User nilima nigam - MathOverflow

Answer by Nilima Nigam for Interesting Applications of the Classical Stokes Theorem?

2011-12-21T03:52:33Z

A student may also learn about the content from Stokes theorem from instances where it failed to hold as expected. For example, one has to exercise care when trying to use the theorem on domains with holes. Turn this around: the failure of Stokes to hold as expected tells you about the cohomology of the domain. I think it is possible via concrete examples to illustrate this point in a multivariate calculus class without using the more technical phraseology.

For a non-standard application of the failure of the Stokes theorem, there's the odd case of the Purcell Swimmer: http://iopscience.iop.org/1367-2630/10/6/063016/fulltext/ Rendering this accessible to a multivariable calculus class may take some work, depending on your students.

Answer by Nilima Nigam for Eigenvalues of Krylov matrices

2011-11-06T16:56:00Z

The short answer is: no. You can see the difficulty if $w$ is an eigenvector of $A$:the Krylov matrix becomes singular, while $A$ may not be.

The Krylov matrix is generated, as you probably know, during the Arnoldi iteration for locating eigenvalues of A. As part of the (stabilized version) of the process, A is partially reduced through orthogonal projections onto $\cal{K}_n$ to Hessenberg form, $H_n$. The eigenvalues of $H_m$, $m

Answer by Nilima Nigam for Textbooks for PDE between Strauss and Folland

2011-10-28T04:11:23Z

'Partial Differential Equations' by Paul Garabedian is an excellent text 'between' Strauss and Folland. The book rewards repeated reading, and contains a wealth of material and insight. It doesn't use the analytical machinery of modern PDE (e.g. does not use Sobolev spaces). It includes topics like the Perron construction, the properties of the Neumann function (as opposed to the Green's function) for a domain, and the Hamilton-Jacobi PDE. My advisor gave me his copy many years ago, saying 'this will be a good friend'.

Answer by Nilima Nigam for What items MUST appear on a mathematician's CV?

2011-10-25T05:08:04Z

I second the advice about modeling your CV on those of others. Personally, when I'm reading CV's of job applicants, here is what I first look at:

Name, employment history, education (the latter two switched in order of importance if the person is a very recent PhD)
research interests
publications
honors, awards, editorial work

At a second pass, I'd look at

talks
teaching
PhDs/postdocs supervised (depending on the position advertised)

Unless these are explicit requirements of the position (senior hires, hires to administration), I find information on grants and department-level service not very helpful. Granting systems in different countries vary wildly, as does the nature of what's service.

Caveat emptor: this is how I read CVs, and is not intended to imply anything of a universal nature.

(added later to provide scope): I've served on hiring committees in Canada for postdoc, junior and senior faculty searches (open and targetted), university senior administrators, prize committees (for research awards) and for granting agencies in North America and Europe. So my experience is limited.

Answer by Nilima Nigam for Why can we allow discontinuity on the interfaces in Discontinuous Garlerkin Method

2011-10-20T16:48:10Z

Hao, the discontinuous Galerkin method allows you to achieve approximation of $C^2$ functions by basis functions which are not globally continuous or differentiable. You write down the variational formulation of your PDE, taking care to account for jumps in the test functions and their derivatives over each of the geometric elements in your domain. Your discrete bilinear form thus includes volumetric integrals, as well as integrals along the inter-element boundaries.

You thus have to solve for not only the coefficients of the basis functions, but also their jumps. If you've set things up correctly, if your solution is globally $C^2$ then the jumps should all be zero, rendering your numerical approximation also $C^2$. There's considerable research into what formulations are stable, and in which contexts. You should look at papers by, for example, Doug Arnold, Bernardo Cockburn, and Dominik Schoetzau (just to mention a few). If you want a quick overview, http://www.cfm.brown.edu/people/jansh/resources/Publications/Lectures/RMMC08-I.pdf

The power of the DG method lies in its ability to achieve approximations for {\it non-smooth} objects, but it's fine (if inefficient?) to use for elliptic PDE on smooth domains as well.

Answer by Nilima Nigam for Am I allowed to do non-rigorous numerical analysis?

