User ed gorcenski - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T15:08:08Z http://mathoverflow.net/feeds/user/5640 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102413/must-read-papers-in-numerical-analysis/102473#102473 Answer by Ed Gorcenski for "Must read" papers in numerical analysis Ed Gorcenski 2012-07-17T18:28:46Z 2012-11-20T07:00:58Z <p>It is not a classic paper, but I would add Xiu, D. and Karniadakis, G.E, "The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations," which can be found <a href="http://www.dam.brown.edu/scicomp/media/report_files/BrownSC-2003-07.pdf" rel="nofollow">here</a>.</p> <p>I mention this paper because Generalized Polynomial Chaos is still underrepresented in many fields, though it has found much use in engineering applications. It is not precisely a magic bullet, but in some cases it can drastically reduce computational demand in uncertainty analysis. The aforementioned paper is a good summary of the method.</p> http://mathoverflow.net/questions/24256/fields-of-mathematics-that-were-dormant-for-a-long-time-until-someone-revitalized/24292#24292 Answer by Ed Gorcenski for Fields of mathematics that were dormant for a long time until someone revitalized them Ed Gorcenski 2010-05-11T21:04:11Z 2010-05-12T00:52:57Z <p>Polynomial Chaos was developed in the late 30s by N. Wiener, but went more or less unnoticed until Ghanem &amp; Spanos picked up on it for use in finite element analysis in the 80s and 90s. In some ways it still may be an under-utilized approach, given the dominance of the Itō and Stratonovich calculi.</p> http://mathoverflow.net/questions/24287/what-is-the-best-algorithm-to-find-the-smallest-nonzero-eigenvalue-of-a-symmetric/24297#24297 Answer by Ed Gorcenski for What is the best algorithm to find the smallest nonzero Eigenvalue of a symmetric matrix? Ed Gorcenski 2010-05-11T21:41:22Z 2010-05-11T21:41:22Z <p>A quick search led me to this paper, which deals specifically with sparse symmetric matrices, although some of its references might be useful.</p> <blockquote> <p>Jang, Ho-Jong, and Lee, Sung-Ho, "NUMERICAL STABILITY OF UPDATE METHOD FOR SYMMETRIC EIGENVALUE PROBLEM," J. Appl. Math. &amp; Computing Vol. 22(2006), No. 1 - 2, pp. 467 - 474.</p> </blockquote> <p>A PDF copy is available here: <a href="http://www.mathnet.or.kr/mathnet/kms_tex/986075.pdf" rel="nofollow">http://www.mathnet.or.kr/mathnet/kms_tex/986075.pdf</a></p> <p>I should also mention that "best" is a difficult superlative to qualify without knowing the structure of your matrices. Probably the best algorithm for a sparse symmetric matrix is not the best algorithm for a symmetric Toeplitz matrix.</p> http://mathoverflow.net/questions/24272/numerical-instability-using-only-heuns-method-on-a-simple-pde/24290#24290 Answer by Ed Gorcenski for Numerical instability using only Heun's method on a simple PDE. Ed Gorcenski 2010-05-11T20:56:09Z 2010-05-11T20:56:09Z <p>There are a couple approaches that I think you could take to avoid this problem.</p> <p>Consider your Euler's method example. With this example, you know that the value at the next time step $W(x_i,t_{j+1})$ will go negative if the delta term, $\frac{c\left[W(x_{i+1},t_{j+1}-W(x_{i-1},t_{j+1})\right]}{2\Delta x}$ is greater than the value at the current time step, $W(x_i,t_j)$. The negative value leads to error inflation, and so on.</p> <p>First, you could try an adaptive solver, which varies the step size to meet certain error tolerances. MATLAB, for example, comes with ode45() which uses a 4th order and 5th order Runge-Kutta solver in conjunction with one another, and adaptively adjusts the step size. </p> <p>A second solution is to use a multi-step method, such as an Adams-Bashforth method. These methods are well-suited for stiff problems, which although your particular issue is not due to stiffness, it does seem to suffer from the same issues as stiff problems -- that is, the method is incapable of approximating the derivative of the function within a desired error tolerance within a neighborhood of a set of points.</p> <p>A third solution is a hybrid approach. Since you know how to evaluate, based on your chosen ODE solver, when the next step will go negative, then you could put in place some conditionals that changes the routine when you encounter these trouble spots either by switching to a different method, or reducing the step size in some ad hoc manner. Alternatively, you could switch to a multi-step method at this point, and use the preceding $m$ steps as the seed for the multi-step method.</p> <p>I haven't put a whole lot of effort into evaluating the region of instability, since I don't know how well the reduced problem applies to your current needs, but if you substituted $e^{(x-ct)^2}$ in your delta term for Euler's method, you could probably determine pretty easily when your solution will dip negative.</p> http://mathoverflow.net/questions/24221/reference-request-for-conceptual-numerical-analysis/24247#24247 Answer by Ed Gorcenski for Reference request for conceptual numerical analysis Ed Gorcenski 2010-05-11T15:06:59Z 2010-05-11T15:06:59Z <p>Are you looking for a reference that links the field of numerical analysis to mathematical concepts moreso than algorithmic concepts? Matrix Computations by Golub and Van Loan is a fairly important book that studies the algebraic structures of matrices and derives algorithms from those properties. If you're looking for an entry-level work, I keep a copy of Michael Heath's book Scientific Computing on my desk. It covers fundamental concepts and algorithms fairly well, in my opinion.