Also called numerical integration, quadrature refers to the approximation of an integral made by evaluating the integrand at a finite number of points.

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### Optimizing a quadratic form integral over unit sphere

I have an optimization problem, which is to maximize the following integral over the unit sphere: $$\max_B \int d\Omega \mathbf{f}^{\dagger}(\theta,\phi) (B^{\dagger} + B) \mathbf{f}(\theta,\phi)$$ ...
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### Variational loss of hp-Variational Physics Informed Neural Networks for 2D-Poisson Equation in Tensorflow

I am trying to reproduce the results from the hp-VPINN paper (https://arxiv.org/pdf/2003.05385.pdf) on tensorflow (v1) for Poisson's equation, particularly the two-dimensional Poisson equation. In one ...
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### How is the integral of a projection over an element $T$ computed in practice? (deal.II related)

I'm studying an error estimator for the equation $\nabla\cdot(\beta u) + cu = f$ and it contains the following term $$||f - cU_h - \Pi(f-c U_h) ||_T$$ where : $\Pi$ is the local orthogonal $L^2$ ...
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### Compute 2D numerical double integration with Boost C++ with parameters

I am trying to compute the double Richardson and Wolf integrals for the focusing of a lens with Boost in C++ (using the Gauss Kronrod method). As a starting point, I used the example presented in this ...
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### Biot-savart numerical integration

I have a mesh structure of tetrahedrals and I know the current density on each node. I want to evaluate the magnetic flux density. Therefore, I use Gaussian quadrature. However, I do not know how to ...
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### How to use numerical integration to calculate the surface area of a superellipsoid?

I am working in an application in which I need to calculate the surface area of a superellipsoid. I have read that there is no closed form solution (see here), so I am trying to compute it using ...
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### Numerical solution of high-dimensional integral involving positive-part function

Consider a potentially high-dimensional (say, $N$ up to 20) integral of the form $$\int_0^\infty \rho_1(x_1)\rho_2(x_2) \cdots \rho_N(x_N) \bigg(x_1+x_2+\cdots+x_N -K\bigg)^+ \, dx_1 \cdots dx_N.$$ ...
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### How many quadrature points should I use?

I am trying to compute the following integration $$\int_0^\infty e^{-y}y^{a/2}L_c^b(y)L_e^d(y)dy$$ using the generalized Gauss-Laguerre quadrature routine in the GNU Scientific Library. Here the $L$'...
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### Algorithm for integrating a 6D function in a Morse-Smale 3D cell

Lets say that one has a scalar field defined in 3D space for whose gradient he wants to find the Morse-Smale Complex for later performing an integration of several hexa-dimensional functions over ...
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### What is the best numerical method for a six dimensional spherical integral?

I am trying to do integrals of the type $$\int d^3\vec{p} \int d^3\vec{p}' e^{-p^2} e^{-{p'}^2}f(\vec{p}, \vec{p}')$$ where $\vec{p}$ and $\vec{p}'$ are three dimensional vectors represented using ...
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### Evaluating an integral numerically at many points

Given a real function $f$, how can one efficiently evaluate $\int_0^{a_i}f(x)dx$ for millions of different $a_i$? Applying a standard quadrature method (such as Simpson's rule or Gaussian quadrature) ...
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### For noisy or fine-structured data, are there better quadratures than the midpoint rule?

Only the first two sections of this long question are essential. The others are just for illustration. Background Advanced quadratures such as higher-degree composite Newton–Cotes, Gauß–Legendre, ...
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### Convergence of Gauss quadrature for a discontinuous function

Is there a known error estimate for Gaussian quadrature when applied to a discontinuous function? For simple one-dimensional experiments, the error appears to be bounded by $C h$, where $C$ is some ...
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### Stability of different quadrature rules in 1st-kind Volterra integral equation

