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I am trying to find the solution of linear advection equation of the form: $\frac {\partial c}{\partial t}+u\frac {\partial c}{\partial x}=0$ $c(x,0)=0$ $c(0,t)=\{c_0 \ \text{for}\ t \leq t_1 \text{... 2answers 89 views ### Damped Harmonic Oscillation. Efficient algorithm to find the parameters resulting in threshold oscillation amplitude Let's assume, that we have damped harmonic oscillation of a body in the form of a cone, immersed in a liquid. Equilibrium condition of the body is: $$m\overrightarrow{a} = \overrightarrow{F_\text{... 1answer 145 views ### Solving n coupled equations numerically in Matlab I would like to solve the following equations simultaneously and numerically for all X, Y, Z, W where i = 1:Nw, j = 1:Nl, k = 1:K. W_\text{net1}, W_\text{net2}... 0answers 63 views ### Solving a non-convex optimization problem using fmincon I am trying to solve a non-convex optimization problem using fmincon(). At each iteration, I am iteratively looking for the optimum value and when the termination ... 0answers 53 views ### Is there a simple way of implementing dark energy into a n-body simulation? I'm working on a gravitational n-body simulator and would like to implement dark energy into it but all I can find is papers with relativistic equations which I don't really understand. Is there a ... 1answer 40 views ### Animation using matplotlib I am trying to animate a plot of two distinct points (blue and green points) moving about the complex unit circle using Python's Matplotlib library. The problem I am having is that the animation does ... 1answer 33 views ### PCJACOBI works but the default PCBJACOBI failed in PETSc I am using PETSc and libmesh to solve a simple linear elastic problem with quite complicated geometry, using linear tetrahedral elements. I am always using the KSP CG as the solver. I noticed that ... 1answer 88 views ### Why would BFGS converge to a local minima of a non-convex function but maintain a large gradient? I'm using BFGS to optimize a smooth but non-convex function f that is computed by a simulation. The simulation also gives me a semi-analytical gradient g, which is verified by the numerical ... 2answers 81 views ### 3d vs 2d finite element method Is the theory of 3d finite element method just an assembly of 2d finite element analysis by putting planes on top of each other, or, a much more comple and different theory applies for 3d, with ... 1answer 46 views ### Definition of Lagrange nodes in Gmsh When gmsh uses higher-order tetrahedral elements, there is an underlying Lagrange basis used to specify the map from reference space to the element. I'm trying to load a gmsh mesh of 3rd degree ... 1answer 63 views ### Rank of Hadamard Product with Masked Matrix I have a matrix A\in\{0,1\}^{d\times n} and rank(A)=d,d<n, and another matrix X\in \mathbb{R}^{d\times n}, but I do not know the rank of X. What can we say about the rank of their Hadamard ... 1answer 66 views ### Missing something fundamental about condition number estimation In Higham's Accuracy and Stability of Numerical Algorithms, Chapter 15, algorithm 15.3 and 15.4: The topic is ostensibly condition number estimation, but these algorithms show how to compute \gamma ... 0answers 69 views ### How can one prove the duality of Voronoi and Delaunay? Hoping I'm not misunderstanding the concept here, but it is my understanding that Voronoi Diagrams and Delaunay Tesselations are 'dual' to one another, owing to the fact that each' solution makes ... 1answer 111 views ### Numerically solving a non-linear PDE I have this non-linear partial differential equation.$$ \frac{\partial C}{\partial t}=\left(\frac{\partial C}{\partial x}\right)^2+C\frac{\partial^2 C}{\partial x^2} $$I want to use the finite ... 0answers 75 views ### Double mach reflection at a inclined wedge I am running into a strange problem when solving the 2D compressible Euler equations on a inclined wedge. To elaborate, my top boundary condition seems to emitting some type of instability. I have ... 0answers 50 views ### Augmented Lagrangian Techniques Frank-Wolfe Algorithm [python] I'm trying to solve the convex quadratic problem (quadratic min cost flow problem) using the Frank-Wolfe algorithm.