Questions tagged [stability]
The study of the propagation of errors in a numerical algorithm.
154
questions
0
votes
0answers
18 views
Online Parameter Estimation
I have a system $\ y = θ^*u+d(t) $ where $\ y $ is the output , $\ u $ is the input which satisfies $\ u\epsilon L_{\infty} $, $\ θ^* $ is an unknown but constant parameter and $\ d(t) $ is an unknown ...
3
votes
1answer
173 views
log-sum-exp trick for signed/complex numbers
I need to evaluate a sum of values that are on many different orders of magnitude in scale but might be signed. I’ve had great luck with the “log-sum-exp” trick for an unsigned version of my problem, ...
2
votes
1answer
93 views
Numerical stability in the product of many matrices
I have to calculate in numpy the matrix-product of many matrices (~400). Are there common practices to increase numerical stability?
If this is relevant, the matrices are $300\times 300$ orthogonal ...
2
votes
1answer
113 views
Modified Equation and Stability for Centred Finite Differences for Wave Equation
I am trying to use the modified equation to derive the stability condition for the finite difference approximation
$$
\frac{u(x,t+\Delta t) - 2 u(x, t) + u(x, t -\Delta t)}{\Delta t^2} = c^2 \frac{...
2
votes
1answer
95 views
Step size and stability of Euler forward method
I'm trying to calculate the maximum step size that provides stability for the following nonlinear IVP using the Euler forward method:
$u'(t) = -200tu(t)^2,\qquad u_0 = 1, \qquad t\in [0,3]$,
with ...
2
votes
3answers
136 views
What is the flaw in my stability analysis?
The ODE $${d^2x\over dt^2}=-kx; k>0$$can be converted in the system of linear equations as
$$\begin{align}
{dx\over dt} & =v\\
{dv\over dt} &= -kx\\
\end{align}$$
Using Euler’s method, ...
2
votes
1answer
46 views
Stability of a finite-difference scheme for the reaction-diffusion equation
I currently need to solve numerically the following reaction-diffusion equation:
$$\partial_tu=\partial^2_xu+u-u^2$$
For this purpose, I use the following numerical scheme (Crank-Nicolson??):
$$ \...
3
votes
1answer
55 views
Bounding error of float32 matrix multiplication
Some numerical debugging led me to the minimal example below. I'm observing relative error of 0.75 on individual elements.
Is there a way to estimate/bound this error without resorting to higher ...
1
vote
0answers
26 views
Finite difference/element method : time step and spatial resolution close to a finite singularity
I'm using the finite element method (FEM), but my question is quite a global question. It's related to this question but it is not the same.
Let's assume we have this equation : $$\partial_t c - u\...
5
votes
1answer
234 views
Stability of hyperbolic PDE and DG-FEM
In the book of Hesthaven and Warburton on discontinuous Galerlkin methods in example 2.3 (regarding solutions of the wave equation), the authors regard the following PDE:
$$\frac{\partial u }{\...
0
votes
0answers
34 views
Testing the SUPG method and other methods for hyperbolic equations
I am interesting in integrating the simple equation
$$
\frac{\partial \phi}{\partial t} + \mathbf{u}\cdot\nabla \phi = 0
$$
with a Dirichlet boundary condition at the influx boundary ($\mathbf{u} \...
1
vote
0answers
59 views
Numerically Approximating the Jacobian and Comparing the Eigenvalues With Analytical Form
I am trying to study the stability of numerical discretization schemes using the Jacobian matrix of the residues with respect to the vector of conserved variables.
For a simple diffusion equation ...
1
vote
0answers
45 views
Stability of SVD, Eigendecompositions, and pseudoinverse procedures in modern LAPACK routines
I have proposed an optimisation algorithm which I claim has improved upon the previous algorithm in a number of ways.
One of these claims is that my proposed solution requires no explicit SVD and ...
0
votes
0answers
60 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
...
0
votes
1answer
70 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 ...
2
votes
2answers
78 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(...
0
votes
0answers
61 views
The final Boundary Condition is Unknown, Is Backward Euler is still valid to be implemented?
I am working on conductive polymer modeling and supposed to do one-dimensional diffusion model in the thickness of the polymer, however, due to the small value of thickness in micro, when I use the ...
3
votes
1answer
56 views
Stability region of explicit midpoint method
Consider the explicit midpoint method, i.e
$$y_{n+1}-y_{n-1} = 2hf(y_n).$$
I'm asked to apply this method to the linear test equation, $f(y_n) = \lambda y_n,$ in order to find the method's stability ...
2
votes
1answer
117 views
Why the numerical solution of advection-dominant problem is challenging
In many CFD text books, usually there is a dedicated chapter for advection term discretization.
Why discretization of such term in advection-dominated problems and near the discontinuities is ...
0
votes
1answer
75 views
Stability of PDEs
I am currently trying to solve some PDEs with FiPy. At page 56, the manual mentions (https://www.ctcms.nist.gov/fipy/download/fipy-3.0.pdf).
The largest stable timestep that can be taken for this ...
2
votes
1answer
80 views
Assessing numerical error in solving a least squares problem
I have a linear system of the type
$$Ax = b$$
I want to minimise $|b - Ax|^2$.
