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How to define $P0-$ Piecewise constant basis function in finite element method?

Suppose if we take $X_h(G)$ as finite element space then this space (space of piecewise constant basis function)is defined as $$X_h=\{v: v|_{T}=c_{T}, T \in \mathbb{T}\},$$ where $\mathbb{T}$ is a ...
0
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0answers
36 views

Numerical solution to the Landau-Zener problem

I tried to use a midpoint method and numerically solve the Schrödinger equation for the original Landau-Zener (LZ) problem: a $2\times 2$ Hamiltonian $$\left(\begin{array}{c} \alpha t\\ \delta \end{...
3
votes
1answer
60 views

What's the terminology for this alternative minimization algorithm?

Say the model is $F(x_1)G(x_2)Z(x_3) = y \in \mathbb{R}^N$, with $F,G,Z$ explicitly known, we are given observation of $y$ as $y_b \in \mathbb{R}^N$ to find the value of $x_1$, $x_2$, $x_3$ for each ...
2
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2answers
37 views

How to include penalty in a Objective Function with Python? GEKKO

I'm trying to include a "great M" penalty in my objective function. I want use the entry x vector values as entry values in a function. A fixed maximum value is took initially for the returned value ...
0
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0answers
22 views

Cubature rule in unit Sphere in $\mathbb{R}^{3}$

I need to find the cubature rule for the following integration $$\int_{S^{2}} f(s,\tilde{s})d\tilde{s} ds,$$ where $S^2$ is the unit sphere in $\mathbb{R}^{3}$.
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0answers
17 views

Calculating the Convolution using FFT

I have the following convolution as part of a numerical simulation. $$T(r)=\int d^3r_2 p(r_2)f(r_2)\alpha(r-r_2)$$ My problem is that the analytical expressions for $f$ and $p$ do exist but, I have ...
1
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1answer
31 views

Finding curves where function goes to zero in two dimensions

Suppose $f(x,y)$ is a complex function of two real arguments with roots* that are not discrete points but lie in curves. (Is there are term for this characteristic?) An example is shown below: the ...
1
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0answers
9 views

Error on the fit parameters when several good fits exist

I am using the reduced chi-squared statistic to determine the goodness of fit. I run several simulations and determine that a parameter 'p' has a certain range of values that all give values between 0....
1
vote
1answer
50 views

Best way of storing numerical data in a compact manner, while leaving it accessible for tools like GnuPlot?

My simulation, written in C++, generates a large amount (roughly ~500) of text files for each set of parameters I try to simulate, with four columns of ~5k double values in each file. Furthermore, to ...
4
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0answers
49 views

Sensitivity of BFGS to the accuracy of the gradient

I am studying how to speed-up the BFGS method using quantum computing techniques. I have used a method of speeding up the gradient of the function, but it sacrifices the precision value of the ...
3
votes
2answers
54 views

MINLP with GEKKO - Modeling discrete variables

I'm trying to define a MINLP optimization problem with GEKKO in Python, and I want to use some variables with fixed values. For my first variable, x1, I need to define the following values (as would ...
1
vote
2answers
558 views

2D Ising Model, heat capacity decreases with lattice size

The problem I'm trying to make a metropolis simulation of the 2D Ising model. Basically, it's the following, for each monte-carlo step: Visit each lattice site, Compute energy required to flip ...
0
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1answer
51 views

ISING2D with Mathematica. Searching a correct way to compute the heat capacity (mean values over several iterations)

I'm trying compute the heat capacity $C_v$ out of my simulation for the 2D-Ising model which is given by $C_v = \frac{\langle E^2 \rangle - \langle E \rangle^2}{T^2N^2}$ ($E$: Energy, $T$: ...
2
votes
1answer
32 views

Givens rotation vs 2x2 Householder reflection

The usual story of Givens rotations vs Householder reflections is that Householder reflections are better if you want to map a long vector to $e_1$, while Givens is better if you want to map a 2-...
1
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0answers
16 views

Fast convergence of smoothing of periodic noise

I have essentially periodic data from a simulation (not exactly periodic but is qualitatively fairly periodic), and I'd like to take an average or noise filter of some sort that I can get a well ...
5
votes
1answer
250 views

Iterative linear solver for “ugly” saddle point system

I am a graduate student majoring scientific computing. The numeric model I made caused a very ugly-looking saddle-point linear system. It is not symmetric at all and I will attach the sparsity pattern ...
0
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1answer
62 views

Software to simulate molten salt flow and thermodynamic operations

I was curious if there was any software (preferably in C++, Java, and/or python) that could be used to simulate the following: Heat capacity of a fluid Heat transfer through a liquid and a solid ...
0
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1answer
72 views

How to set an initial guess for the iterative solver in Comsol?