</p> <p>Do you have a specific problem domain in mind?</p> http://mathoverflow.net/questions/24132/what-are-examples-of-mathematical-concepts-named-after-the-wrong-people-stigler/24142#24142 Answer by Ed Gorcenski for What are examples of mathematical concepts named after the wrong people? (Stigler's law) Ed Gorcenski 2010-05-10T19:52:00Z 2010-05-10T21:53:16Z <p>If you search for almost any eponymous topic in Wikipedia, you'll find that it was first studied by someone else. For example, the Gaussian distribution (according to Wikipedia) was first studied by de Moivre. It seems that in many cases, naming the body of work was given to the person who first applied its study to some other field (using the earlier example, Gauss used the distribution in astronomy).</p> <p>The common story goes that L'Hôpital bought "the rights" to L'Hôpital's rule, as he was a nobleman and not a mathematician by trade, although I am not sure about the veracity of that story.</p> <p>Although I am no expert on the history of Mathematics, it seems as though ideas or formulae assumed their names from certain mathematicians due either to a.) the more notable application or publication of the theory or b.) attribution by mathematicians of a later generation to pay tribute to (or garner attention from) the work of their predecessors.</p> http://mathoverflow.net/questions/22634/nice-solution-to-repeated-integral "Nice" Solution to repeated integral Ed Gorcenski 2010-04-26T18:49:32Z 2010-04-27T18:40:11Z <p>I have a problem wherein I have defined a function $I_r(t) = \int e^{(2r-1)at} \int e^{(2r-3)at} \cdots \int e^{at} dt\cdots dt$, and $I_r(0) = 0$, for $r = 1,2,3,\ldots$.</p> <p>I find that $e^{-ar^2t} I_r(t) = \left(1-e^{-at}\right)^r q(t)$, where $q(exp(-at))$ is a polynomial of $e^{-at}$. Is there a general technique for evaluating repeated integrals of this type that allows me to write $q$ in a nice clean way?</p> <p>If I took $I^*_r(t) = \int e^{at} \int e^{at}\cdots \int e^{at} dt\cdots dt$ with $I^*_r(0) = 0$, and multiplied by $e^{-art}$, I would get $e^{-art}I^*_r(t) = (1-e^{-at})^r$. I am looking for a nice closed-form solution where I have a quadratic in $r$.</p> <p>This is related to the derivation of a discrete probability distribution where the transition rate function is quadratic w/r.t. the number of events per cell.</p> http://mathoverflow.net/questions/40525/principal-eigen-vector-of-a-matrix Comment by Ed Gorcenski Ed Gorcenski 2010-09-29T20:31:38Z 2010-09-29T20:31:38Z You may wish to seek some references on &quot;Matrix Completion&quot; techniques. Depending on the rank of M, I suppose the answer is &quot;it depends.&quot; http://mathoverflow.net/questions/24272/numerical-instability-using-only-heuns-method-on-a-simple-pde/24290#24290 Comment by Ed Gorcenski Ed Gorcenski 2010-05-13T16:22:16Z 2010-05-13T16:22:16Z Another thought... you gain some marginal stability by increasing the order of your solver. This gain is analogous to reducing the step size of a lower order solver (by a very large margin). Does changing \Delta x have any effect on your system? http://mathoverflow.net/questions/24256/fields-of-mathematics-that-were-dormant-for-a-long-time-until-someone-revitalized/24292#24292 Comment by Ed Gorcenski Ed Gorcenski 2010-05-12T02:05:12Z 2010-05-12T02:05:12Z My typo rate for mathematicians' names is 2/2 the last couple days... :( http://mathoverflow.net/questions/24132/what-are-examples-of-mathematical-concepts-named-after-the-wrong-people-stigler/24142#24142 Comment by Ed Gorcenski Ed Gorcenski 2010-05-11T15:01:57Z 2010-05-11T15:01:57Z Oops! Thanks for fixing the typo. Here is the Wikipedia page regarding the relationship between L'Hopital and Bernoulli: <a href="http://en.wikipedia.org/wiki/Johann_Bernoulli#L.27H.C3.B4pital_controversy" rel="nofollow">en.wikipedia.org/wiki/&hellip;</a> http://mathoverflow.net/questions/22634/nice-solution-to-repeated-integral/22757#22757 Comment by Ed Gorcenski Ed Gorcenski 2010-04-27T19:01:24Z 2010-04-27T19:01:24Z Indeed, I had hoped for something of the sort. I am trying to derive a probability distribution for a discrete random process wherein the transition rate function is a nonlinear polynomial in terms of the number of events per cell. For instance, if the transition rate function was f(r,t) = c+br, (without writing out all the equations behind it), we could easily derive the negative binomial distribution, which leads to a clustered grouping of points (the existence of k events in a cell has a linearly positive influence on the induction of another event in that cell). http://mathoverflow.net/questions/22634/nice-solution-to-repeated-integral/22757#22757 Comment by Ed Gorcenski Ed Gorcenski 2010-04-27T18:22:59Z 2010-04-27T18:22:59Z Brilliant, I should have known to use the Laplace transform, foolish me. Thank you! http://mathoverflow.net/questions/22634/nice-solution-to-repeated-integral Comment by Ed Gorcenski Ed Gorcenski 2010-04-27T13:12:07Z 2010-04-27T13:12:07Z Yes, I apologize, it was a poor choice of nomenclature! I have not had much luck in determining a general rule for the coefficients, but they are annoyingly close to matching certain sequences. Thank you for your comments. http://mathoverflow.net/questions/22634/nice-solution-to-repeated-integral Comment by Ed Gorcenski Ed Gorcenski 2010-04-26T18:57:01Z 2010-04-26T18:57:01Z I am seeking a solution where I do not have to evaluate $I_{r-1}(t)$ to find $I_r(t)$. Notice that I'_r(t) is linear in r, but for any r I know what the repeated integral evaluates to. I am seeking the same &quot;happy&quot; solution for I_r(t) which is quadratic in r.