2011-10-20T00:15:26Z

When I review papers with such assertions, here is what I look for:

A clear description of the problem, and any known features of the quantity one is interested in (unique root, local minimizer, etc);
A clear description of the method used;
Information on the stopping/ error criteria used. This latter is rather important - one may stop an algorithm when the successive approximations are 'close' in some norm, or when some residual measure is smaller than some threshold (presuming one's not exceeded a specified total number of iterations.)

With this information, and a sufficiently modest claim "the computed quantity 'a' appears to provide a good approximation to the desired result'', this reviewer would be happy.

Answer by Nilima Nigam for Fast root finding for strictly decreasing function

2011-10-16T02:31:56Z

In the absence of any other information, you'd need to use bisection as described above (let's avoid the issue, for the moment, that your guesses need to bracket the root.) The {\bf order of convergence} of bisection is linear. Indeed, if you already have a good set of such guesses, and know the function is decreasing, take a couple of steps of the Secant method.

However, if you have knowledge that the function is differentiable AND you know initial guesses to bracket the root, I'd use the Regula Falsi method. This couples the Secant method with bisection. Since you have a strictly decreasing function, the usual problems with rounding may be ameliorated, since you know the sign of the derivative. The order of convergence should be around 1.67.

Finally, if you have the derivative handy, I'd start off with Newton's method. This converges quadratically, you don't need a bracketing guess, and you'll find out within a few iterations if gradient-based methods are going to fail.

In other words: use the information you have, and build in safe-guards to default to the bisection method if needed.

Answer by Nilima Nigam for Convergence of finite element method: counterexamples

2011-09-13T02:18:03Z

The maximum and minimum angle conditions for meshes are needed to prove various bounds on the error of interpolation. In other words, the solution of the PDE is a secondary concern; what goes wrong is that one cannot control the interpolation error.

Of the two, the minimum angle condition is less restrictive. What one may observe is that deteriorating conditioning of the linear system to solve for the unknown coefficients of the approximant as the minimum angle condition is violated.

There's a famous paper by Babuska and Aziz in a 1976 SIAM J. Numerical Analysis v. 13, no. 2, 'On the angle condition in the finite element method'. This also has a nice counter example showing why the interpolation error cannot be bounded unless the maximum angle is bounded away from $\pi$.

Please see http://www.bcamath.org/documentos_public/archivos/publicaciones/BraHanKorKri-SeMA.pdf for a survey and discussion of these ideas.

Answer by Nilima Nigam for Properties of the quadruple layer potential

2011-09-11T00:13:21Z

Here's a reference to a paper by Shidong Jiang which may be useful as regards jump relations : http://web.njit.edu/~jiang/Papers/jump.pdf

Answer by Nilima Nigam for Interesting mathematical topics arising from Biology

2011-09-02T01:33:39Z

As regards interesting mathematics arising in biology:

A mathematically fascinating class of integro-PDEs arise in the study of age-structured population models. The independent variables are age $a$ and time $t$ ; the systems are first-order PDE; and the boundary conditions on the curve age=0 are given in terms of integrals of the dependent variables, $u$. That is, $u(0,t)= \int_{a=0}^T \phi(u(s,t)) ds$ where $\phi$ may be a nonlinear function. Such models arise frequently in physiology. It's my impression that this is a field with many interesting open mathematical questions to be asked.
PDEs arising in pattern-forming systems in biology exhibit interesting mathematical behavior; questions about long-time regularity of such PDE are mathematically interesting. One may wish, for example, to characterize finite-time blow-up, or development of geometric singularities on interfaces.
Dynamical systems with delays (functional differential equations) for the form $\frac{dy}{dt} = A(y(t-\tau),t)$ arise naturally in biology. This is a field which is not as mathematically developed as the theory of ODE.

Answer by Nilima Nigam for Applications of PDE in mathematical subjects other than geometry & topology

2011-09-01T03:16:52Z

Not sure if this counts, but aren't there a whole bunch of neat results in combinatorics on discrete analogs of PDE? I'm thinking, for example, of Stone's theorem for the discrete Schrodinger equation; the characterization of graph spectra; and how some solutions of the discrete Laplacian/discrete Helmholtz/discrete modified Helmholtz lead to special instances of ADE Dynkin diagrams.