How to set the initial guess for the iterative solver GMRES or FGMRES for linear problems (Helmholtz equation of RF module) in Comsol?
0
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0answers
21 views

Question about the visible and hidden neurons in neural networks methods

My problem is the following : I found the ground state energy (for the Ising model) with neural networks (more specifically RBM). I reproduced the same result but by increasing every time the ratio $=...
1
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2answers
62 views

Fast iterative approximate order-oblivious Orthogonalization algorithm?

I have set of N m-dimensional vectors $\{\phi_i\}$ which gradually loose mutual orthogonality in an algorithm. => I have to re-orthogonalize them every few iterations. But if I do e.g. Gram–Schmidt ...
2
votes
1answer
607 views

Local truncation error of Dufort Frankel Scheme

The scheme is given by $$\frac{v_m^{n+1}-v_m^{n-1}}{2k} + b\frac{v_m^{n+1}+v_m^{n-1}-v_{m-1}^n-v_{m+1}^n}{h^2} = 0$$ where $v_m^n$ is the numerical solution at the $m^\text{th}$ spatial coordinate ...
1
vote
1answer
175 views

(FEM) Nodes reordering for sparse matrix storing techniques

Is it necessary to reorder nodes (using Reverse Cuthill-Mckee algorithm, for example) if I am already using a CSR or CSC storing technique? Because since CSR/CSC stores only non-zero elements I guess ...
3
votes
1answer
68 views

Reference request: Riks method (Nonlinear FEM)

I'm struggling to find a good detailed reference explaining the Arc-length method or, more generally, Riks method and its derivations. I looked for the classical books in nonlinear mechanics (the ones ...
15
votes
7answers
830 views

Robust computation of the mean of two numbers in floating-point?

Let x, y be two floating-point numbers. What's the right way to compute their mean? The naive way ...
1
vote
0answers
60 views

How to compute the following Crank-Nicolson scheme for the viscous Burgers' Equation?

I am attempting to replicate results from this article. I'm not sure why but my results are completely different and wrong. For example, the exact solution with parameters ($x=0.1$, $T=0.01$, $Re=0....
0
votes
1answer
88 views

Elliptic PDE finite volume method with Dirichlet boundary condition

I want to discretize the following equation using a Finite Volume Method $$\nabla \cdot (a(x)\nabla u)=f(x)\\x\in \Omega \subset \mathbb{R}^2 \\u_{|\partial\Omega}=g$$ I'm using Voronoi cells here: ...
0
votes
1answer
134 views

How to make a less diffusive code to solve 2D advection equation?

I would like to solve the following differential equation numerically in 2D, $$\frac{\partial z^-}{\partial t}+(\vec{B}\cdot\vec{\nabla})z^-=0,$$ see Wikipedia if you are curious about what the ...
2
votes
1answer
75 views

Incorporating a potential barrier in a wave-packet simulation (Fourier Transform method)

I'm trying to simulate the scattering of a wave-packet at a potential barrier in Python. I'm using a Fourier Transform method (not sure if its the same as the Split-Step method), where I apply Fourier ...
6
votes
0answers
67 views

Numerically estimating expected value of f(x) when x is normally distributed

I need to estimate $$ \mathbb{E}_x[f_i(x)] = \int_{\mathbb{R}^n} f_i(x) p(x) dx $$ for many functions $f_i(x)$, where $p(x)$ is the density of a normal distribution. The evaluation of all the ...
0
votes
0answers
25 views

Why does accessing a memory cell that I didn't allocate does not crash the system? [on hold]

In C programming language, if I allocate char buffer[2], but I store inside buffer something bigger (for example size 5), using for example buffer[4]=3; the program does not crash. Is it normal ? ...
0
votes
2answers
44 views

Verifying convergence of a stationary solution to a PDE with the Runge-Kutta method

I am numerically solving a nonlinear wave PDE using the Runge-Kutta method, and I know the solution I am looking for is constant in time, but I do not know the solution. What is a good way of ...
4
votes
0answers
40 views

Evaluate Nth root of a rational to a correctly rounded float

Excuse my lack of vocabulary for I have no formal training in this field, which is also why I ask this question - it may be trivial or it may be impossible. I want to evaluate an expression in the ...
3
votes
1answer
51 views

Convex optimization with constraints involving matrix inverse

I have the following convex optimization problem. I would like to ask is there any efficient way to solve it in Python? Can I use CVXOPT package? If so, any detailed instruction? Thanks a lot. $$ \...
1
vote
1answer
63 views

What is a good library in Python for correlated fits in both the $x$ and $y$ data?

