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11
votes
3answers
1k views

How should non-constant coefficients be treated with finite-volume first order upwind scheme?

Starting with the advection equation in conservation form. $$ u_t = (a(x)u)_x $$ where $a(x)$ is a velocity which depend on space, and $u$ is a concentration of a species which is conserved. ...
11
votes
5answers
2k views

Global maximization of expensive objective function

I am interested in globally maximizing a function of many ($\approx 30$) real parameters (a result of a complex simulation). However, the function in question is relatively expensive to evaluate, ...
9
votes
2answers
449 views

How much regularization to add to make SVD stable?

I've been using Intel MKL's SVD (dgesvd through SciPy) and noticed that results are are significantly different when I change precision between ...
9
votes
5answers
963 views

How can I automate the process of optimizing the design of a physical object?

I'm trying to optimize a flow distributor in a tank such that the velocity and temperature distribution across any cross-section is relatively uniform. There are many parameters I can adjust to the ...
6
votes
2answers
4k views

How can I prove numerical diffusion in upwind scheme for transport equation

I was just implementing the upwind scheme for a linear transport equation $u_t + cu_x = 0$ where $c=0.5$ and I saw that the solution was indeed advected but over time it starts to diffuse. Can anyone ...
5
votes
2answers
2k views

Does the finite element method impose any restrictions on the Peclet number for numerical stability?

Background on finite volume method When discretising the flux with a central difference stencil of the the advection-diffusion equation restriction $\frac{ah}{d} < 2$ must be observed for the ...
22
votes
6answers
3k views

Future of OpenCL?

The OpenCL programming paradigm promises to be a royalty free opens standard for heterogenous computing. Should we invest our time in developing software based on OpenCL? Pros/cons?
16
votes
3answers
4k views

Finding which triangles points are in

Suppose I have a 2D mesh consisting of nonoverlapping triangles $\{T_k\}_{k=1}^N$, and a set of points $\{p_i\}_{i=1}^M \subset \cup_{k=1}^N T_K$. What is the best way to determine which triangle each ...
14
votes
2answers
2k views

FeniCS: Visualizing high order elements

I've just started messing around with FEniCS. I am solving Poisson with 3rd order elements and would like to visualize the results. However, when I use plot(u), the visualization is just a linear ...
11
votes
1answer
693 views

Smallest eigenvalue without inverse

Suppose $A\in\mathbb{R}^{n\times n}$ is a symmetric, positive definite matrix. $A$ is big enough that it's expensive to solve $Ax=b$ directly. Is there an iterative algorithm for finding the ...
11
votes
4answers
1k views

Matrix exponential of a skew-Hermitian matrix with fortran 95 and LAPACK

I'm just getting tucked into fortran 95 for some quantum mechanics simulations. Honestly, I've been spoiled by Octave so I've taken matrix exponentiation for granted. Given a (small, $n\leq 36$) skew-...
8
votes
2answers
254 views

calculating eigenvector components of a given vector

I have some vector $V$ which can be decomposed into the eigenspace of the hermitian sparse operator $M$: $V = \sum_i v_i \hat{m}_i$ Is there a way to find the $\hat{m}_i$ (the eigenvector itself) ...
6
votes
2answers
319 views

What guidelines should I use when choosing a scalable linear solver?

There are many different linear solvers, some which work best for diagonally dominant matrices, some for symmetric, some for positive definite ones, some for banded matrices, etc... There are direct ...
18
votes
1answer
2k views

Diagonal update of a symmetric positive definite matrix

$A$ is a $n \times n$ symmetric positive definite (SPD) sparse matrix. $G$ is a sparse diagonal matrix. $n$ is large ($n$ >10000) and the number of nonzeros in the $G$ is usually 100 ~ 1000. $A$ has ...
14
votes
1answer
528 views

Is there a multigrid algorithm that solves Neumann problems and has a convergence rate independent of the number of levels?

Multigrid methods usually solve Dirichlet problems on levels (e.g. point Jacobi or Gauss-Seidel). When using continuous finite element methods, it is much less expensive to assemble small Neumann ...
12
votes
3answers
2k views

What's the current state of the art regarding algorithms for the singular value decomposition?

