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209 views

What are the most important theorems in computational science? [closed]

I was reading the book: The Finite Element Method: Theory, Implementation, and Applications by Mats Larson and Fredrik Bengzon, in page 140 of this book they say this: "The Lax-Milgram Lemma is ...
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2answers
137 views

Why minimizing with respect to A-norm?

Assume solving the linear system $A \textbf x = \textbf b$, with an $A$ so large that nothing but iterative methods may be employed. Assuming $A$ induces a norm, I realized that it is often desired to ...
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4answers
180 views

What scale problem can be simulated on an PC using Smooth Particle Hydrodynamics?

I want to simulate tidal features in a large body of water using was smooth particle hydrodynamics. For a system composed of an incompressible fluid, what general scale of problem can be tackled with ...
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3answers
212 views

How to determine global stiffness matrix is constrained or not

Background In solid fem, we often solve $$\mathbf{Ku}=\mathbf{p}$$ where $\mathbf{K}$ is global stiffness matrix, $\mathbf{u}$ is displacement, $\mathbf{p}$ is global load vector. If displacement not ...
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4answers
588 views

Which Linux OS for computer science and numerical analysis

I am switching from OSX to Linux, which I am familiar with but have no greater experience. There are several different alternatives: archlinux, gnome, ubuntu etc. Which would you recommend for a PhD ...
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2answers
690 views

Time step relationship with number of elements or material properties

When looking at the output file of my solver, I have been told that the time-step taken by the solver depends on parameters like the total number of elements and their relative size in my geometry, or ...
0
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2answers
3k views

FEM Stiffness Matrix is always close to Singular or Badly Scaled

I am making a code for an 18-node (3x3x2) 3D element FEM. However, even though I am (pretty) sure that all the shape functions are correct and whatnot, whenever I try and invert the stiffness matrix ...
0
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3answers
272 views

How to find closed form $C$ such that $CC^T = AA^T + BB^T$

How to find $C$ such that $CC^T = AA^T + BB^T$, $A$ and $B$ are known. $A = \left(\begin{matrix}X\\Y\end{matrix}\right)$, $B = \left(\begin{matrix}0\\cY\end{matrix}\right)$, $c$ is a constant. To ...
0
votes
1answer
101 views

Confusion related to P and NP problems

I have this confusion related to P and NP problems. Why is P a subset of NP? I didn't get it. P problems can be solved in polynomial time. However, NP problems cannot but only verify if a solution is ...
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2answers
132 views

Computing infinite series with iterated functions

I found this question (linked here) which asks to find what this infinite series converges to $$ \sum_{n=1}^{\infty} \int_0^{\pi} f_n(x) dx $$ where $f_{n+1}(x) = \sin(f_n(x)) $ and $f_1 = \sin(x)$. ...
0
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2answers
337 views

Matlab, Mathematica & LAPACK returning 3 different eigenvectors

(I'm not sure which of math.se / stackoverflow / scicomp.se is the right place to ask this question) I have a C++ code which generates a complex matrix and then calculates its eigenvalues and ...
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1answer
104 views

Writing a non-square linear system in standard form $A\cdot{x}=b$

I have spend the last few days working my way through an interesting paper and I'm building a numerical model so I can apply the method. However, I am getting stuck at an "it can be shown" step. I am ...
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2answers
1k views

Help implementing finite difference scheme for heat equation

I am trying to solve the following problem via a finite difference approximation: $u_t = k \, u_{xx}$, on $0 < x < L$ and $t > 0$; $u(0,t) = u(L,t) = 0$; $u(x,0) = f(x)$. I take $u(x,0) = ...
0
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2answers
316 views

MPI+OpenMP Scalability

I have a numerical code which is MPI+OpenMP (hybrid) parallelized and an available computational resource of 32 nodes with 16 cores on each node. The code has been tested for MPI scalability up to 16 ...
0
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2answers
236 views

On Boyd et al.'s convergence analysis of ADMM: Why do we need the convexity assumption?

Please refer to Boyd et al.'s convergence analysis of ADMM (Chapter 3 and Appendix A). My question is: Why do we need $f$ and $g$ to be convex? I don't see the need of this assumption. If the ...
0
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3answers
913 views

Determining efficiency in MFLOPS/s of a parallel program

I am running some scientific (parallel) code and would like to obtain some performance profiling measurements. I want to obtain the "efficiency" of the code in terms of flops/s over theoretical (peak) ...
0
votes
1answer
127 views

What are the benefits of using machine learning for interpolation over traditional interpolation methods?

