Ferrara Rudolph (Snel van Royen) Snellius 1572 Universität zu Köln / Ruprecht-Karls-Universität Heidelberg Georg Joachim von Leuchen Rheticus 1535 Martin-Luther-Universität Halle-Wittenberg Sebastian (Theodoricus) Dietrich 1544 Martin-Luther-Universität Halle-Wittenberg Caspar Peucer 1545 Martin-Luther-Universität Halle-Wittenberg Johann Hoffmann Ludolph van Ceulen Joseph Justus Scaliger 1563 Collège de France Balthasar Kaeuffelin 1521 Eberhard-Karls-Universität Tübingen Jacobus (Jacques Dubois) Sylvius 1530 Université de Paris / Université de Montpellier Rutger Rescius 1513 Université de Paris Jacobus (Jacques Masson) Latomus 1502 Collège de Montaigu 1519 Katholieke Universiteit Leuven Georgius Hermonymus Janus Lascaris 1472 Università di Padova Gaetano da Thiene Alessandro Sermoneta Pelope Ognibene (Omnibonus Leonicenus) Bonisoli da Lonigo Università di Mantova Jacob ben Jehiel Loans Johannes Argyropoulos 1444 Università di Padova Marsilio Ficino 1462 Università di Firenze Valentin Thau 1555 Universität Leipzig Heinrich Maius Matteo Realdo (Renaldus Columbus) Colombo 1544 Università di Padova Immanuel Tremellius 1549 University of Cambridge 1561 Ruprecht-Karls-Universität Heidelberg Valentine Naibod Martin-Luther-Universität Halle-Wittenberg / Universität Erfurt Nicolaus (Mikołaj Kopernik) Copernicus 1499 Uniwersytet Jagielloński / Università di Bologna / Università degli Studi di Ferrara / Università di Padova Johannes Volmar 1515 Martin-Luther-Universität Halle-Wittenberg Jean Tagault François Dubois 1516 Université de Paris Girolamo (Hieronymus Aleander) Aleandro 1499, 1508 Università di Padova Jan Standonck 1474, 1490 Collège Sainte-Barbe / Collège de Montaigu Desiderius Erasmus 1497 /1506 Collège de Montaigu / University of Turin Matthaeus Adrianus Basilios Bessarion 1436 Mystras Johannes Müller Regiomontanus 1457 Universität Leipzig / Universität Wien Demetrios Chalcocondyles 1452 Mystras / Accademia Romana Vittorino da Feltre 1416 Università di Padova Theodoros Gazes 1433 Constantinople / Università di Mantova Erasmus Reinhold 1535 Martin-Luther-Universität Halle-Wittenberg Thomas Cranmer 1515 University of Cambridge Domenico Maria Novara da Ferrara 1483 Università di Firenze Leonhard (Leonard Vitreatoris z Dobczyc) von Dobschütz 1489 Uniwersytet Jagielloński Bonifazius Erasmi 1509 Martin-Luther-Universität Halle-Wittenberg Moses Perez Scipione Fortiguerra 1493 Università di Firenze Georgios Plethon Gemistos 1380, 1393 Rudolf Agricola 1478 Università degli Studi di Ferrara Guarino da Verona 1408 Jakob Milich 1520, 1524 Albert-Ludwigs-Universität Freiburg im Breisgau / Universität Wien Gemma (Jemme Reinerszoon) Frisius 1529, 1536 Université Catholique de Louvain Luca Pacioli Angelo Poliziano 1477 Università di Firenze Demetrios Kydones Manuel Chrysoloras Elissaeus Judaeus Ulrich Zasius 1501 Albert-Ludwigs-Universität Freiburg im Breisgau Petrus (Pieter de Corte) Curtius 1513, 1530 Université Catholique de Louvain Georg von Peuerbach 1440 Universität Wien Cristoforo Landino Nilos Kabasilas Alexander Hegius 1474 Maarten (Martinus Dorpius) van Dorp 1504, 1515 Université Catholique de Louvain Johannes von Gmunden 1406 Universität Wien Thomas à Kempis Leo Outers 1485 Université Catholique de Louvain Heinrich von Langenstein 1363, 1375 Université de Paris

Academic Genealogy of Martin Stynes

The Mathematics Genealogy Project is a service of North Dakota State University and the American Mathematical Society http://www.genealogy.math.ndsu.nodak.edu

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 25/25

Summary

Thanks, and happy birthday Martin!

