10

The integrand is periodic modulo $\pi$ in each variable, so it suffices to integrate each variable over $[0, \pi]$ and replace the constant factor by $2^{-7}$. If we were to apply a change of variables (e.g. set $x = \cos(s_1)$ and similarly for the other five variables), we would have an integral of a piecewise-algebraic function which thus belongs to ...


6

Nemo's representation of $F(\eta)$ in terms of a hypergeometric function can be evaluated without difficulty for large $\eta$: $$F(\eta)=\frac{\sqrt{3} {\eta}^2 \Gamma \left(\frac{2}{3}\right) \; _1F_2\left(\frac{2}{3};\frac{4}{3},\frac{5}{3};-\frac{{\eta}^3}{27}\right)}{6\pi }-\frac{12 {\eta} \; _1F_2\left(\frac{1}{3};\frac{2}{3},\frac{4}{3};-\frac{{\eta}^3}...


5

Some buzzwords that should lead to some non-textbook examples: In the fields of uncertainty quantification, statistical inverse problems or Bayesian inference one wants, for example, compute conditional expectations for posterior distributions. The domain of integration has as many dimensions as the the quantity of interest has degrees of freedom, and this ...


4

For small $n$ Monte Carlo integration is not needed. For $n$ up to 100 see Kolmogorov-Smirnov Tests when Parameters are Estimated with Applications to Tests of Exponentiality and Tests on Spacings (table 3). These exact results were used to construct the Graphs for Use with the Lilliefors Test for Normal and Exponential Distributions. There are also ...


4

For every square matrix $C$, let $r(C)$ denote its spectral value. We say that a complex number $\lambda$ is a dominant eigenvalue of $C$ if $\lambda$ is the only eigenvalue of $C$ with modulus $r(C)$. a semisimple eigenvalue of $C$ if it is an eigenvalue of $C$ and its algebraic multiplicity coincides with its geometric multiplicity. Theorem. The ...


3

The answers are yes and yes. Consider the Hermite Gauss functions : $\psi_n(x)=e^{-x^2/2}H_n(x) $, we have two properties: $$\psi_n''(x)+(2n+1-x^2)\psi_n(x)=0 $$ and $$\psi_{n+1}=\psi_n'(x) +x\psi_n(x) $$ For $n$ large, we take a lenght $L>0$ which is very small (for example $L=1/\log{n}$). On any $[x_0-L/2,x_0+L/2]\subset ]- \sqrt{2n},\sqrt{2n}[$, $\...


3

Do you know the Riemann zeta function ? $\displaystyle\frac{\log \zeta(\sigma+2i\pi \xi)}{\sigma+2i\pi \xi}$ is the Fourier transform of the prime counting function $\displaystyle J(e^u) e^{-\sigma u} = e^{-\sigma u}\sum_{p^k \le e^u} \frac{1}{k}=e^{-\sigma u}\sum_{k\ge 1} \frac{\pi(e^{u/k})}{k}$


3

It looks to me like you are searching for what are called monomial cubature rules on the simplex in the literature on numerical integration. As an example, a 4th-degree monomial in 2-D $(x,y)$ would be any of: $x^4, x^3y, x^2y^2, xy^3, y^4$. A 4th-degree monomial cubature rule will exactly integrate any of those, or any linear combination of them. There are ...


2

Here are a few papers that discuss high-dimensional Monte Carlo integrals, together with quotes from Math Reviews. MR2719643 (2011i:65038) Griebel, Michael; Holtz, Markus; Dimension-wise integration of high-dimensional functions with applications to finance, J. Complexity 26 (2010), no. 5, 455–489. "In addition to error bounds, the authors also ...


2

I must premise that I am not a specialist in numerical analysis, therefore I may be not right when talking about more popular methods in this field pertaining the solution of ITEs. Said that, however, I think I can be of some help. Could you recommend me any articles or book with a brief overview of some methods (maybe classical one), please? Perhaps a ...


2

The key here is the simple change-of-interval/rescaling formula, found e.g. at the link in the OP, according to which \begin{equation} T_n(f)(x)=T_{n,[0,x]}(f)=x\sum_1^n w_i f(xx_i), \tag{*} \end{equation} where the $w_i$'s and $x_i$'s are such that $T_n(f)(1)=T_{n,[0,1]}(f)=\sum_1^n w_i f(x_i)$ for all $f$. From here, one immediately has the affirmative ...


1

You want the relation between $\phi(x)$ evaluated at a point $x$ on the boundary and the derivative $\partial\phi/\partial n$ evaulated at the same point $x$, which is given by $$\phi(x)=\frac{\partial\phi(x)}{\partial n}\int_{-\Delta s/2}^{\Delta s/2}\ln |s|\,ds=[\ln(\Delta s/2)-1]\Delta s\frac{\partial\phi(x)}{\partial n}.$$ Hence $G_{ii}=[\ln(\Delta s/2)-...


1

Following Michael Renardy's suggestion, write $g=\phi\psi$, and write $$ g(x)=g(0)+x\cdot G(x) , $$ where $G$ is a smooth function. Then the integral becomes $$ \int_\Omega fg = g(0)\int_\Omega f + \sum_i\int_\Omega \frac{x_i}{|x|}G_i(x)d^3x . $$ The integrands in the latter integrals are bounded, and the first integral can be written as $$ \int_\Omega f = \...


1

It is not hard to show, assuming that $b>0$, that the smooth solutions of the Euler-Lagrange equation are given by the formula $$ y(x) = \frac{a}{b} - c_1^2 - \left(\frac{x-c_0}{2c_1}\right)^2 < \frac{a}{b} $$ where $c_0$ and $c_1\not=0$ are constants. For example, see my answer to the MO question Riemannian surfaces with an explicit distance function?...


1

Under the assumption that your solution is smooth enough, you can use the calculus of variations to turn your minimization problem into a differential equation. The differential equation is then given by the Euler-Lagrange equation, and can be solved by standard methods for numerical simulation of differential equations.


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