I'm working on a header-only matrix library to provide some reasonable degree of linear algebra capability in as simple a package as possible, and I'm trying to survey what the current state of the ...
11
votes
3answers
1k views

Numerically stable explicit solution of small linear system

I have an inhomogeneous linear system $$ Ax=b $$ where $A$ is a real $n\times n$ matrix with $n\leq 4$. The nullspace of $A$ is guaranteed to be of zero dimension so the equation has a unique ...
9
votes
3answers
832 views

Iterative methods for indefinite systems without block structure

Indefinite systems of matrices appear for example in the discretization of saddle point problems by mixed finite elements. The system matrix can then be put in the form $$\begin{pmatrix} A & B^t \...
9
votes
2answers
496 views

Which novel data structures are used in adaptive FEM?

A lot of adaptive FEM libraries use more advanced mesh data structures to handle adding/removing nodes, edges, triangles, tetrahedra, etc. For example, the p4est library uses octree data structures ...
9
votes
1answer
2k views

Sensitivity of BFGS to initial Hessian approximations

I'm trying to implement the Broyden-Fletcher-Goldfarb-Shanno method to find the minimum of a function. I need two initial guesses $x_{-1}$ & $x_0$ and an initial Hessian Matrix approximation $B_0$...
7
votes
1answer
484 views

How to check the correctness of my implementation of a numerical scheme for differential equation

I know the method of constructing solutions. For example I have a BVP: $$ u_{xx} + u = 0 $$ subjected to: $$ u_{x}(0) = f_1, $$ $$ u_{x}(1) = f_2 $$ If I want to check the correctness of my ...
7
votes
0answers
485 views

Numerical implementation of the Dirichlet-to-Neumann map

I am solving the Dirichlet problem $$ \begin{cases} \Delta u = 0, \\ u|_{\partial D} = f, \end{cases} $$ in a $2d$ domain $D$ using the finite element method. What I want to get is the ...
7
votes
3answers
367 views

Strong coupling of a non-linear multiphysic problem: failure with Newton Raphson method

I am trying to solve a multiphysic problem using finite elements and a Newton Raphson solution scheme. I have two non-linear subsystems that are coupled bi-directionally. The first subsystem includes ...
6
votes
2answers
400 views

Matrix free finite elements method for visualization in process tomography

I am Computer Scientist and now I am interested in matrix multiplication on GPUs. My research are focused on matrix free finite elements method where I multiply sparse matrix. Sparse matrix could ...
6
votes
1answer
399 views

Fourth order IMEX Runge-Kutta method

I am looking for the Butcher tableau of a fourth order accurate Runge-Kutta method with IMEX splitting. I have been reading the ''classical'' paper on the subject by Ascher, Ruuth and Spiteri as well ...
6
votes
4answers
2k views

Numerical integration of non-uniform acceleration samples

I'm given a stream of acceleration data with timestamps. The sampling is non-uniform. Apart from Euler, is there a way to integrate the acceleration into velocity? Something more accurate or of ...
5
votes
1answer
310 views

Working with multi-dimensional functions

How would you represent functions of type $[-1, 1]^n \to \mathbb R \;$ for moderate $n$? How would you integrate them? For small $n$ (1-2) such functions can be represented as histograms, vectors in ...
4
votes
2answers
7k views

Tikhonov regularization in the non-negative least square - NNLS (python:scipy)

I am working on a project that I need to add a regularization into the NNLS algorithm. Is there a way to add the Tikhonov regularization into the NNLS implementation of scipy [1]? [2] talks about it, ...
3
votes
3answers
2k views

A good, simple book/resource on Parallel Programming in C++ for scientific computing

I am a Mechanical Engineering grad student, currently working on a project which will be scaled up in the new future to require quite some processing power. I am using C++ for the code that I have, ...
3
votes
1answer
294 views

Random placement of euclidean points with constrained inter-point distances in a fixed area

I'd like to place as many random points as possible in a 2D square $S=[0,1]x[0,1]$ such that the euclidean distance $d$ between any two points $d$ is greater than a given value $b$ (b is small). I'm ...
2
votes
1answer
153 views

Finite element method for odd order DE

What are theoretical hurdles in applying Galerikin method on, say, first order time dependent ODE? Is there no way we can form an inner product??
1
vote
1answer
392 views

More Smearing with decreasing timestep in advection problems

I find it kind of counter intuitive, that the result of an advection gets more smeared out at the borders when decreasing the timestep (which should make it more accurate). Let there be a equally ...
0
votes
1answer
280 views

Find cfl condition

We have the advection equation $u_t+a u_x=0, a>0, 0<t<T_f, x \in \mathbb{R}$ with initial condition $u(0,x)=u_0(x)$. Suppose that we have the following sheme: I want to find the CFL ...
-1
votes
1answer
535 views

Pde problem with robin boundary condition

I have my pde 2D problem with robin condition (form: du/dn +ku=g) to solve with matlab. i have the exact function u and I want to find the function g in robin condition. How can i do it? thanks for ...
15
votes
3answers
2k views

Why isn't my Matrix-Vector Multiplication Scaling?