I have $x$ and $y$ data, both of which have their own covariance matrices. scipy.optimize.curve_fit will accept a covariance matrix for the $y$ data, called ...
0
votes
2answers
242 views

Boundary conditions for streamlines in enclosed flow

I am trying to solve Lid driven square cavity flow problem of Stokes equation using finite element method. I have boundary conditions for velocity as zeros on every boundary but u=1 on top boundary. ...
-1
votes
0answers
36 views

How one can simulate a system given by differential equation?

I want to simulate a diffusion environment given by the differential equation $$\frac{\partial u(x,y,t)}{\partial t}=D\left(\frac{\partial^2 u(x,y,t)}{\partial x^2}+\frac{\partial^2 u(x,y,t)}{\...
5
votes
2answers
112 views

Is there an efficient way to form this block matrix with numpy or scipy?

Is there an efficient way to form this block matrix with numpy or scipy? $$ \left[ \begin{array}{cccc} \mathbf{B} & \mathbf{0} & \cdots & \mathbf{0}\\ \mathbf{AB} & \mathbf{B} & \...
1
vote
1answer
65 views

Product of rank one updates as a low rank update for quasi newton/BFGS

I'm trying to improve the speed of the following iteration to calculate $s_k$: $$B_k^{-1} = \Bigg( I + \frac{s_{k}s_{k-1}^T}{||s_{k-1}||^2}\Bigg)...\Bigg(I+ \frac{s_1s_0^T}{||s_0||^2}\Bigg) B_0^{-1}\\...
0
votes
1answer
121 views

Finite Element - Flux Calculation

I am solving an advection-diffusion equation using the FEM and am having trouble calculating my fluxes. I start with the equation, $$\frac{\partial n}{\partial t} = \frac{\partial j_{n}}{\partial x}\...
10
votes
1answer
119 views

Are there testing frameworks for numerical software development

I found that a lot of my computational science programming has testing requirements that are not covered by standard test frameworks: Computation time testing To make sure that algorithms don't get ...
7
votes
1answer
527 views

Time discretization of the variational formulation of the Navier-Stokes equation

I've asked this question on mathoverflow too. Let $T>0$ $I:=(0,T]$ $d\in\mathbb N$ $\Lambda\subseteq\mathbb R^d$ be nonempty and open, $$\mathcal V:=\left\{\phi\in C_c^\infty(\Lambda,\mathbb R^d):...
1
vote
0answers
30 views

3D log density plot in ParaView

I have a .csv file of thousands of (x,y,z) point particles that I would like to visualize in ParaView. I am able to plot a 3D scatterplot using the TableToPoints, and by decreasing point opacity I can ...
0
votes
1answer
174 views

Proving solution existence and uniqueness of the Helmholtz equation with Robin boundary conditions with complex coefficients

I am trying to solve the Helmholtz equation with Robin boundary conditions with complex coefficients and the weak formulation $$ \iint_\limits\Omega\nabla p_0(x,y)\nabla\left(\overline{v(x,y)}\right)...
0
votes
0answers
21 views

Need suggestions about technical difficulties - Drying + Pyrolysis of coal particle

In OpenFOAM by default, the FireFOAM is well supported for solid pyrolysis modeling. With that in mind, I managed to built my solver for a modified version of pyrolysis (for dry coal - without ...
3
votes
1answer
48 views

Optimizing for multiple objectives

Optimizing two models here, each model having its own set of parameters and an objective, but both models run on the same data which is difficult to compute, and which is computed based on both models'...
1
vote
2answers
75 views

Simulating the heat equation with insulating material

My plan is to solve the heat equation in the right half portion of the domain, while having the left half completely isolated with constant temperature. To do so, I model the left half with a very low ...
2
votes
3answers
482 views

Finite volume piecewise linear 2D advection develops instability

I'm developing a finite volume solver for the simple twodimensional advection equation with constant velocities $u, v$ and constant mesh spaces $\Delta x$: $$ \frac{\partial \rho}{\partial t} + u \...
0
votes
1answer
555 views

Finite Difference Method Limitations/Stability Criteria

Is it possible to solve an equation with only a single derivative such as: $$\frac{\partial U(x,t)}{\partial t} = A - BU(x,t)$$ with finite difference methods? I ask as I am trying to solve the ...
2
votes
1answer
148 views

Non-linear flux interface condition - variational formulation

Context: I am working on implementing this paper and I am struggling to come up with a variational formulation for the Butler-Volmer interface conditions. To simplify my question I consider the ...
1
vote
0answers
32 views

How to deal with a huge system of ODEs in Boost ODEINT?

I am using the C++ library ODEint, which is part of Boost, to solve an extremely large system of coupled ODEs - in particular 1975 equations with large rational functions in the coefficients. In the ...

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