# Questions tagged [high-dimensional]

A high-dimensionality space is one that can only be spanned by a basis set with a large number of elements. High-dimensional problems often suffer from the *Curse of Dimensionality*, which is exponential growth in the problem size as a function of the number of dimensions.

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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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### Efficient projection of a vector onto matrix kernel

Given an $m \times n$ matrix $A$ and a vector $x\in\mathbb R^n$, with $m<n$, what's an efficient way of computing the projection of $x$ onto the kernel of $A$?
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### Solver for large dense BVP system in python

I have a large system of boundary value problems of the form $$\frac{d^2 y }{dt^2} = C(t) y + b(t),$$ where the variable $y$ is a vector that has anywhere from 50 to around 500 components, $C$ is a ...
108 views

### Methods to approximate obective function gradients from point cloud

Problem statement: Assume that I have an objective function $f(x)$ which takes as input a $D$-dimensional vector $x\in\mathbb{R}^D$, and that $f(x)$ is sufficiently smooth. Assume further that I ...
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### Numerical solution to N-dimensional diffusion on simplex?

Assume I have a system of at least (but generally only) $N+1$ points in an $N$-dimensional space ($N > 3$ is possible). At each of these points $x_i, i=1,...,N+1$ I know an initial potential/...
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1 vote
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### Data structure for efficient high dimensional histogramming

What data structure (or C++ library implementing it) is most suitable for efficient high dimensional histogramming? I have an application where I need to compute something similar to a histogram in a ...
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### Meshing software: connectivity between elements and boundary

I am implementing an algorithm which produces a 4d mesh for a cylinder with a given 3d base. This means, I have a 3d mesh and I want to generate a 4d mesh for the corresponding space-time cylinder. ...
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### Solving Poisson equation while suffering from the curse of dimensionality

I have a heat transfer equation in a cube in $R^{100}$: $[0,1]\times[0,1]\times[0,1]\dots$: $$\nabla^2 \varphi = f,$$ with boundary conditions set in a form that in the number of points $p_i$, ...
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### most efficient way to calculate eigen states of a 2D or 3D potential (Matlab)

I know of several ways to calculated the eigen states of 1D potentials (i.e. DVR, Crank–Nicolson, etc). However I wonder what is the most efficient way to do the same for a N-Dimensional potential? ...
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### Dimensionality reduction of the domain of f(x)

I'm wondering if there is something analogous to a PCA for data sets where there is a dependent variable. (Though I am interested in any method of dimensionality reduction, PCA is just an example.) ...
358 views

### What is the current state of the art in solving higher dimensional parabolic PDEs (multi-electron Schrödinger equation)

What is the current state of the art for solving higher dimensional (3-10) parabolic PDEs in the complex domain with simple poles (of the form $\frac{1}{|\vec{r}_1 - \vec{r}_2|}$) and absorbing ...
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### K-nearest neighbours search in subspaces of a high-dimensional space

I'm looking for a good way to partition a large, fairly high-dimensional dataset in order to perform fast kNN searches not just in the full $N$-dimensional space, but also in lower-dimensional ...
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### Best incremental multidimensional Delaunay tessellation algorithm

I'm looking for a specific type of Delaunay tessellation algorithm. The algorithm should be: incremental so that I can add new sites inside known simplexes (i.e. no searching for the right simplex ...
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### Multivariate Orthogonal Polynomial Generation

I'm trying to apply the stochastic galerkin method to partial differential equation with multiple uniform random coefficients. I'm puzzled as to how to extend the corresponding orthogonal (legendre) ...
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### Is there an Implementation of the Hilbert curve from $[0,1]$ to $[0,1]^n$, where $n$ is large? ($n=10,000$, say)

I would like to map each point in $[0,1]$ to $[0,1]^n$ with a Hilbert curve, where $n=10,000$. That is $$f: [0,1] \to [0,1]^n,$$ is the $n$-dimensional Hilbert curve. I found the library of Cortesi,...
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### N-dimensional Delaunay Tesselation Software Libraries

I have a set of known points/nodes irregularly spaced in N-Dimensional space (N>=2), and I would like a way to generate the Delaunay triangulation of these points, and return the corresponding ...
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