Sorry for the long post but I wanted to include everything that I thought was relevant in the first go. What I want I am implementing a parallel version of Krylov Subspace Methods for Dense Matrices....
14
votes
3answers
649 views

PDEs in Many Dimensions

I know that most methods of finding approximate solutions to PDEs scale poorly with the number of dimensions, and that Monte Carlo is used for situations that call for ~100 dimensions. What are good ...
9
votes
2answers
667 views

How to model a fishing rod (or a rope)?

I wish to model a fishing rod (or a rope) by joining short segments. (The segments may have equal (short) length but each segment should be assigned its own individual mass.) One segment will ...
8
votes
3answers
2k views

How to use polylogarithm function in c++?

Is there any preprocessor directives that could be used to use the polylog function? Or is it included in cmath? If so, do you call it by Li or by polylog? EDIT: What I really am trying to do is give ...
8
votes
1answer
597 views

Newton iteration applied to nonlinear PDE

I'm having difficulty understanding how to apply Newton iteration to nonlinear PDEs and then use a fully implicit scheme to time step. For example, I want to solve Burgers equation $$u_{t} + u u_{x} -...
7
votes
1answer
1k views

Solving for null space of a matrix with mkl LAPACK

I want to find a solution for $xA=0$, where $A$ is a square matrix. I know that most of the LAPACK routines solve for $Ax=b$. So I take $A^T$ as a, and set $b=0$. I have an additional restriction of $\...
6
votes
0answers
416 views

How to do FEM in sector elements?

Suppose we have an element in the polar coordinate as $0\leq r\leq 1$, $0\leq\theta\leq \alpha$. What are the correct basis functions in this element? For this kind of mesh, there are a lot of ...
6
votes
3answers
137 views

Accurate evaluation of the sign of a polynomial

Let $p$ be a polynomial with floating-point coefficients and let $a$ be a floating-point value. Is there a method for accurately evaluating the sign of $p(a)$ in floating-point arithmetic? I don't ...
5
votes
1answer
2k views

Finite Difference Method Neumann Boundary Condition with Variable Coefficients

Disclaimer In the process of typing up this question, I determine its solution. Since I went through the trouble of typing up the question in its entirety, I will post its answer as well. It may ...
5
votes
2answers
464 views

How to parallelize a banded direct solver?

I have a linear system whose matrix that is diagonally dominant, non-symmetric, but banded. Since the band-radius is 2 (producing only 5 variables per equation), a banded direct solver (gaussian ...
5
votes
1answer
1k views

method of frozen coefficients and its relation to von Neumann stability analysis

I am considering two equations $$u_t=a(x)u_{xx}$$ and $$v_t=b(x)v_x$$ as classical representatives of the parabolic and hyperbolic family of equations. If $a(x)=a$ and $b(x)=b$ were constants, to show ...
5
votes
1answer
1k views

PDE - Conservative form, conservative methods and discrete conservation

I cannot find a reference explaining clearly and rigorously the links between the notions of conservative form for a PDE, a conservative numerical method and discrete conservation. I would be very ...
4
votes
1answer
262 views

adjoint method for reaction-diffusion problem

I'm trying to code a parameter estimation for a reaction-diffusion problem. Namely, knowing the distribution of tumor density $u$ at time $0$ and $T_f$ ($u^0$ and $u^f$), what are the best ...
4
votes
1answer
2k views

Is there a mesh generator that will generate zero thickness elements for interfaces?

I have a geometry and a finite element method where there are a couple internal interfaces across which I want to enforce some fairly simple jump conditions. One recommendation I've gotten has been ...
4
votes
3answers
501 views

Quality Measures for Various Pseudo-Random Number Generators

According to this paper, Ideally, a pseudorandom number generator would produce a stream of numbers that: are uniformly distributed, are uncorrelated, never repeats itself, ...
3
votes
1answer
904 views

Implementation of gradient zero boundary conditon in advection-diffusion equation

My question is about Finite Element Method. I want to know how to implement "gradient zero" conditions to advection-diffusion equations in conservative form like, $\frac{\partial \rho}{\partial t} + ...

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