Martin Joseph Stynes 1977 Oregon State University

Wen Guo 1993 National University of Ireland, Cork Guangfu Sun 1993 National University of Ireland, Cork Torsten Olaf Linss 1998 National University of Ireland, Cork Niall Madden 2000 National University of Ireland, Cork Maria Caridad Natividad 2001 University of the Philippines Aidan Naughton 2006 National University of Ireland, Cork Martin Viscor 2011 National University of Ireland, Cork Sebastian Franz 2008 Technische Universität Dresden Meghan Stephens 2009 National University of Ireland, Galway Philip Marshall Anselone 1957 Oregon State University Arvid Turner Lonseth 1939 University of California, Berkeley Hans Lewy 1925 Georg-August-Universität Göttingen Richard Courant 1910 Georg-August-Universität Göttingen David Hilbert 1885 Universität Königsberg

• C. L. Ferdinand (Carl Louis) Lindemann

1873 Friedrich-Alexander-Universität Erlangen-Nürnberg

• C. Felix (Christian) Klein

1868 Rheinische Friedrich-Wilhelms-Universität Bonn Julius Plücker 1823 Philipps-Universität Marburg Rudolf Otto Sigismund Lipschitz 1853 Universität Berlin Christian Ludwig Gerling 1812 Georg-August-Universität Göttingen Gustav Peter Lejeune Dirichlet 1827 Rheinische Friedrich-Wilhelms-Universität Bonn Martin Ohm 1811 Friedrich-Alexander-Universität Erlangen-Nürnberg Carl Friedrich Gauß 1799 Universität Helmstedt Simeon Denis Poisson 1800 École Polytechnique Jean-Baptiste Joseph Fourier École Normale Supérieure Karl Christian von Langsdorf 1781 Georg-August-Universität Göttingen / Justus-Liebig-Universität Gießen / Universität Erfurt Johann Friedrich Pfaff 1786 Georg-August-Universität Göttingen Joseph Louis Lagrange 1754 Università di Torino Pierre-Simon Laplace Abraham Gotthelf Kästner 1739 Universität Leipzig Johann Elert Bode Handelsakademie Hamburg Leonhard Euler 1726 Universität Basel Giovanni Battista (Giambattista) Beccaria Jean Le Rond d'Alembert Christian August Hausen 1713 Martin-Luther-Universität Halle-Wittenberg Johann Georg Büsch 1752 Georg-August-Universität Göttingen Johann Bernoulli 1690, 1694 Universität Basel Johann Christoph Wichmannshausen 1685 Universität Leipzig Johann Andreas Planer 1686, 1709 Martin-Luther-Universität Halle-Wittenberg / Eberhard-Karls-Universität Tübingen Johann Andreas Segner 1726, 1734 Friedrich-Schiller-Universität Jena Jacob Bernoulli 1676 Universität Basel 1684 Universität Basel Nikolaus Eglinger 1660, 1661 Universität Basel Otto Mencke 1665 Universität Leipzig Johann Pasch 1683 Martin-Luther-Universität Halle-Wittenberg Rudolf Jakob Camerarius 1684, 1686 Eberhard-Karls-Universität Tübingen Georg Erhard Hamberger 1721 Friedrich-Schiller-Universität Jena Simon Paul Hilscher 1704 Friedrich-Schiller-Universität Jena Nicolas Malebranche 1672 Peter Werenfels 1649 Universität Basel Emmanuel Stupanus 1613 Universität Basel Johann Caspar Bauhin 1649 Universität Basel Georg Balthasar Metzger 1644, 1650 Friedrich-Schiller-Universität Jena / Universität Basel Franciscus de le Boë Sylvius 1634, 1637 Universiteit Leiden / Universität Basel Jakob Thomasius 1643 Universität Leipzig Gottfried Wilhelm Leibniz 1666 Universität Leipzig 1667 Universität Altdorf 1676 Académie royale des sciences de Paris Michael Walther, Jr. 1661, 1687 Martin-Luther-Universität Halle-Wittenberg Elias Rudolph Camerarius, Sr. 1663 Eberhard-Karls-Universität Tübingen Johann Adolph Wedel 1694 Friedrich-Schiller-Universität Jena Rudolf Wilhelm Krause 1671 Universiteit Leiden Theodor Zwinger, Jr. 1630 Universität Basel Petrus Ryff 1584 Universität Basel Friedrich Leibniz Aegidius Strauch 1651, 1657 Martin-Luther-Universität Halle-Wittenberg Johann Andreas Quenstedt 1643, 1644 Universität Helmstedt / Martin-Luther-Universität Halle-Wittenberg Johann Georg Macasius 1638, 1640 Friedrich-Schiller-Universität Jena Georg Wolffgang Wedel 1667 Friedrich-Schiller-Universität