Answer by Nilima Nigam for Computational methods for dealing with geometrically complicated solid boundaries in fluid-air interface problems

2011-09-01T02:41:07Z

You may also consider the immersed boundary method, and its variants. It was specifically developed for situations involving complex fluid/structure interactions. The method is quite successful for solid structures (complexity is not a problem); it's a bit trickier for porous media. In essence one write down the interaction forces experienced by the particles in the solid body, and integrates.

The methods do indeed rely on the computation of integrals, but fast quadratures work quite well here.

A really great starting point for this field is the Acta Numerica paper by Charles Peskin (2002). He also has some nice course notes, and code, online:

http://math.nyu.edu/faculty/peskin/ib_lecture_notes/index.html

Answer by Nilima Nigam for Changing field of study post-PhD

2011-07-12T04:05:29Z

When I see an applicant for a postdoctoral or tenure-track position who has switched/is switching fields, I ask: 'why'? You'll face this question, and will need to be clear to your interlocutor about your reasons. I read CVs for evidence of mathematical strength. In young researchers, I also look for evidence of intellectual independence from the PhD/postdoc supervisors and their research program. A well-conceived move of fields is a positive here.

Consider the following:

(1) Candidate A is interested in spectral theory and microlocal analysis, and moves into numerical analysis and fast integral equation techniques because a specific problem demanded a computational approach. Candidate A then begins to contribute to NA, and takes care to publish substantial papers in reputed journals. During an interview Candidate A can precisely describe the motivating problem, and why they needed to move into numerical analysis.

(2) Candidate B starts off in microlocal analysis, and switches to math biology with no apparent link. The publications in math bio don't signal deep engagement with the new field, and aren't in the better journals. During an interview Candidate B is not quite clear about why they moved, but funding and jobs come up often.

Speaking only for myself, Candidate A's move is viewed as intellectually courageous and nimble, whilst Candidate B appears cynical. No one in field Y wants to hear that a candidate moved from field X to Y because of the money. Instead, they want to hear why the candidate finds Y an appealing, natural field to work in.

Answer by Nilima Nigam for Exponential of large matrices

2011-06-23T05:24:42Z

I've asked for some clarification in a comment. In the meanwhile, if you're looking for software, I'll assume you've tried PETSc or Trilinos already? Here's a link to the freeware by Jiri Pittner, which links to BLAS routines as well: http://www.pittnerovi.com/la/

Here's a site from INRIA http://verdandi.gforge.inria.fr/doc/linear_algebra_libraries.pdf

Answer by Nilima Nigam for What could be some potentially useful mathematical databases?

2011-06-22T03:42:39Z

For numerical analysts and scientific computing folks, a database of 'standard problems' with given geometry, parameters, tolerances and input/output specifications, and a mechanism for storing and comparing (curated) computational attacks on these. For example, the problem of lid-driven cavity flow is considered a major test for computational fluid dynamics, and it would be great to have an agreed set of 3 or 4 sub-problems (laminar flows, angled walls, incompressible flows, nearly incompressible flow) on which the performance of algorithms could be compared.

Comparisons of the performance of algorithms on a given problem according to criteria such as accuracy, storage needed, and efficiency would be useful. Code in a given language would be useful, but one probably cannot insist on this.

There's a dearth of such 'standard problems', and thus algorithms purportedly approximating solutions for the same problem are rarely compared. NIST has an example of such standard problems for models of micromagnetics. http://www.ctcms.nist.gov/~rdm/mumag.org.html

Answer by Nilima Nigam for Going to graduate school for mathematics next year, need some advice

2011-06-18T19:45:39Z

Steppenwolf, the strictly pragmatic advice would be:

1) Identify the graduate programs in differential geometry you'd like to be a part of, and look at their first year coursework. Many programs have written qualifying or comprehensive examinations, so the coursework may be structured around it. I would imagine analysis, algebra and topology would be key ingredients in most of these program.