I am trying to get a better understanding of the application of function approximation with machine learning. My question is simple, how does function approximation with ML compare to traditional ...
0
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2answers
83 views

Stability of the Forward-Time Central Space method, section 9.3 in LeVeque

I am reading section 9.3 in Leveque about stability. The explanations are very short and brief so I am asking for some help to understand and elaborations. The result of the section is that for FTCS ...
0
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3answers
86 views

Changing variables in integral to avoid infinity

I want to write a code in Fortran to solve this integral numerically: $$\int_{1095}^\infty \frac{dx}{x\sqrt{(x+644.153)(4.17 \cdot 10^{-5} x+0.145)}}$$ What is the best method for it? I tried Monte-...
0
votes
1answer
163 views

Why the magnetisation shows abrupt behaviour for this 3D ising spin system

I am trying to simulate a 3D Ising spin system (+1 & -1) using Monte Carlo Metropolis Algorithm. I want to get different physical quantities from this simulation like magnetization, Average Energy,...
0
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2answers
396 views

Non-linear Boundary Value Problem. How to compute the Jacobian?

Consider a Boundary Value Problem: $$ \delta u''+u(u'-1) =0 \Leftrightarrow u''=\frac{-u(u'-1)}{\delta}=:f(t,u',u), \\ u(0)=a, u(1)=b $$ $\delta,a,b$ are known parameters. I want to implement Newton'...
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3answers
76 views

Rudin lecture — if f(x) is not integrable on some interval, does it not have a Fourier Series expansion on that interval?

I found an old lecture on YouTube given by Walter Rudin (1990, in Wisconsin), and towards the beginning he mentions that if $f(x)$ were not integrable, on some interval, it would be obvious that it ...
0
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3answers
1k views

Why is the test function space in FEM chosen with homogeneous boundary conditions?

It is so confusing, especially when I learns discontinuous galerkin method in broken Sobolev space and weak Dirichlet boundary condition. If the trial function is chosen with homogeneous boundary ...
0
votes
2answers
63 views

Finding parameters numerically

I suspect that a function $f(x,y)$ is of the form $f(x,y)=a(bx+c)^{dy+e}$. I have access to several values of $f(x,y)$. How do I proceed numerically to find the parameters $\{a,b,c,d,e\}$? By ...
0
votes
1answer
257 views

How can an engineering student become a computational scinece expert in a short time [closed]

How can a student with zero computing or programming language knowledge, few engineering mathematics knowledge, understand computational science especially Finite Element Modelling (FEM) from ...
0
votes
2answers
463 views

Matlab: ode45 loop

I have to solve this equation $$\ddot{x}x = -\frac{3}{2}\dot{x}^2 + \frac{\dot{x}}{h} + \sin(t)$$ where $h$ is defined by $$h = \left(\frac{\dot{x}}{h}\right)^{1/3} + \frac{\dot{x}}{h}$$ My idea ...
0
votes
1answer
1k views

Numerical integration of sharp peaked function (position of peak known)?

What methods are available to integrate a sharply peaked function (position of peak known) on a finite interval (the interval includes the peak)? Currently I am getting underflows using some of GSL's ...
0
votes
1answer
1k views

Difference between eigendecomposition and singular value decomposition for Hermitian matrices

Let consider the following Hermitian matrix ...
0
votes
2answers
87 views

Simplest solver for linear equation systems

Normally, this boards sees a lot of traffic about the most efficient and most powerful solvers for huge linear equation systems. But this time, I have the opposite problem: I need to implement a ...
0
votes
2answers
81 views

Validating that a code is a good spherical code

Apologies if this is a trivial question. If that is the case I imagine I would benefit from someone explaining where my understanding is lacking. I am having some trouble interpreting the (putatively ...
0
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2answers
116 views

Given a symmetric matrix, is it ok to apply Cholesky decomposition to see if it has negative eigenvalues?

I intend to check the diagonal of L, where A = L'L, for negative elements. However, I don't know if Cholesky is meaningful in theoretical / computational sense if there are some negative eigenvalues.
0
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3answers
120 views

How to store a TB size array in C++ on a cluster

I want to do a huge simulation that requires ~ 1 TB of data to describe a bunch of interacting particles (each has different interactions). Is it possible to store this data in an array in C++, so ...
0
votes
1answer
313 views

Time complexity of numerical finite differences

I have a function $f:\mathbb R^N\to \mathbb R$ and I would like to compute all the partial derivatives of $f$ w.r.t. the $N$ input. What is the computational complexity using the (ones-sided) finite ...
0
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2answers
211 views

Update for QR factorization least squares

I found after some research that the most numerically stable way to solve the least squares problem is through QR factorization. For $n$ number of observations and $p$ number of parameters it takes ...
0
votes
2answers
194 views

What's a nice simple PDE to play with in Matlab?