Jena / Universiteit Leiden Erhard Weigel 1650 Universität Leipzig Christiaan Huygens 1647, 1655 Universiteit Leiden / Université d'Angers Bartholomäus Leonhard Schwendendörffer Sebastian Beck 1610 Universität Basel Theodor Zwinger 1553 Collège de France 1559 Università di Padova John Craig 1580 Universität Basel Abraham Klein (Calovius) 1632 Universität Rostock Andreas Kunad Christoph Notnagel 1630 Martin-Luther-Universität Halle-Wittenberg Georg Calixt 1607 Universität Helmstedt Balthasar Widmarcter 1640 Friedrich-Schiller-Universität Jena Johannes Musaeus 1634 Universität Erfurt Werner Rolfinck 1625 Martin-Luther-Universität Halle-Wittenberg / Università di Padova Adolph Vorstius 1619, 1622 Universiteit Leiden / Università di Padova Philipp Müller 1604 Universität Leipzig Jan Jansz Stampioen, Jr. Frans van Schooten, Jr. 1635 Universiteit Leiden Johann Jacob Grynaeus 1564 Eberhard-Karls-Universität Tübingen Petrus (Pierre de La Ramée) Ramus 1536 Collège de Navarre Vittore Trincavelli Università di Padova Bassiano Landi 1542 Università di Padova Ambrosius Rhodius 1600, 1610 Martin-Luther-Universität Halle-Wittenberg Paul Röber 1613 Martin-Luther-Universität Halle-Wittenberg 1617 Martin-Luther-Universität Halle-Wittenberg Johannes Caselius 1560, 1566 Martin-Luther-Universität Halle-Wittenberg / Universität Leipzig / Università di Pisa Cornelius Martini 1592 Universität Helmstedt Jacobus Martini 1596 Universität Helmstedt Georg Großhain 1629 Martin-Luther-Universität Halle-Wittenberg Adriaan van den Spieghel 1603 Université Catholique de Louvain / Università di Padova Daniel Sennert 1594, 1599 Martin-Luther-Universität Halle-Wittenberg Gilbert Jacchaeus 1601/1603/1611 University of St. Andrews / Universität Helmstedt / Universiteit Leiden Christoph Meurer 1582 Universität Leipzig Jacobus Golius 1612, 1621 Universiteit Leiden Marin Mersenne 1611 Université Paris IV-Sorbonne Jacob Andreae 1553 Eberhard-Karls-Universität Tübingen Johannes (Johann Sturm) Sturmius 1527 Université Catholique de Louvain Jacques Toussain 1521 Université de Paris Adrien Turnèbe 1532 Collège de France Pietro Pomponazzi Università di Padova Giovanni Battista della Monte Università degli Studi di Ferrara Università di Padova Andreas (Andries van Wesel) Vesalius 1537 Université Catholique de Louvain / Università di Padova Melchior Jöstel 1583, 1600 Martin-Luther-Universität Halle-Wittenberg Ernestus Hettenbach 1576, 1591 Martin-Luther-Universität Halle-Wittenberg Philipp Melanchthon 1511, 1514 Ruprecht-Karls-Universität Heidelberg / Eberhard-Karls-Universität Tübingen Johannes Hommel 1543 Martin-Luther-Universität Halle-Wittenberg Duncan Liddel 1582,1596 Universität Viadrina Frankfurt an der Oder / Universität Breslau / Universität Helmstedt Hieronymus (Girolamo Fabrici d'Acquapendente) Fabricius 1559 Università di Padova Jan Jessenius 1588, 1591 Universität Leipzig / Università di Padova Salomon Alberti 1564, 1574 Martin-Luther-Universität Halle-Wittenberg / Università di Padova Jacobus (Jacob Harmensz.) Arminius 1582 Philipps-Universität Marburg / Universiteit Leiden Moritz Valentin Steinmetz 1550, 1567 Universität Leipzig Willebrord (Snel van Royen) Snellius 1607 Universiteit Leiden Thomas Erpenius 1608 Universiteit Leiden Jakob Beuerlin 1551 Eberhard-Karls-Universität Tübingen Johannes Winter von Andernach 1527, 1532 Université Catholique de Louvain / Collège de Tréguier Nicolas (Nicolaes Cleynaerts) Clénard 1515, 1521 Université Catholique de Louvain Guillaume Budé 1486, 1491 Université d'Orléans / Université de Paris Pietro Roccabonella Università di Padova Niccolò Leoniceno 1453 Scuola Pubblica di Vicenza 1464 Università di Padova Nicoletto Vernia Università di Padova Antonio Musa Brasavola 1520 Università degli Studi di Ferrara Marco Musuro 1486 Università di Firenze Valentin (Valentinus Otho) Otto 1570 Martin-Luther-Universität