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37 views

Find alternative path closest to original

I have a set of points in a 2D space. I want to connect the outer points so I get the convex hull. The problem here is that there is a limit to the distance between two points. Let me clarify that ...
1
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0answers
82 views

A linear polynomial that is the minimax (best) approximation to $f(x)$ on the interval $[-1,1]$

Suppose you have the following analytical information about a function $f(x)$ on $[-1,1]$: \begin{align*} f(x) &= \frac{1}{x+3}\\ f'(x) & = -\frac{1}{(x+3)^2}\\ f^{''}(x) &= \frac{2}{(x+3)...
1
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0answers
42 views

Formation sketching; trouble with sampling unit optimization

I'm looking for assistance in understanding a concise portion of a paper that involves a quadratic equation, so that I can properly implement it. There are 2 versions of the paper. Each paper ...
1
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0answers
129 views

How to get the eigenvalues of Hamiltonian in an over complete basis

Let $|\psi_i\rangle$, $i=1...N+m$, be a set of overcomplete basis vector in a $N$-dim Hilbert space. The following are known: (Einstein's summation convention assumed) $$\hat{H}|\psi_i\rangle=H_{ji}|\...
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0answers
282 views

Mutual information of Multivariate Gaussian Distribution [closed]

Consider N random variables $\mathbf{x}=[x_1,x_2,...,x_N]^T$ with joint density $p(\mathbf{x})$, and let $\mathbf{R}\in \mathbb{R}^{N\times N}$ denote the covariance matrix. How to derive the mutual ...
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0answers
85 views

Spherical Harmonics: band-limited representations of a vector field on a sphere

I have used pyShtools in the past to expand scalar functions to spherical harmonics and to synthesize band-limited representations of them. However, I am not too sure how to achieve this for a vector ...
1
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0answers
75 views

Direction-splitting for SSP-RK schemes

What are the implications of applying a direction-splitting within each stage of an SSP-RK scheme? For instance, given a standard advective transport type equation: $$ \partial_{t}Q + \operatorname{...
1
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0answers
133 views

Generate gaussian random field in spherical polar coordinates

I would like to generate an isotropic gaussian random field described by a power spectrum $P(k)$ on a 3D grid which represents spherical polar coordinates (i.e., the angular separation between pixels ...
1
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0answers
558 views

Rank sufficiency concept in Finite Element

From section 19.4.1, Rank Sufficiency: The element stiffness matrix must not possess any zero-energy kinematic mode other than rigid body modes. This can be mathematically express as follows....
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0answers
656 views

Seismic Wave modelling: Elastic Wave or Acoustic Wave?

I am modelling a seismic wave equation using FEM. In the few papers that I read, I understand the following: (Kindly correct me if you disagree) A shear (secondary wave - no change of volume) is more ...
1
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0answers
147 views

Updating factorization of Laplacian (add/remove edges)

For a graph $G=(V,E)$, recall that the unweighted Laplacian is $L:=D^\top D$, where $D\in\{-1,0,1\}^{|E|\times|V|}$ is the graph "gradient" operator that subtracts adjacent vertex values onto edges. ...
1
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0answers
171 views

Numerical method to find stationary solution of an PDE in the phase space

I'm relatively new in numerical methods for PDEs, so I have a question about solving the following PDE numerically for $W(x,p)$ in the phase space: $$-p\frac{dW}{dx}+x\frac{dW}{dp}+\frac{d}{dx}\left(...
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0answers
62 views

Book Recommondation for the time-dependent Schrödinger equation [duplicate]

Can you recommend a good book that discusses several methods for the numerical solution of time-dependent and time-independent Schrödinger equation? I have searched the internet several times but ...
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0answers
317 views

Package for Discontinuous Galerkin method

I am trying to find some package of Discontinuous Galerkin (DG) method for solving hyperbolic and parabolic equations. In my research, I focus on designing new schemes for some very simple equations ...
1
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0answers
47 views

Best way to to find fitting parameters for time series of decaying-growing oscillator type

I have discrete time series emerging from the numerical simulations. It means that the time series can be slightly noisy. The time series should "obey" to the following formula: $$ \psi(t) = \sum_{i=1}...
1
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2answers
97 views

What is meant by "operations"?