2) Identify some of the people you'd like to work with, and maybe ask them in person? This would have the side-effect of learning about these people as potential supervisors.

Best wishes for your graduate career! I hope you find it a pleasurable (even if unpredictable) journey.

Answer by Nilima Nigam for Extension theory with bump function

2011-06-15T04:04:53Z

Short answer: yes.

Let $\psi_\epsilon(x):=\frac{1}{\epsilon^n}\exp{\epsilon^2/(\epsilon^2-|x|^2)}$ for $|x|<\epsilon$, and $\psi_{\epsilon}(x)=0$ for $|x|\geq \epsilon$. Set $\epsilon=2$, and define $\phi$ is the convolution of $C\phi_{\epsilon}$ with the characteristic function of $B_{r+3/2}(0)$, that is,

$\phi(x):= C\psi_\epsilon(x)* \chi_{B_{r+3/2}(0)}$. Here $C$ is a normalizing constant (this may not be needed, but I haven't checked).

This yields a smooth cut-off function which is 1 in the ball $B_{r+1}(0)$, and zero outside $B_{r+2}(x)$.

To see this does the trick, one can use a localization theorem, for example, Theorem 3.20 in 'Strongly Elliptic Systems and Boundary Integral Equations' by W. McLean. This theorem states:

'Suppose that $\phi \in C^r_{comp}(\mathbb{R}^n)$ for some integer $r\geq 1$, and let $|s|\leq r$. If $u\in H^s(\Omega)$ then $\phi u \in H^s(\Omega)$, and $||\phi u||_{H^s(\Omega)} \leq C_r||\phi||_{W^{r,\infty}(\mathbb{R}^n)}Q_u$ where $Q_u=||u||_{H^s(\Omega). }$ (Apologies, I encountered trouble while trying to typeset the LaTeX here).

The same result holds with $H^s(\Omega)$ replaced with $\tilde{H}^s(\Omega)$.'

The proof proceeds using $\Omega = A_2$, and then

either by (a) considering the situation for $s=r$, using duality to see it holds for $s=-r$, and the intermediate $s$ by interpolation. This is suggested by Yakov above.

or (b) by examining $\hat{\phi u}$ and using Peetre's inequality.

Since the constructed $\phi \in C^\infty$ and has compact support, it will satisfy the inequality you seek. In my comment I asked whether you wanted a $\phi$ of minimal regularity (relative to $\tau$); my construction works but may be overkill.

There must be a good introductory numerical analysis course out there!

2011-05-19T02:39:18Z

Background As a numerical analyst, I've frequently taught the 'Introductory Numerical Analysis' class. Such courses are found in many major universities; the audience typically consists of reluctant engineering majors and some majors of mathematics.

The structure of the course is very similar in many of the institutions whose syllabi I've looked at: one begins with finite-precision arithmetic, then fixed-point methods for root-finding (usually 1-D problems),interpolation by polynomials, quadrature, numerical differentiation, some standard ODE methods, and perhaps some finite difference methods for PDE. Any rationale for this particular sequence of topics is obscured in the course.

The truly deep and interesting aspects - approximation theory, error analysis, computational complexity - are either not discussed, or not dwelt on. Instead, the typical introductory course is a collection of algorithms for problems which seem contrived. This is a pity. The stronger mathematics student comes away believing numerical analysis is boring and shallow, and the engineer comes away thinking mathematics has nothing to offer a real problem.

The question: Are there examples (links to course outlines or course webpages preferred) of introductory numerical analysis courses which avoid the above-described tedium, and which have a history of attracting strong mathematics students?

The constraints: The courses should be aimed at students with a background in multivariate calculus, linear algebra, undergraduate dynamical systems and PDE. One example per answer, please.

The motivation: The eventual goal is to compile such a list, and based on these courses suggest a better curriculum at my institution.

Answer by Nilima Nigam for Classification of PDE

2011-06-04T20:20:59Z

I am unsure of the etiquette surrounding multi-part questions. Here are answers to two sub-parts. Since your Q5. invites opinion, I've addressed that in a comment instead.