Halle-Wittenberg Andreas Schato 1562, 1578 Martin-Luther-Universität Halle-Wittenberg Johann (Johannes Kapnion) Reuchlin 1477, 1481 Universität Basel / Université de Poitiers Jan (Johannes Campensis) van Campen 1519 Université Catholique de Louvain / Universität Ingolstadt Johannes Stöffler 1476 Universität Ingolstadt Paul Wittich 1566 Universität Leipzig / Martin-Luther-Universität Halle-Wittenberg Wolfgang Franz 1590 Martin-Luther-Universität Halle-Wittenberg Gabriele Falloppio 1547 Università di Padova / Università degli Studi di Ferrara Rudolph (Snel van Royen) Snellius 1572 Universität zu Köln / Ruprecht-Karls-Universität Heidelberg Georg Joachim von Leuchen Rheticus 1535 Martin-Luther-Universität Halle-Wittenberg Sebastian (Theodoricus) Dietrich 1544 Martin-Luther-Universität Halle-Wittenberg Caspar Peucer 1545 Martin-Luther-Universität Halle-Wittenberg Johann Hoffmann Ludolph van Ceulen Joseph Justus Scaliger 1563 Collège de France Balthasar Kaeuffelin 1521 Eberhard-Karls-Universität Tübingen Jacobus (Jacques Dubois) Sylvius 1530 Université de Paris / Université de Montpellier Rutger Rescius 1513 Université de Paris Jacobus (Jacques Masson) Latomus 1502 Collège de Montaigu 1519 Katholieke Universiteit Leuven Georgius Hermonymus Janus Lascaris 1472 Università di Padova Gaetano da Thiene Alessandro Sermoneta Pelope Ognibene (Omnibonus Leonicenus) Bonisoli da Lonigo Università di Mantova Jacob ben Jehiel Loans Johannes Argyropoulos 1444 Università di Padova Marsilio Ficino 1462 Università di Firenze Valentin Thau 1555 Universität Leipzig Heinrich Maius Matteo Realdo (Renaldus Columbus) Colombo 1544 Università di Padova Immanuel Tremellius 1549 University of Cambridge 1561 Ruprecht-Karls-Universität Heidelberg Valentine Naibod Martin-Luther-Universität Halle-Wittenberg / Universität Erfurt Nicolaus (Mikołaj Kopernik) Copernicus 1499 Uniwersytet Jagielloński / Università di Bologna / Università degli Studi di Ferrara / Università di Padova Johannes Volmar 1515 Martin-Luther-Universität Halle-Wittenberg Jean Tagault François Dubois 1516 Université de Paris Girolamo (Hieronymus Aleander) Aleandro 1499, 1508 Università di Padova Jan Standonck 1474, 1490 Collège Sainte-Barbe / Collège de Montaigu Desiderius Erasmus 1497 /1506 Collège de Montaigu / University of Turin Matthaeus Adrianus Basilios Bessarion 1436 Mystras Johannes Müller Regiomontanus 1457 Universität Leipzig / Universität Wien Demetrios Chalcocondyles 1452 Mystras / Accademia Romana Vittorino da Feltre 1416 Università di Padova Theodoros Gazes 1433 Constantinople / Università di Mantova Erasmus Reinhold 1535 Martin-Luther-Universität Halle-Wittenberg Thomas Cranmer 1515 University of Cambridge Domenico Maria Novara da Ferrara 1483 Università di Firenze Leonhard (Leonard Vitreatoris z Dobczyc) von Dobschütz 1489 Uniwersytet Jagielloński Bonifazius Erasmi 1509 Martin-Luther-Universität Halle-Wittenberg Moses Perez Scipione Fortiguerra 1493 Università di Firenze Georgios Plethon Gemistos 1380, 1393 Rudolf Agricola 1478 Università degli Studi di Ferrara Guarino da Verona 1408 Jakob Milich 1520, 1524 Albert-Ludwigs-Universität Freiburg im Breisgau / Universität Wien Gemma (Jemme Reinerszoon) Frisius 1529, 1536 Université Catholique de Louvain Luca Pacioli Angelo Poliziano 1477 Università di Firenze Demetrios Kydones Manuel Chrysoloras Elissaeus Judaeus Ulrich Zasius 1501 Albert-Ludwigs-Universität Freiburg im Breisgau Petrus (Pieter de Corte) Curtius 1513, 1530 Université Catholique de Louvain Georg von Peuerbach 1440 Universität Wien Cristoforo Landino Nilos Kabasilas Alexander Hegius 1474 Maarten (Martinus Dorpius) van Dorp 1504, 1515 Université Catholique de Louvain Johannes von Gmunden 1406 Universität Wien Thomas à Kempis Leo Outers 1485 Université Catholique de Louvain Heinrich von Langenstein 1363, 1375 Université de Paris