$$ \min\{x^TQx+qx: Ex=b,\quad 0\leq x \leq u\} $$The standard algorithm is okay ... 0answers 41 views ### Singular Spectrum Analysis Explanation I need you to help me understand the Singular Spectrum Analysis algorithm. I already read a lot of articles about the subject but they never answered my questions like what is the mathematical reason ... 1answer 75 views ### How to compute the determinant of Hessian of a multivariable function? I have a function F(\vec x) of many variables (let's say in the order of hundreds of thousands). I need to compute the determinant of the Hessian matrix at the point x_0. Is there a way to ... 0answers 54 views ### Bound for Expectation of Singular Value In my case, X_{\boldsymbol{\delta}}\in\mathbb{R}^{d\times M} is a function of Rademacher variables \boldsymbol{\delta}\in\{1,-1\}^M with \delta_i independent uniform random variables taking ... 2answers 67 views ### Convexity of Sum of k-smallest Eigenvalue If I have a real positive definite matrix A\in\mathbb{R}^{n\times n}, and denote its eigenvalues as \lambda_1\leq \lambda_2 \leq ... \leq \lambda_n . Define the function as f(A)=\sum_{i=1}^{k} \... 1answer 99 views ### Split-step Fourier method applied on Schrodinger equation I'm trying to solve a Schrodinger equation of the form i\frac{\partial}{\partial t}\psi=-\frac{\partial^2}{\partial x^2}\psi + (V(x)+\alpha|\psi|^2)\psi using the split-step Fourier method ... 1answer 59 views ### Numerical Stability of a Generalized Spatial Discretization Scheme After reading the matrix stability chapter (10) of Hirsch [1], I decided to dive in the reference list of the chapter. One of the papers [2], which is cited as reference shows an very interesting ... 1answer 46 views ### An Upper Bound of \left<H,X\right>_F with Constrainted Rank We have X=[\mathbf{x}_1,\mathbf{x}_2...,\mathbf{x}_n]\in\mathbb{R}^{d\times n}, H=[\mathbf{h}_1,\mathbf{h}_2...,\mathbf{h}_n] \in\mathbb{R}^{d\times n}, and d<n. H has rank r\leq d and X... 0answers 41 views ### Computationally obtaining the convergence rate of upwind scheme for Advection equation The Advection equation (with velocity = 1) is$${\partial u \over \partial x} + {\partial u \over \partial t} = 0$$I am trying to solve the equation with periodic BC. One of the ways to numerically ... 0answers 49 views ### Calculating depth mask from different lighting I have a object which is static, the camera is static and light source is moving. How can the depth mask be calculated ? Concept is to use - calculate height from shadow length Lets imagine a have ... 0answers 59 views ### Quadrupole moment for a right triangle The authors define a quadrupole moment for a right triangle in Lazić, Predrag, Hrvoje Štefančić, and Hrvoje Abraham. “The Robin Hood Method – A Novel Numerical Method for Electrostatic Problems ... 1answer 86 views ### Dormand–Prince 5(4): How to update the stepsize and make accept/reject decision? https://en.wikipedia.org/wiki/Dormand–Prince_method I want to implement the Dormand-Prince 4(5) version to solve Initial Value problems. Using regular notation I have A matrix and the c,b,\hat{b} ... 1answer 116 views ### Why do we use hermite interpolation for finite element method in beams? Why not just Lagrange polynomials basis functions 2answers 40 views ### denoting a variable as a matrix using octave syms package I'd like to use the syms package to do some algebra for me, but the baseline assumption seems to be that variables are scalars. I would like to denote some variables as matrices. This will change the ... 0answers 68 views ### Writing python program for Weierstrass function with Monte Carlo Our assignment asks us to create a python program to plot the Weierstrass function: Weierstrass function is an example of a pathological real-valued function on the real line. The function has the ... 1answer 40 views ### Monte Carlo - Random Walk Simulation - polyfit the log log data points? This is part of the code in matlab for a random-walk simulation. To test the code, I'm using steps=[30]; there will be more values, but I want to run it for 1 trial to decrease code processing. <... 2answers 109 views ### Choose a subset of m columns that maximize |A^T A|? I have a set of n-dimensional vectors, and would like to choose m of them to become the columns of an n\times m matrix. I would like to choose the subset that maximizes |A^T A|, where A^T is ... 