Q1: yes, the definition of ellipticity via the non-vanishing of the principle symbol is a useful characterization for elliptic PDE, see Hörmander's book. All manners of existence and regularity properties can be examined from here.

Q2: This is murkier. If the principal symbol of a linear PDE, order q, with smooth coefficients is a hyperbolic polynomial, then the PDE is hyperbolic. This doesn't generalize easily to nonlinear cases, and is not an easy condition to check. See an extensive discussion here: http://math.stackexchange.com/questions/21525/mathematical-precise-definition-of-a-pde-being-elliptic-parabolic-or-hyperbolic

L.C. Evans, in his preface to his AMS text on PDE, mentions that he finds it unsatisfactory to classify PDE, since it creates the false impression that a general classification is available. Several equations change type (eg. Tricomi's equation) and many PDE of interest are highly nonlinear.

Answer by Nilima Nigam for Reference request for conceptual numerical analysis

2011-06-01T05:07:07Z

To augment Timur's answer:

Claes Johnson's introductory book on FEM
Braess's book on FEM
Iserles's book on numerical analysis of DE
Gottlieb and Orzsag's book on spectral methods
Nick Trefethen's book on spectral methods for spectral collocation ideas.
Quarteroni, Sacco, Saleri on numerical methods.
From 'the horse's mouth', the Cleve Moler book on numerical computing using Matlab.

I've picked these books for their balance of important algorithms and key insights, delivered with clear prose.

I also like Strikwerda's book on finite difference methods, and the Hairer-Wanner books on numerical methods for ODE. But these focus a lot on error analysis, which may not be what you wish.

Answer by Nilima Nigam for Characterize where the Dirichlet Problem for the Laplacian is always solvable

2011-05-25T04:14:43Z

I assume you are asking about strong solutions (so u is actually $C^2(G)\cap C(\partial_\infty G))$. In this case, the characterization via barriers, or equivalently, as Will says, using Perron's method, cannot be improved upon, I think.

Here's what I remember, please correct me if there are flaws in the argument.

Define a regular point as a point $a$ on the boundary of $G$ such that a barrier exists at $a$ (with respect to G).Conway's characterization is saying domains with boundaries consisting of only regular points are Dirichlet regions.

Now for the converse: if a region is a Dirichlet region, it must have a boundary of regular points. Suppose there is a domain $G$ which is a Dirichlet region. Let $y$ be a point on the boundary of the domain. Consider a continuous function $f:\partial_\infty G \rightarrow \mathbb{R}$ such that $f(y)=0$, and $f>0$ for all other parts of the boundary.The solution $u$ of the Dirichlet problem with $f$ as data is, by the strong maximum principle, a barrier at $y$. Hence $y$ is a regular point.

The question of the boundary regularity necessary for solvability of the Dirichlet problem has indeed been studied, and the answer may vary depending on the specific notion of solvability (strong solution? weak solution? solution a.s.?). Gilbarg and Trudinger, H\"ormander, Maz'ya have all written nice books on this and related topics.

Answer by Nilima Nigam for Do the Euler method's approximations always approach the true solution?

2011-05-24T17:39:56Z

You may also want to look at the literature on backward error analysis for ODE methods.

Answer by Nilima Nigam for Approximation in $L^2$ by piecewise constant functions

2011-05-24T00:01:28Z

Dmitry's answer addresses your question. If you're looking for the rate of convergence in h, then -you need to say a bit more about the regularity of w. I'd recommend the book by Brenner and Scott.

Answer by Nilima Nigam for What notions are used but not clearly defined in modern mathematics?

2011-05-04T16:05:20Z

'Applied Mathematics' is a much-used term in modern mathematics, but I've yet to find a universally-agreed upon definition. Given its use as a major category ('pure' vs 'applied') and repository of sundry generalizations ('non-rigorous','relevant', 'not deep', 'critical to science', etc.), surely a precise definition is in order.