I want to learn a very simple PDE to gain physical and mathematical intuition—a PDE with just one spatial dimension $x$ and a time dimension $t$. And, I want to write code for it in Matlab and use a ...
0
votes
1answer
392 views

I've developed a derivative-free optimization method, looking for comments

Here is the URL: https://github.com/avaneev/biteopt I've tested it on numerous global optimization benchmarking functions (included), and on real-world hyperparameter optimization problems I have. ...
0
votes
2answers
132 views

How do you apply boundary conditions in a time-stepping problem?

It looks to me like a very common problem, yet I haven't been able to find any practical guide on the subject despite many hours searching. Here is a clearer statement of my question: I have a ...
0
votes
2answers
90 views

Which image filter acts like “surface tension”?

I'm looking for an image filter that does the following operation: In my first image, I have two spheres. After applying the filter, I'd like to have them "glued together" which something that kind ...
0
votes
1answer
479 views

How to create an optimal pizza delivery plan and how to visualize it

This question is quite open, and the actual problem comes from something you would probably consider an everyday niche (something you'd probably take for granted without really thinking about it). ...
0
votes
1answer
308 views

Euler-Bernoulli beam element versus continuum beam element

I am using OpenSees to model a simply supported beam with a point load in the middle. The model is in consistent units. The beam is made up of bilinear quad elements. I have used 30 elements along the ...
0
votes
2answers
122 views

Looking for a particular algorithm for numerical integration

Consider the following differential equation \begin{equation} p(t) = \frac{\partial q(t)}{\partial t} \end{equation} where $t \in (0,\infty)$. I have a build a code that spits out values of the ...
0
votes
2answers
149 views

Relationship between FEM solutions of PDE with different spatial resolutions

I use FEM to simulate deformations of elastic objects for animation applications in computer graphics. The governing equation is generally with the form: $$ \mathbf{M}\ddot{\mathbf{u}} +\mathbf{C}\dot{...
0
votes
2answers
103 views

Explain this multivariate differential identity

$$ \frac{\partial|\nabla\phi|^2}{\partial\phi}=-2\nabla\cdot\nabla\phi$$ I would very appreciate that you help me . Please do it in detail, I am quite not good at such problems. There is something ...
0
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2answers
3k views

Successive over-relaxation not converging (when not done in-place)

I'm trying to find the potential given some boundary conditions using the successive over-relaxation method. I have 2 solutions: -One iterates over all elements and applies the formula ...
0
votes
2answers
133 views

Validating conservation laws from physics

Say I'd like to implement a numerical model of a physical system (e.g. a simple wave equation). Of course the energy conservation law is valid for this system. Is there a way to validate that my ...
0
votes
1answer
191 views

Where do I find engineering problems to practice solving computationally?

I'm an engineer and I'm planning to get a bigger toolbox than Excel to solve difficult problems. I started learning Python (as that seems the script language to go for math intense jobs, and runs in ...
0
votes
3answers
183 views

Equivalence of linear systems, solving one instead of the other

This question is related to recently posted one, but I guess it deserves a separate attention. Suppose a symmetric matrix $L\in\mathbb{R}^{n\times n}$ is given, and a rectangular matrix $A\in\mathbb{...
0
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2answers
107 views

Impact of irregularity at the boundary on the error analysis

I am looking at the pde of the type $u_t=\mathcal{L}u$ for some elliptic operator $\mathcal{L}$ and on some domain $D$. Assume I am solving that with a finite difference method and want to estimate ...
0
votes
2answers
317 views

Implementing routine for $-\nabla\cdot (k(x,y) \nabla u)=f$ in Matlab

I am solving the Poisson Equation for 2D given by the following expression: $$-\nabla\cdot (k(x,y) \nabla u)=f$$ in a rectangle with Dirichlet conditions on the boundary using Matlab. In principle I ...
0
votes
1answer
61 views

How much space store a matrix of numbers?

When we work in numerical analysis, when the matrix is too big the computer runs out of memory (i guess it's ram memory). But how much space does a matrix of numbers (integer, single, double) use? I ...

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