In this paper, on page 243, we have $$d_{jk} = \frac{a_j}{a_k(x_j - x_k)}$$ where $$a_k = \prod_{l = 0;l\neq k}^{N}(x_k-x_l)$$ Now, $a_k$ requires evaluating N multiplications. Why does the author ...
1
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1answer
84 views

RNG float range for metropolis monte carlo

I have a robust RNG that generates random 32-bit (unsigned) ints. As is probably well known, for metropolis MC simulation, a random number between 0 and 1 is needed to determine acceptance/rejection ...
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0answers
71 views

Gauss Seidel moving mesh AMR hamilton Jacobi

I was trying to implement a moving mesh algorithm using Gauss Seidel, so: I have a pde like this (Hamilton - Jacobi eq) $\begin{cases}\phi_t+H(\phi_x)=0 & [-1,1]\times [0,T]\\\phi(x,0)=\phi_0 &...
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0answers
55 views

Square error estimate for adaptive mesh refinement

In a particular implementation(Finite volume advection using upwind) of adaptive mesh refinement the error square estimate for a cell C is given as $$ \sum_{i = x,y,z} vol * \frac{1}{12} * h^{2} * (\...
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0answers
77 views

Time integration for elastodynamics

I'd like to solve the elastodynamics equation for a problem with fast scales from an impact (although no shocks). $\rho\ddot{\mathbf{u}} - \nabla \cdot \sigma(\mathbf{u}) = \mathbf{f}$ In structural ...
1
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0answers
107 views

RCWA for Non-Periodic Structure?

I have a quick question regarding Rigorous Coupled Wave Analysis (RCWA) for numerically solving problems (in optics). I am trying to simulate a device which is non-periodic in x, y directions (but is ...
1
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0answers
955 views

linearly interpolate and determine gradients for data on non-uniform grid

I have measurements of a quantity on a 3d grid. My measurements are distributed on four x-y planes similar to what is shown in the image below. The measurements roughly follow a Cartesian grid but ...
1
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2answers
567 views

Can I model laminar incompressible fluid flow and heat transfer in MATLAB's PDE toolbox?

I have a system of PDEs in cylindrical coordinates that needs to be solved: 1. Continuity equation 2. Incompressible Navier stokes ( in r & z coordinates) 3. Heat transfer equation with both ...
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0answers
34 views

Difference between Chebyshev first and second degree iterative methods

Consider linear equation $Au = f$. We want to solve it with iterative method (assuming $A$ is good). First order iterative method is: $$ u^{k+1} = u^k - \alpha_{k+1}(Au^k - f), $$ The second degree ...
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0answers
65 views

Computing 3-term Connection Coefficients for Wavelets

I am trying to calculate the three-term connection coefficients $$ Λ_{l,m}^{d_1,d_2,d_3} = ∫_{-∞}^∞ φ^{(d_1)}(x) φ^{(d_2)}_l(x) φ^{(d_3)}_m(x) dx $$ for Daubechies wavelets numerically using Python. ...
1
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0answers
85 views

Mesh partitioner with user-defined overlap

I am looking for a mesh partitioner, where I can specify overlap, for example h = 3. I have looked into metis, but I wasn't able to find such a functionality. Is there any other package which ...
1
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0answers
87 views

Oscillations in Chorin's method due to the BC

I am pretty new to the CFD and I wanted to start with Chorin's projection. The starting problem is just a free jet flowing in the investigated area. I got terrible oscillations almost immediately and ...
1
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1answer
330 views

Matlab: Creating a matrix that represents a colored, sinusoidally-modulated grating

I would like to draw a texture (in Matlab - Psychtoolbox) which is composed of a sinusoidally-modulated isoluminant (magenta & cyan) grating. I do have the RGB matrices of the two phases (for ...
1
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0answers
36 views

explicit scheme stability restriction

Looking at the plain heat equation $u_t=u_{xx}$ the explicit scheme for it would look like the following iteration: $$u_{m,n+1}=\rho u_{m-1,n}+(1-2\rho)u_{m,n}+\rho u_{m+1,n}$$ I noticed this equation ...
1
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0answers
31 views

Size of the support

I am working on moving least square approximation. To well-define of this method, the following matrix of size $n \times m$ must be of rank $m$. For $m=3$ $${\bf A }=\begin{pmatrix} 1&x_1&y_1\\...
1
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0answers
133 views

Finding boundary intersection points with Cartesian grid?

I would like to find the intersection points e.g. $M_x$ and $M_y$ as in the attached figure. The boundary (solid line) is defined by the Lagrangian points. I am working in C++ (basic - medium ...
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0answers
339 views

How to add a Ricker Wavelet (Mexican Hat) to a 2D/ 3D fem mesh?

