integration – Deterministic Approximation of Products of Integrals

Suppose that I am interested in approximating the product of two integrals. For simplicity, let $P$ be a probability density function, let $f, g$ be two $P$-integrable functions, define

begin{align}
phi = int P(x) f(x) , dx \
gamma = int P(x) g(x) , dx,
end{align}

and say that I am interested in approximating the product $phi cdot gamma$.

I am interested in how a numerical analyst would approach this problem. A naive solution to this would be to do e.g. Gaussian quadrature with $P$ as the base measure, use the GQ nodes to form approximations of $phi, gamma$, and then multiply the approximations together. Can / does one ever do fancier things than this in practice?

I ask because I come from a statistics / applied probability background, where one often places a premium on having unbiased estimates of integrals. As such, taking the product of two estimates which use the same set of function evaluations will generally induce a statistical bias, and so there are simple and popular techniques which allow for this bias to be removed. I am curious about whether similar approaches are considered in numerical analysis.

To compress the question further: from the point of view of numerical analysis, are there better ways of approximating a product of integrals than taking the product of approximations to each of the integrals?

integration – How can you be sure that an integral does not exist? Not that it diverges, but that you cannot complete it?

Say you have the integral $displaystyleint_1^infty{frac{1}{x^{1+frac{1}{x}}}};mathrm{d}x$

This integral cannot be completed. Not that it goes to infinity, but it physically just cannot be completed. How can you realize this if you encounter it? How can you prove it?

integration – Find the mass of the solid S made out of material with density f

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Integration architecture: data categorization – Software Engineering Stack Exchange

I’m currently in the process of analyzing our company’s interfaces, which are not document nor centrally managed and as a result, very heterogeneous. As this is a huge task and it’s part of a master thesis, I’m concentrating on three conceptual information:

  • Overview of the system (subdomain)
  • Logical interfaces (specific attributes like direction, transformation complexity etc.)
  • Data categorization

The last step gives me a bit of a hard time; as it’s a scientific work, I would need some reasoning, how and why I would categorize data, but I didn’t find anything about that topic in literature. Checking enterprise frameworks like TOGAF and Zachmann, but also known books like “Enterprise Integration patterns”, “Microservices patterns” etc., I didn’t find any good source about that topic.

The only hint I found was in a PluralSight course with this image:
enter image description here

But I didn’t find any more information where this is from nor does the author give details, how to proceed hereby.

Is there a known data categorization strategy around? Or is this just not worth the effort and it makes sense to not categorize the data at all?

Integration with Kontent Machine and DeepL

I have contracted DeepL and Kontent Machine. I want to integrate them into GSA so that it automatically extracts articles and translates them. How could you do it?
I also have contracted GSA Content Generator

numerical integration – NIntegrate blowing up/behaving weirdly at the turning point of the integrand

I’m performing (what should be) a straightforward numerical integration (Fourier transform) of a function with no poles / singularities (at least in a particular parameter regime):

<< FourierSeries`

ClearAll("Global`*")

l = 1;
ϵ = 10^-4;
r = 0.1; (*radial coordinate*)

p = 0;
P = 80;

Data1 = ConstantArray(Null, {P, 2});

While(p < P, p++;
  
  w = -10 + 20 p/P;
  
  σ = (
   2 (-2 l^2 r^2 Sin(s/(2 Sqrt(l^-2 - r^2)))^2 + (1 - l^2 r^2) 2 Sinh(
        s/(2 Sqrt(l^-2 - r^2)) - I ϵ)^2))/l^2;
  
  RF = NInverseFourierTransform(-1/(4 π^2) E^(-I w s)/σ, s,
      w, Method -> {"LocalAdaptive", "SymbolicProcessing" -> 0}, 
     MinRecursion -> 6) // Chop;
  
  Data1((p, 1)) = w;
  Data1((p, 2)) = RF;
  
  );

ListLinePlot(Data1, PlotRange -> All, 
   LabelStyle -> {FontSize -> 16, FontFamily -> "CMU Serif", Black}, 
   PlotLabel -> StringForm("r = ``.", {r // N})) // Print;

When the parameter r is small, the numerics seem to work fine/as expected, producing plots like this (r = 0.1):
enter image description here

(x-axis is the energy, y-axis is the transition rate of a two-level system – this shows a roughly thermal Planck spectrum).

When increasing r beyond some critical point, the numerics seem to blow up, giving a weird answer. For example, r = 0.8 yields:

enter image description here

(see especially the magnitude of the y-axis).

