most recent 30 from http://mathoverflow.net 2013-05-22T17:34:40Z http://mathoverflow.net/feeds/user/1783 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/64701/estimating-an-integral-with-a-singularity-at-the-intervals-endpoint tristes_tigres 2011-05-11T21:53:52Z 2011-05-11T23:36:07Z

<p>I am trying to obtain an analytic estimate of this integral:</p> <p>$\int_0^1\frac{1}{\sqrt{x}}\exp\left(-a(x-x_0)^2\right) dx$,</p> <p>where $a\gg1$, $x_0\in[0,1]$. Saddle-point approximation doesn't work due to infinite derivative of $1/\sqrt{x}$ at 0. Any tips on how to get a handle on this will be much appreciated.</p>

http://mathoverflow.net/questions/54795/concentration-of-measure-and-bounds-on-variance tristes_tigres 2011-02-08T17:25:32Z 2011-02-08T22:30:54Z

<p>I am trying to characterize the sensitivity of a function $f: R^N\to{}R$ to the perturbations in the input vector $\mathbb{x}=\left[x_1,\dots{}x_N\right]$. For that purpose, I evaluate Cramer-Rao bound for Gaussian i.i.d. arguments.</p> <p>The function has a singularity at the point $\mathbb{x}_0$ where $f(\mathbb{x}_0)=0$, in the sense that $\|\nabla{}f\|\sim{}1/f$ as $\mathbb{x}\to\mathbb{x}_0$</p> <p>The Cramer-Rao bound then doesn't make sense, because it diverges at $\mathbb{x}_0$, while the variance of $f$, obviously, remains bounded. What I am looking for, I guess, some type of "concentration of measure"/"deviation inequality"-type sharp bound on the variance of $f$. </p> <p>The literature on concentration of measure phenomenon is extensive and deals with fairly advanced topics, whereas I am looking for something rather more basic. If you could point towards some starting point, your help will be appreciated.</p>

http://mathoverflow.net/questions/35643/conjugate-gradient-for-a-slightly-singular-system/47789#47789 tristes_tigres 2010-11-30T13:00:49Z 2010-11-30T13:00:49Z

<p>You can add a linear constraint, stipulating that your solution should be orthogonal to the null space, assuming it is known to you ahead of time.</p>

http://mathoverflow.net/questions/41994/basic-software-libraries-for-numerical-analysis-using-modern-programming-language/47785#47785 tristes_tigres 2010-11-30T11:40:57Z 2010-11-30T12:08:12Z

<p>A number of numerical libraries in Java can be found at <a href="http://math.nist.gov/javanumerics/" rel="nofollow">JavaNumerics</a></p> <blockquote> <p>"Modern" in this context means to me: object oriented (not C or Fortran)</p> </blockquote> <p>Fortran has object-oriented features since 2003, see, for instance <a href="ftp://ftp.nag.co.uk/sc22wg5/N1551-N1600/N1579.pdf" rel="nofollow">Fortran working group note 5</a> (IMHO object-orientation has been oversold, in general, and does not belong in Fortran, specifically, but the Fortran standards committee didn't ask me, alas)</p> <p>Commercial libraries that have been ported to Fortran 2003 include IMSL and NAG. Opensource library <a href="http://netlib.org/lapack/index.html" rel="nofollow">LAPACK</a> is F90, but if you are going to do numerical work, chances are, you'll <strong>have</strong> to use it.</p> <blockquote> <p>A software library in a "modern" programming language could attract more people (maybe more contributors if it is open source), for this reason, than a library in a much better suited, but less known, programming language. </p> </blockquote> <p>It is much harder to write robust numerical code, than to contribute to an average opensource project. I believe, that the choice of language is a tertiary concern, - understanding the mathematical subject matter and error anlysis takes more work than learning another (procedural) programming language.</p>

http://mathoverflow.net/questions/64701/estimating-an-integral-with-a-singularity-at-the-intervals-endpoint/64712#64712 tristes_tigres 2011-05-12T19:01:27Z 2011-05-12T19:01:27Z

OK, I missed that this integral evaluates exactly. Thank you for your answer

http://mathoverflow.net/questions/64701/estimating-an-integral-with-a-singularity-at-the-intervals-endpoint/64712#64712 tristes_tigres 2011-05-12T13:15:55Z 2011-05-12T13:15:55Z

Put another way, there's still problem of analytic estimate for $\int_0^\infty\frac{1}{\sqrt{y}}\exp(-(y-y_0)^2)dy$ and the saddle-point approximation doesn't work for $у_0\in[0,1]$. Sure, it's easy to get the value $2\ \Gamma\left(5/4\right)$ for this integral when $y_0=0$, but that wasn't my question.

http://mathoverflow.net/questions/64701/estimating-an-integral-with-a-singularity-at-the-intervals-endpoint/64712#64712 tristes_tigres 2011-05-12T10:03:31Z 2011-05-12T10:03:31Z

