# Questions tagged [convergence]

Questions related to whether the sequence of iterates generated by an iterative method has one or more limit points, and if those limit points have the correct properties.

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Let's consider an optimal-step gradient algorithm and assume that: $g(α) := f(X_k - α∇f(X_k)) = 2α^2-4α+17$, how can we determine the optimal $α_k$? Here is my simple solution: $g(α) = 2α^2-4α+17$ $g'(... • 25 0 votes 1 answer 47 views ### step-fixed algorithm first iterates let us have the fixed-step gradient algorithm, with$p = 2$and we assume that for$X = (x, y)$,$∇ f(X) = \begin{pmatrix} x -1\\ y -2 \end{pmatrix}$Let me assume we intialize with$X_0 = (0,0)$what ... • 25 0 votes 1 answer 66 views ### step-fixed algorithm to minimize f, which step to ensure convergence? If we want to apply the fixed-step gradient algorithm to the minimization of$f(x) = \frac{1}{2}(Ax, x)$where$A$is a symmetric 2x2 matrix with eigenvalues$\lambda_1 > \lambda_2 > 0$, for ... • 25 0 votes 1 answer 40 views ### Estimating the rate of convergence of Projected Gradient Descent on constrained polynomial objectives I am estimating the order of convergence of Projected Gradient Descent (GD) on quadratic polynomials with random coefficients independently drawn from Uniform(-1,1) and bounded by a unit hypercube. I'... • 23 1 vote 1 answer 136 views ### How to refine$h$and$\Delta t$for convergence tests on evolution PDE Setting I am solving for$u(x,y,t)$the wave equation$\partial_{tt} u - \partial_{xx} u = f$on$(x,y)\in\Omega=[0,1]\times \mathbb{R}$by splitting it into an equivalent first order system: $$\... 0 votes 0 answers 117 views ### Is there a fast matrix-free inverse power iteration? Problem: I want to solve the eigenvalue problem$$x=Ax$$to the eigenvalue 1 for a large matrix (roughly N^3\times N^3 and N ranges from 10 to 100) where A is stochastic (i.e. all entries are ... 0 votes 0 answers 58 views ### Calculating a 2D Ewald sum for a multipolar expansion I am attempting to calculate the potential of a particle at the center of an infinite two-dimensional lattice as per the following reference: Reference: Lambin, PH & Senet, P. Ewald Summation of ... • 43 3 votes 1 answer 199 views ### Stability of Euler forward method I am trying to solve a linear system of ODEs of the form:$$ \frac{du}{dt} = A u, \quad u(0)=k$$where A is a 2x2 matrix and u(t) is a 2x1 column vector. I want to solve this numerically, using ... • 31 3 votes 1 answer 151 views ### Role of rotation's pivot point in optimization? In this paper, the authors describe how to use locally rigid transformations (sampled on nodes in space) to deform mesh vertices. In the paper, rotations are relative to the pivot point, which ... • 31 1 vote 2 answers 81 views ### Why is the definition of convergence different for root finding algorithms as compared to sequences? The definition of convergence for root finding algorithms is given in a few sources as: A sequence {x^k} generated by a numerical method is said to converge to the root \alpha with order p\geq 1 ... 2 votes 1 answer 315 views ### Why do we use modified pressure in incompressible multiphase solvers with gravity? The context of my question is two-phase incompressible solvers such as interFoam in OpenFOAM, but I have seen this trick used ... 0 votes 0 answers 67 views ### Faster convergence for minimizing least squares of forward modelling problems This specific question was raised from optimizing parameters of column experiments in the hydrogeological context. I want to optimize a parameter of interest (in this case D), based on experimental ... 3 votes 1 answer 245 views ### Convergence-test for ODE approximates wrong limit I am trying to numerically solve a differential equation but I am having trouble getting the convergence test to run properly. The problem is as follows: Consider an ODE$$y'(t) \enspace = \enspace f(... • 185 6 votes 1 answer 342 views ### Why FEM for incompressible materials is ill-posed? I am an engineer who is trying to get a deeper understanding of FEM. I have been using the Zienkiewicz texts as my bible. It touches on the issue of incompressibility but I need a more intuitive way ... 0 votes 0 answers 78 views ### Convergence rate for not smooth solution with classical$P^1$Lagrangian FEM I'm using classical$P^1$finite elements to solve$- \Delta u = f$with Dirichlet BC in a 2D domain$\Omega$. I know from theory that the solution is not in$H^1$for my particular choice of$f$, so ... • 405 1 vote 0 answers 144 views ### Convergence of Evolutionary Algorithms When it comes to Evolutionary Algorithms (e.g. Genetic Algorithm), I have often heard people make the following broad statement: "Evolutionary Algorithms Do Not Converge." I was curious ... • 139 5 votes 3 answers 185 views ### Why in scientific papers convergence of finite difference and finite volume schemes is tested using multiple norms ($l_1$,$l_2$and$l_{\infty}$)? In peer reviewed numerical papers, the order of accuracy of finite difference and finite volume for PDEs is computed in multiple norms, usually$l_1$,$l_2$and$l_{\infty}$, and other times$l_{\...
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Is there an "Unspoken Tradeoff" in Convergence Rates for "Quasi-Newton Methods vs. Gradient Descent"? As a quick summary: Gradient Descent based algorithms try to find the minimum ...
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### Getting euclidean distance between vector A and C without anyway of retrieving them when their distances with a common vector B is known

Motivation: My plan is to get the overall euclidean distance matrix for all the vectors in N number of dataset. Each dataset is basically an array of n-dimensional points. For e.g: A dataset can be ...
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### Confused on how Method of Manufactured solutions works?

I am new to computational science and I am trying to wrap my head around how MMS works. I am solving the time independent Helmholtz equation as a simple test of the technique so my starting equation ...
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### Manufactured solution for $-\operatorname{div}(a(x) \nabla{u}) = f$ when $\alpha(x)$ is discontinuous

I'm studying the dealii tutorial number 4,5 and I understand the workflow. I've also been able to find the EOC by using manufactured solution where $f$ is a smooth r.h.s. and $\alpha(x)$ smooth too. ...
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### Step3 in deal.II - Convergence of the mean

I'm trying to understand the Convergence of the mean part of the Step-3 tutorial in deal.II. The authors say that $\frac{1}{|\Omega|}\int_{\Omega} u_h(x)dx$ converges with $\mathcal{O}(h^2)$, but I ...
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The Gauss-Kuzmin distribution gives the probability of an integer appearing as a partial denominator in the continued fraction of a real number $x$ as $$P(a_k = k) = -\log_2\left(1 - \frac{1}{(k+1)^2}... • 2,165 1 vote 0 answers 53 views ### Effect of reducing flux consistency order at boundary on convergence order Consider the 1D nonstationary convection-diffusion PDE$$ \begin{alignat}{2} \partial_t u &= -a \partial_x u + D \partial_{xx}u, &\qquad x \in (0,1), t \in (0,T), \\ f(t) &= \left.\left( a ...
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Consider the PDE $$u_t = b_{11}u_{xx} + 2b_{12}u_{xy} + b_{22}u_{yy},$$ where $b_{11}, b_{22} > 0$, and $b_{12}^2 < b_1b_2$. In Strikwerda's book, the ADI scehme \begin{align*} \left( 1 - \frac{...