I plotted the integrand for these two values below. The numerics seem to blow up when the function bifurcates from having one turning point to two turning points:

r=0.1 (integrand plotted as a function of s)

enter image description here

and r =0.8

enter image description here

Why does NIntegrate not like this seemingly inconspicuous change in behaviour of the integrand? Any help would be greatly appreciated! (If there’s an analytic solution, even better :P)

integration – Associative Error of Composite Trapezoid Rule over 2 Domains

In general, I have found literature suggesting the error is calculated via

${epsilon} = -frac{(b-a)^3}{12 n^2} f”(xi)$ ; $xi in(a,b)$

however I have seen varying interpretations of

$f”(xi)$ and $xi in(a,b)$

Including

$f”(xi) = {frac{1}{n}sum_{i=0}^{n-1} f”(xi_i)}$

$f”(xi) = maxbig|{f”(xi)}big|_{xi in(a,b)}$

$f”(xi) = int_{a}^{b}f”(xi_i),dxi_i$

and unfortunately it is still not clear to me from derivations what method this actually assumes.

Additionally, I have problem in which I am performing an integral over two domains (where three domains total describe my relationship). I.e:

$g(x) = int_{c}^{d}int_{a}^{b} {j(x, y, z)}, dy ,dz$

Thus I am at a loss for determining how to quantify my error over two domains.

Please help!

numerical integration – How to improve the solution of the following differential equation?

Consider a PDE
$$
tag 1 frac{partial f_{n}}{partial t}- frac{1}{2t}Efrac{partial f_{n}}{partial E} = 0
$$

I choose the initial condition $f_{n}(E, t_{0}) = e^{-E/T_{0}}$, where $T_{0}$ is some constant (plus the boundary condition $f_{n}(E_{max},t) = 0$ for some large $E_{max}$).

Eq. (1) may be transformed into a trivial ODE by performing a change of variables. For the given initial condition, the $f_{n}(E,t)$ is
$$
tag 2 f_{n}(E,t) = e^{-E/(T_{0}sqrt{t0/t})}
$$

I want to obtain this result by solving the PDE. This is my code: first, I discretize the E domain, then turn the PDE into ODE by discretizing the derivative, and then solve the set of ODEs:

imax = 500;
t0 = 0.1;
(*E grid*)
DEvalue = 0.05;
Emin = 0.01;
EStep(k_, DE_) = Emin + DE*k;
(*Discretization of the derivative*)
fnEnDerivative(i_) = 
  If(i != imax, (fn(i + 1)(t) - fn(i)(t))/DEvalue, -fn(imax)(t)/
    DEvalue);
Tt(t_)=0.84/Sqrt(t);
(*Table with equations, initial conditions and functions*)
InitialConditionTable = 
  Join({Tg(t0) == Tt(t0)}, 
   Table(fn(i)(t0) == Exp(-(EStep(i, DEvalue)/Tt(t0))), {i, 0, imax, 
     1}));
functionsTable = Table(fn(i), {i, 0, imax, 1});
EquationsTable = 
  Table(fn(i)'(t) -
     1/(2*t)*EStep(i, DEvalue)*
      fnEnDerivative(i)== 0, {i, 0, 
    imax, 1});
sol = NDSolve({EquationsTable, InitialConditionTable}, 
    functionsTable, {t, t0, tmax}, 
    Method -> {"EquationSimplification" -> "Solve"})((1));

After obtaining the solution, I compare the behavior of the exact solution $(2)$ and the numerical solution. I found a discrepancy due to low precision:

fnv(t_) := 
     Table({EStep(i, DEvalue), (fn(i) /. sol)(t)}, {i, 1, imax, 1})
    BoltzmannDistrDiscrete(t_) := 
     Table({EStep(i, DEvalue), Exp(-EStep(i, DEvalue)/Tt(t))}, {i, 1, 
       imax, 1})
    ListLogPlot({fnv(0.3), BoltzmannDistrDiscrete(0.3)}, 
     PlotRange -> {{0.003, 25}, All}, Joined -> True, Frame -> True, 
     ImageSize -> Large)

enter image description here

Improving Accuracy/Precision Goals does not help. For example, with PrecisionGoal ->30 I get this:
enter image description here

Could you please help me with how to resolve this problem?

integration – Is this function bounded below by a non-integrable function?

I am struggling to discern if the following function is integrable on $(0,2)$ with respect to the lebesgue measure.

$$h(x)=frac{1}{(cosh(x)-1)^{1/3}}$$

How does one go about finding a lebesgue integrable function that is $leqslant h(x)$ on $(0,2)$? I have tried to use the monotone convergence theorem on intervals the functions $h(x)chi_{(epsilon,2)}$ but calculating the integral and it’s limit as $epsilon rightarrow 0$ is difficult.

angularjs – SharePoint application with Angular Integration

We have a fully functional Angular-7 web-application up and running.

Now we have lots of static content which can be seamlessly loaded using a SharePoint application. we are looking at a solution of integrating these two so that we have everything in one place.

Is this a good solution to begin with? what can be possible issues that we many run in to?