I am dealing with a integral equation $$f'(t) = -\int_0^t K(s) f(t-s)\quad t\in [0,t_\max] \tag{1}$$ in which $f(t)$ and $f'(t)$ are known, well-behaved functions of $t$ and $K(t)$ is the unknown. In ...
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### Kernel-based differentiation

Consider a $\mathcal{C}^1$ function $V:\Omega\rightarrow\mathbb{R}$ where $\Omega\subset\mathbb{R}^n$. If a random vector $X$ has a parametric density $p_\theta(\textbf{x})$ that's smooth in its ...
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### Building Gaussian-type quadrature schemes with Zernike polynomials

The abscissas for Gauss quadrature are given by the zeros of the Legendre polynomials. The Legendre polynomials form an orthogonal set over $[-1, 1]$, and it is shown in (for instance) Kress that the ...
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### Modulus (absolute) of a function, its quadrature, and relevance of zeros

Modulus of a (discretized) function, $|f_h(x)|$, where $h$ refers to the mesh spacing, would, in general, have zeros, and those zeros would not necessarily lie exactly at the mesh points. A naive <...
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### Reference for Dunavant Quadrature Implementations

I am using Dunavant quadrature in my software, specifically this file by John Burkardt. Recently, I wanted to convert the code into a constexpr code in C++. But ...
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### 2D numerical integration with infinite limit (C++)

In order to integrate a two dimensional function of the form $$\int_{1}^\infty \int_{-\sqrt{x^2-1}}^{\sqrt{x^2-1}} e^{-x} \rm{d}y \rm{d}x,$$ I have been attempting to use the following code (written ...
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### Line integral along the edge of an isoparametrically mapped triangle

I need to integrate the following function on the line segment from $P_{1} = \begin{bmatrix} -2\\-1 \end{bmatrix}$ to $P_{2} = \begin{bmatrix} 1\\2 \end{bmatrix}$: $$\int_{P_{1}}^{P_{2}} 4x + y \ ds$$...
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### Computation of stiffness matrix with variable coefficient

I am implementing a finite element solver (in 2D) to solve the generic differential equation : $$-\nabla(a(x) \nabla u) = f$$ Brief explanation By integrating and multipling by a test function, the ...
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### Numerical integral of oscillating function with known zeros

I have a function that I need to numerically integrate from $0$ to $+\infty$, given by: $$I = \int_0^{+\infty} \mathrm{d}x\,x\,T^2(x)f(x)$$ where $T^2$ is an interpolated function that goes to $1$ ...
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### Spherical volume integral from pre-calculated points - which algorithm is best?

I need a fast and accurate method to calculate 3d spherical volume integrals. I have pre-calculated data of high precision that just needs a few trivial manipulations before each integration step - ...
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### Vandermonde matrix DG Hestaven

I am trying to understand the nodal and modal basis formulation from the book of Hesthaven (Nodal Discontinuous Galerkin Methods, Hesthaven, Jan S., Warburton, Tim). For $N=2$, I get the Vandermonde ...
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### solving numerically a 2D integral by using simps and quad combined

I have the following function $$f(x,y) =\left(\frac{1}{\exp(x-E_f)+1}-\frac{1}{\exp(x-y-E_f)+1} \right)\frac{1}{\sqrt{4-xy}}$$ I want to integrate it using Python 3. The domain of the $y$ variable ...
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### Methods for integration of oscillatory complex vectors as a function of time

I'm attempting to solve a problem of the form: $$\mathbf{a}^{(n+1)}(t) = \int_{0}^{t}d\tau e^{i\mathbf{H}\tau} \mathbf{D}(\tau)e^{-i\mathbf{H}\tau}\mathbf{a}^{(n)}(\tau)$$ Where $\mathbf{D}(\tau)$ ...
Let $\Delta\subseteq\mathbb R^2$ denote the triangle spanned by $(0,0)$, $(1,0)$ and $(0,1)$ and \mathbb P_r(\Delta):=\left\{p:\Delta\to\mathbb R\mid p(x)=\sum_{|\alpha|\le r}\lambda_\alpha x^\alpha\...