I am interested in this integral as a function of $x_0$ for fixed large, but finite $a$. The interval of values $x_0\in[0, \sqrt{a}]$ is of particular interest.

http://mathoverflow.net/questions/64701/estimating-an-integral-with-a-singularity-at-the-intervals-endpoint/64712#64712 tristes_tigres 2011-05-11T23:51:18Z 2011-05-11T23:51:18Z

No, $x_0$ doe not depend on $a$. I mean that I need the value of the integral for all $x_0\in [0,1]$, including the region $x_0\sim 1/\sqrt{a}$. Sorry if my comment was unclear.

http://mathoverflow.net/questions/64701/estimating-an-integral-with-a-singularity-at-the-intervals-endpoint/64710#64710 tristes_tigres 2011-05-11T23:25:05Z 2011-05-11T23:25:05Z

This doesn't produce the correct behaviour in the region $x_0\sim 1/\sqrt{a}$, which I do need.

http://mathoverflow.net/questions/64701/estimating-an-integral-with-a-singularity-at-the-intervals-endpoint tristes_tigres 2011-05-11T23:07:35Z 2011-05-11T23:07:35Z

That substitution doesn't help with respect to saddle-point approximation.

http://mathoverflow.net/questions/54795/concentration-of-measure-and-bounds-on-variance tristes_tigres 2011-02-09T12:44:43Z 2011-02-09T12:44:43Z

Didier - this is quite correct, the gradient does go to infinity at $\mathbb{x}_0$. That's what makes the question interesting, because the &lt;b&gt;variation&lt;/b&gt; remains bounded. It is qualitativerly clear why, but I would like to produce quantitative bound.

http://mathoverflow.net/questions/54795/concentration-of-measure-and-bounds-on-variance/54827#54827 tristes_tigres 2011-02-09T00:06:56Z 2011-02-09T00:06:56Z

Although, may be I get, what you mean with Poincare inequality. Perhaps, it could do the job.

http://mathoverflow.net/questions/54795/concentration-of-measure-and-bounds-on-variance tristes_tigres 2011-02-08T23:22:06Z 2011-02-08T23:22:06Z

Didier - It's not a hypothesis, it's a property of the specific function that I am analyzing.

http://mathoverflow.net/questions/54795/concentration-of-measure-and-bounds-on-variance/54827#54827 tristes_tigres 2011-02-08T23:06:29Z 2011-02-08T23:06:29Z

Right off the bat, the trouble with using Poincare inequality is that in my case, $\|\nabla{}f\|\to\infty$ at $\mathbb{x}_0$, so it's hard to see, how to get a useful bound from it. The actual analytic expression for $\mathbb{E}|\nabla{}f|^2$ doesn't seem to be easier to find than that for the $\mathbb{E}|f|^2$

http://mathoverflow.net/questions/54795/concentration-of-measure-and-bounds-on-variance/54827#54827 tristes_tigres 2011-02-08T22:40:47Z 2011-02-08T22:40:47Z

I mean, thank you for citing freely accessible articles

http://mathoverflow.net/questions/54795/concentration-of-measure-and-bounds-on-variance/54827#54827 tristes_tigres 2011-02-08T22:39:24Z 2011-02-08T22:39:24Z

Thank you. I'll need some time to digest the references, and not citing articles behind the paywall is definitely welcome :)

http://mathoverflow.net/questions/54795/concentration-of-measure-and-bounds-on-variance tristes_tigres 2011-02-08T20:33:24Z 2011-02-08T20:33:24Z

The function is defined implicitly, so I mixed different gradients

http://mathoverflow.net/questions/54795/concentration-of-measure-and-bounds-on-variance tristes_tigres 2011-02-08T20:26:11Z 2011-02-08T20:26:11Z

Didier - I mean that $f(\mathbf{x})=(\nabla{}f)\cdot{(\mathbf{x}-\mathbf{x}_0)}+O((\mathbf{x}-\mathbf{x}_0)^2)$ near $\mathbf{x}_0$, but my notation was somewhat erroneous. Fixed that.

http://mathoverflow.net/questions/41994/basic-software-libraries-for-numerical-analysis-using-modern-programming-language/47785#47785 tristes_tigres 2010-12-01T12:33:21Z 2010-12-01T12:33:21Z

There's an article [&quot;Matrix Multiply: A Case Study&quot;](<a href="http://stellar.mit.edu/S/course/6/fa08/6.197/courseMaterial/topics/topic2/lectureNotes/Intro_and_MxM/Intro_and_MxM.pdf" rel="nofollow">stellar.mit.edu/S/course/6/fa08/6.197/&hellip;</a>) which is highly illuminating wrt cost one may pay for using popular prescriptions, such as OO or immutability. They end up with 4 orders of magnitude speedup obtained by &quot;playing dirty&quot;.