In the MSC, there is only one MSC code with this phrase (00A69). Based on this, maybe 'Applied Mathematics' is a field of inquiry which is not important

Answer by Nilima Nigam for Lie Groups and PDEs

2011-05-04T01:55:09Z

I found a solid background in PDE, together with some physics, to be a useful entry point to Olver's nice book. There's the 'Lectures on Partial Differential Equations' by V.I.Arnold which is fun to read alongside, if not before. Any solid book on mathematical methods in classical mechanics and quantum mechanics should prove useful as well. Finally, I agree with Deane- the most efficient path is to start reading the book, and learn the material you need as you proceed.

Answer by Nilima Nigam for The first eigenvalue of a graph - what does it reflect?

2011-05-01T15:48:05Z

I fear my answer may not directly address the question, but I like the question!

Suppose we wish to numerically approximate solutions of the Poisson problem on a given domain. One strategy is to 'mesh' the region by simplices, and seek information on the nodes. One can approximate the Laplacian either strongly (finite differences) or weakly (finite elements) on the resultant graphs. The resultant matrices are symmetric and positive definite. Their largest eigenvalue reflect their condition number, which usually scales as $1/h^2$ as the length of edges $h \rightarrow 0.$ This condition number $\kappa:=|\lambda_{max}|/|\lambda_{min}|$ tells us how sensitive the computed solution will be to small errors in data, eg. due to rounding. Were I to change the data locally on one of the simplices, for example by marginally changing the location of one node, $\kappa$ predicts the worst amount by which computed solutions may change.

I conjecture there is a similar result in graph theory: given a graph, its largest eigenvalue provides a measure of how small changes to the graph structure influence flows on the graph. But I don't know enough about graph theory to even frame the conjecture precisely.

Answer by Nilima Nigam for Trace theorem for $C^{k,1}$ domains

2011-04-30T02:22:24Z

I also believe the Kim paper cited above has the relevant result. On a related matter, there's a very nice paper by Buffa, Costabel and Sheen in J.Math.Anal.Appl. (2002) on trace theorems for H(curl) fields in Lipschitz domains. And Luc Tartar may have the relevant result you seek for part 2 of your question.

Approximations of negative Sobolev norms

2011-04-29T03:59:58Z

Consider the standard Cahn-Hilliard free energy, augmented by a nonlocal interaction term which measures the $H^{-1}$ norm of a zero-mean function. Could someone point me to a reference where this nonlocal term is numerically approximated for a function on a compact domain, but without assuming periodicity? The periodic case is handled, for example, in a paper by Choksi et al in SIAM J. Appl.Math., 2009. Specifically, any strategies which avoid a Poisson solve would be welcome.

Answer by Nilima Nigam for Geodesic Triangles in Finite Element Method

2011-04-29T02:04:04Z

Interesting question! There's recent work on isogeometric analysis (eg. Hughes et al in CMAME 2005), who seek to combine FEM with NURBS. This may be a useful literature to look at, for potential applications of your own ideas.

In essence, if one is to study PDE on surfaces, then an intrinsic (geodesic) mesh may confer advantages in terms of accuracy.

Answer by Nilima Nigam for Demonstrating that rigour is important

2011-04-28T01:54:26Z

A rich source of examples may be found in the study of finite element methods for PDEs in mixed form. Proving that a given mixed finite element method provided a stable and consistent approximation strategy was usually done 'a posteriori': one had a method in mind, and then sought to establish well-posedness of the discretization. This meant a proliferation of methods and strategies tailored for very specific examples.

In the bid for a more comprehensive treatment and unifying proofs, the finite element exterior calculus was developed and refined (eg., the 2006 Acta Numerica paper by Arnold, Falk and Winther). The proofs revealed the importance of using discrete subspaces which form a subcomplex of the Hilbert complex, as well as bounded co-chain projections (we now call this the 'commuting diagram property). These ideas, in turn, provided an elegant design strategy for stable finite element discretizations.

Comment by Nilima Nigam

2011-12-23T06:17:11Z

I'd be cautious while using the built-in routines for high-order Bessel functions. I believe they are computed using recurrence relations (numerically, not symbolically), so there are issues of cancellation. How are you computing the special functions, and how high do you go in n? Ben Adcock points you to a good reference.