0answers 30 views ### Extracting raw data from a graph I am supposed to do black-box modeling, I need the input and output data for training, however, the authors didn't agree to share the raw data with me, I can only use their publication where these ... 1answer 54 views ### How to find a pair of divisors as close as possible to each other? For a given integer n\in\mathbb{N}^*, I want to find a pair (x,y)\in{\mathbb{N}^*}^2 such that x*y=n and |y-x| is as small as possible. A naive algorithm I found is : ... 1answer 174 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 ... 1answer 59 views ### Fast Poisson solver (with Dirichlet BC zero) on a *truncated* Cartesian 3D grid I find myself in the position of having to solve -\Delta u = f on a subset of Cartesian grid points that don't necessarily form a cuboid domain subject to a homogenious Dirichlet boundary condition ... 1answer 335 views ### Is there a database/website with Butcher tableaus? I have started investigating in mostly Runge Kutta and Runge Kutta Nyström methods and there one of the only differences between the methods of the same type is their Butcher tableu. For the most ... 1answer 55 views ### Poisson image blending artifacts I am trying to implement Poisson image blending as in the paper Poisson Image Editing. This is the task of filling in a masked region of an image by minimizing$$\min_f\int_\Omega \left | \nabla f - \... 0answers 19 views ### work/memory ratio for product of two square matrices From Scientific Parallel Computing by Scott Ridgway: Definition: The work/memory ratio of an algorithm is the ratio$\rho_{wm}$of the number of floating point operations to the number of memory ... 1answer 55 views ### Problem of multiplication of big (sparse) matrix with numpy (python) I wanted to multiply two simple (big and sparse) matrix with numpy. And I saw that the calculation fails when matrices are too big. If i take$X$a random vector (size$n$). With pandas, I ... 1answer 108 views ### Time integration of wave equation My question is: how come that certain formulations of the wave equation can be time integrated more efficiently then others? Le me expand a bit on that. Consider the wave equation: $$\frac{d^2 p(t,... 0answers 29 views ### ILUTP in sparse.linalg.spilu? In Matlab, an ILU with threshold and pivoting (ILUTP) can be passed by default as: setup.type = 'ilutp'; [L, U] = ilu(A, setup); Looking for an equivalent in ... 0answers 45 views ### Cover a polygon with least amount of parallelograms [closed] I am solving the task that is as follows: Input: a polygon. Can be any kind of polygon without self intersections. Can be a non-convex and with holes inside. Goal: to cover it with 2 (at least) or ... 1answer 44 views ### Gnuplot: How can I determine the maxima of a fit function in gnuplot? I have a set of data data.txt which can be fit to a Gaussian function, f(x). I want to determine the coordinates of the point of ... 1answer 40 views ### Gnuplot: How can I fit a range of points (out of the entire data) to a function? I have a set of data obtained for the I-V characteristics of an LED. ... 2answers 70 views ### Stability of Crank-Nicolson for u_t = iu_{xx}+2iu I want to use the Crank-Nicolson scheme to solve the equation$$u_t = iu_{xx}+2iu$$Here's the analysis: Suppose we make a grid, with k = dt and h = dx, the usual notation, and also u_j^n = u(... 0answers 44 views ### How to integrate the contents of a vector using an adaptive quadrature routine [duplicate] I have a function which requires the return type to be a container. The problem is that I need to integrate the contents of the container as efficiently as possible and was hoping to use adaptive ... 2answers 119 views ### Positive root of x^q + bx - b Is there either a closed-form expression or fast/elegant algorithm for computing the positive root of the polynomial$$f(x)=x^q + \beta x - \beta,$$where$\beta>0$and$q\geq2$? How about the$q\...
Let us assume that we have a function, $f(A)=\text{vec}(A^{-1})^\intercal B$, dependent on $A^{-1}$. However, due to some machine-precision limitations, the programming language I'm using cannot ...