Academic Genealogy of Martin Stynes

The Mathematics Genealogy Project is a service of North Dakota State University and the American Mathematical Society http://www.genealogy.math.ndsu.nodak.edu

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References

Ansari, A. R. and Hegarty, A. F. (2003). A note on iterative methods for solving singularly perturbed problems using non-monotone methods on Shishkin meshes.

• Comput. Methods Appl. Mech. Eng., 192(33-34):3673–3687.

Chen, Y., Davis, T., and Hager, W. (2009). Algorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate. ACM Trans. Math. Software, 35(3). Clavero, C., Gracia, J., and O’Riordan, E. (2005). A parameter robust numerical method for a two dimensional reaction-diffusion problem.

• Math. Comput., 74(252):1743–1758.

Duff, I. S. (2004). MA57 – a code for the solution of sparse symmetric definite and indefinite systems. ACM Trans. Math. Softw., 30(2):118–144. Kellogg, R., Madden, N., and Stynes, M. (2008). A parameter-robust numerical method for a system of reaction-diffusion equations in two dimensions.

• Numer. Methods Partial Differ. Equations, 24(1):312–334.

Liu, F., Madden, N., Stynes, M., and Zhou, A. (2009). A two-scale sparse grid method for a singularly perturbed reaction–diffusion problem in two dimensions. IMA J. Numer. Anal., 29(4):986–1007. Madden, N. and Stynes, M. (2003). A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction-diffusion problems. IMA J. Numer. Anal., 23(4):627–644. Matthews, S., O’Riordan, R., and Shishkin, G. I. (2002). A numerical method for a system of singularly perturbed reaction-diffusion equations.

• J. Comput. Appl. Math., 145:151–166.

Numerical Analysis for Singularly Perturbed Problems, 16–18 Nov 2011 26/25

References

Roos, H.-G. (1996). A note on the conditioning of upwind schemes on Shishkin meshes. IMA J. Numer. Anal., 16(4):529–538. Shishkin, G. I. (1995). Mesh approximation of singularly perturbed boundary-value problems for systems of elliptic and parabolic equations.

• Comp. Maths. Math. Phys., 35(4):429–446.

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