Comment by Nilima Nigam

2011-12-21T16:10:47Z

Daniel, thank you for the reference!

Comment by Nilima Nigam

2011-12-21T03:24:09Z

This is a wonderful example!

Comment by Nilima Nigam

2011-11-29T03:32:01Z

@Gilead, thanks - of course, you are correct. The constraint that the solution consist of integers renders it (very) hard. I think Xiao-wen Chang has some papers in this area, including one on box-constrained integer least squares: cs.mcgill.ca/~chang/pub/ChaH08.pdf

Comment by Nilima Nigam

2011-11-28T05:53:19Z

I suggest rephrasing this as locating the minimizer of $x^T A x - bx +c$, and then using the fact that $A$ is symmetric, and positive semi-definite, to use a Krylov method to solve the associate linear problem.

Comment by Nilima Nigam

2011-11-25T16:21:31Z

Marius Mitrea has a bunch of papers on the regularity of the Dirichlet problem on manifolds.

Comment by Nilima Nigam

2011-11-18T00:25:30Z

Is there any additional information you can provide on this problem (in terms of $A,B,C,D.E,F$) - is there any reason to expect unique solutions for this system? Trivially, one would interpret this question as a system of $N^2$ equations for the entries of the $N\times N$ matrix $X$. One could then use a host of algorithms including the family of Newton methods. Which algorithm to use will depend on the structure of the equations.

Comment by Nilima Nigam

2011-11-02T14:41:41Z

There is a typo here- as stated, the inequality is false (take M=identity, N=matrix of zeros).

Comment by Nilima Nigam

2011-10-29T14:36:51Z

The matrix appears nearly rank deficient, so I'd suggest using methods for rank-deficient QR decompositions with column pivoting. The key would be Householder/Givens rotations rather than projections. As Igor suggests, Golub and van Loan's book has lots on the numerical analysis of this. Demmel's book will point you to algorithms for your particular situation.

Comment by Nilima Nigam

2011-10-28T03:54:31Z

How are you computing the orthogonal vectors? Pure Gram-Schmidt is the obvious incorrect choice; have you tried using Householder reflections? Those are going to be stabler for a given precision than standard Gram-Schmidt when columns are near-orthogonal. Trefethen and Bau's book would be a good place to look, and Demmel's book would have a comprehensive collection of algorithms for specific situations.

Comment by Nilima Nigam

2011-10-20T18:47:25Z

Mcandril, algori's solution tells you a lot. For example, if $k=0$, your matrix $A(t)$ is constant in time; the solution has simple features. If $k\not=0$, there are three regimes to consider: $|k|\approx 0$, $|k|\rightarrow +\infty$, and the intermediate regimes. The solution in the asymptotic regimes can, and should, be analysed asymptotically (see, eg. Bender and Orszag for inspiration). Also, a word of caution re numerical simulations for large $k$: are you sure you don't have a very stiff ODE system (large separation of time scales)? Did you use a DAE solver?

Comment by Nilima Nigam

2011-10-18T15:18:33Z

Nice! As an aside, I admire your posts (questions and answers) on MO a lot for their clarity and the occasional cool graphics.

Comment by Nilima Nigam

2011-09-28T01:41:58Z

I'd have to agree- the problem, as stated, may not admit well-defined solutions. Set $c=0, d=1,a=-1,b=0$, and one finds Burger's equations. I can think of boundary data on the edges of the square which would lead to a lack of well-posedness. A numerical attack via finite differences would be reasonable to try.

Comment by Nilima Nigam

2011-09-27T02:07:51Z

I'm not sure I am clear about the set up. Are a,b,c,d, smooth functions whose values you know only at the grid points? And is the relation you describe only satisfied at the grid points? If the latter, then I don't think the method of characteristics makes much sense, since we don't know anything about the invariants away from those points.

Comment by Nilima Nigam

2011-09-20T02:30:43Z

You are correct- I was thinking about the $H^1$ norm, in which the counterexample above fails to converge. If you are interested only in the failure of convergence in the $L^\infty$ norm, this may be hard to show under the hypotheses you have ($u\in C^2$, $\Omega$ convex).