I have a 2D square mesh and a 3D beam shaped mesh and I want to propagate a seismic wave in them. I am trying to simulate them using Open source FEM codes (fenics). I have left the top surface to be ...
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0answers
237 views

Reaction-diffusion equations

I'm simulating a biological phenomena with reaction diffusion equations. There are multiple diffusing materials and there are some complex relations about consumption and production of such materials. ...
1
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0answers
192 views

Weak form for elastoplastic wave propagation

I am trying to simulate elastoplastic seismic wave propagation using Fenics Solid Mechanics Application. The app. provides some quasi-static demos to show elastoplastic behaviour in a cube/ beam/ ...
1
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0answers
33 views

How to calculate multi-class prediction probabilities using One vs All and One vs One classifiers

To be specific, let's consider Support Vector Machines. In the 1 vs all case, one has $n$ binary classifiers where $n$ is the number of classes. In each classifier one has two labels 0, 1 and for a ...
1
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0answers
37 views

Inner Products vs Discretizations of Functions

Let $f:(0,1)\rightarrow [0,1]$ be a $\mathcal{C}^{\infty}$ function. We say that a function $D_r^f:\{1,...,r\}\rightarrow [0,1]$ is an $r$-discretization of $f$ if $$D_r^f(j) = \frac{1}{|a(j)-b(j)|}\...
1
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0answers
101 views

Choose of basis set [closed]

I am engaged with the syntheses and DFT calculations of coordination complexes. I wonder how can I set up the basis set for carried out using B3LYP method and mixed basis sets of LanL2DZ for Pd and 6-...
1
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0answers
56 views

Stiff ODEs coupled with PDEs (computational efficiency)

I am simulating in COMSOL a system of 3 coupled PDEs (parabolic & elliptic) along with 10 stiff ODEs. In order to have the system working, I am downsizing the time step size too much to achieve ...
1
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0answers
154 views

Quasi Newton taking very small steps

I have implemented a Quasi-Newton method based on the Hessian approximation. I am noticing that the algorithm takes too many iterations to converge, even though it does converge. What I am not able to ...
1
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0answers
36 views

BC's for intermediate velocities in Implicit Fractional Step Methods

Lately, I was reading some seminal papers on Fractional Step Algorithms and I found this one: Kim, D., Choi, H. A Second-Order Time-Accurate Finite Volume Method for Unsteady Incompressible Flow on ...
1
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0answers
135 views

Conservative Short Sales Portfolio Optimization with MatLab code help

I'm trying to solve the optimization problem below with MatLab, but I'm unsure of how to modify the constraints in the quadprog function (or how to add the constraints with the Portfolio object). In ...
1
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0answers
81 views

PDE solver for drift-diffusion with generation

I am trying to solve the drift-diffusion model with generation, $$\frac{\partial N_e}{\partial t} = \alpha(x) \left| \Gamma_e (x,t) \right| - \frac{\partial \Gamma_e}{\partial x} (x,t)$$ $$\frac{\...
1
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0answers
227 views

Implementing pressure inlet boundary condition

I am interested in implementing pressure inlet boundary conditions for the 2D compressible Euler equations. My equation of state is an ideal gas ($p=\rho R T$) which is thermally prefect but not ...
1
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0answers
117 views

Stiffness emerges as number of ODEs increases

I want to solve a system of ordinary differential equations with Matlab. I need this to solve a mechanical engineering related problem. If $n$ is the number of degrees of freedom of my mechanical ...
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0answers
264 views

Numerical solution of non-linear advection equation other than inviscid burgers

I am solving a non-linear advection equation of the form $u_t + f(u)_x = 0$ where $f(u)$ is a complicated function of $u$. I am solving this equation using a first order fully implicit scheme (...
1
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0answers
360 views

Solving system of constrained linear and non-linear equations in MATLAB

Solving system of constrained linear and non-linear equations in MATLAB I'm solving a FEM problem in MATLAB with use of the direct stiffness method. The problem is now formulated as a system of nn ...
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0answers
76 views

Manufacturing a solution for non-smooth coefficients in elliptic problems

This question is a continuation of this answer (I can't comment) If we were going to manufacture a solution for a problem with discontinuous coefficients, I understand that the solution should have ...
1
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1answer
199 views

How to make objective elastic SPH model?

I have implemented a constitutive equation of elastic materials (Hooke's law) in my 3D weakly compressible SPH solver based on [1]. The coding seems to be correct. To verify the implementation I ...
1
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0answers
607 views

Issues with self-consistent Poisson-Schrodinger solver

I'm currently in the process of writing a self-consistent Schrodinger-Poisson solver for a device heterstructure (High Mobility Electron Transistor). The algorithm is based off of this journal(1). I ...
1
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0answers
64 views

How to make the following data separable for the classification into three classes?

The figure below shows the PCA projections of inputs which are 14 meteorological features, (i.e. wind, temperature, humidity, pressure, and so on.) I would like to use any technique to make it more ...

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