I know there are different approaches to directly solve the system (Normal equation + Cholesky, QR decomposition, SVD ...
3
votes
2answers
252 views
Generate high n quantum harmonic oscillator states numerically
How can I generate the higher $n$ quantum harmonic oscillator wavefunction (in position space) numerically? Here, higher means around $n=500$, or say $n=2000$, where $n$ is the $n$th oscillator ...
5
votes
0answers
211 views
Any way to avoid catastrophic cancellation when computing the discriminant of a quadratic function?
Homework disclaimer...
The task:
We are using the following algorithm to solve the quadratic equation $x^2+2px+q=0$:
$x_1=|p|+\sqrt{p^2-q}\mathtt{;}$
$\mathtt{if}\,p>0\,\mathtt{then}\,...
3
votes
2answers
82 views
1+x not backwards stable?
If you compute 1+x for x less than the machine precision, the answer will be 1 which is the ...
5
votes
2answers
566 views
Fast and Numerically Stable Pairwise Distance Algorithms
I'm looking for resources on fast, numerically stable pairwise euclidean distance algorithms. In particular, suppose $A \in \mathbb{R}^{M \times D}$ and $B \in \mathbb{R}^{N \times D}$ are two sets of ...
2
votes
1answer
87 views
Stability Analysis
The partial differential equation,
\begin{align}
\dfrac{\partial f}{\partial x} + a(x)\dfrac{\partial f}{\partial y} = 0
\qquad & f(0,y) = f(L_1,y) = c_0e^{-y} \\
& f(x,0) = c_0 \;,\; f(x,L_2) ...
4
votes
0answers
67 views
Automatic differentiation of barycentric rational functions
By a barycentric rational interpolant we understand a function of the form
\begin{align*}
r(t) := \frac{\sum_{i=0}^{n-1} \frac{w_i y_i}{t-t_i} }{ \sum_{i=0}^{n-1} \frac{w_i}{t-t_i}}
\end{align*}
In ...
4
votes
2answers
218 views
Why is my simulation of a first-order wave equation not stable?
According to the equation
$$ \frac{\partial y}{\partial t} = -a\frac{\partial y}{\partial x} $$
I simulated this in python. I used center differentiation, and I determined step size based on Von-...
2
votes
1answer
401 views
Von Neumann's stability analysis on non linear and coupled equations
I'd like to know if is possible to make a Von Neumann's stability analysis on an system of coupled equations, featuring quadratics:
$$\begin{aligned}
\frac{\partial u_1}{\partial t}&=D_1\Delta ...
3
votes
2answers
639 views
Numerical Sensitivity in Density of States of Tight-binding model
I'm working with the tight-binding model, and I'm trying to learn the basics of how to compute the Density of States (DOS) $N(E)$ numerically.
The DOS is given by
$$N(E) = \frac{1}{N}\sum_k \delta(...
1
vote
1answer
19 views
Implementation of a ratio with a well-defined limit
I'm implementing a function $f(x_1, x_2, \dots x_n)$ given by the ratio of two closed form equations.
$$ f(x_1, x_2, \dots, x_n) = \frac{g(x_1, x_2, \dots, x_n)}{h(x_1, x_2, \dots, x_n)} $$
...
1
vote
0answers
37 views
Stability of different quadrature rules in 1st-kind Volterra integral equation
I am dealing with a integral equation
$$ f'(t) = -\int_0^t K(s) f(t-s)\quad t\in [0,t_\max] \tag{1}$$
in which $f(t)$ and $f'(t)$ are known, well-behaved functions of $t$ and $K(t)$ is the unknown. In ...
3
votes
0answers
129 views
Crank-Nicolson integrator: oscillations with complex matrix
I'm working on a Real-Time TDDFT implementation and I am currently comparing different propagation schemes for the propagation of the Kohn-Sham wave function,
$$
\phi(t+\Delta t) = \hat{\mathcal{U}}\...
8
votes
2answers
309 views
Lagrange multipliers space is too rich in a mathematical view
Background:
Lagrange multiplier method has been employed in numerous fields, such as contact problems, material interfaces, phase transformation, stiff constraints or sliding along interfaces.
It is ...
1
vote
1answer
186 views
Method to solve linear, first order ODE of generalized matrix matrix form
The equation and its meaning:
Consider two sets $(A)_{l=0,...,m_a},$,$(B)_{l=0,...,m_b}$ of hermitian matrices and a set of positive semidefinite matrices $(C)_{l=0,...,m_c}$. Each matrix has the ...
2
votes
0answers
225 views
Von Neumann analysis for coupled PDE's
I am interested in understanding how to perform stability analysis for coupled (to keep things do-able, lets say linear) PDE's
In the case of a single PDE, i understand the logic behind the VN ...
4
votes
1answer
329 views
Clenshaw-type recurrence for derivative of Chebyshev series
The naive summation of a Chebyshev series
\begin{align*}
f(x) = \frac{c_0}{2} + \sum_{k=1}^{n-1} c_{k}T_{k}(x)
\end{align*}
which employs the three-term recurrence for evaluation of the Chebyshev ...
0
votes
1answer
650 views
Finite Difference Method Limitations/Stability Criteria
Is it possible to solve an equation with only a single derivative such as:
$$\frac{\partial U(x,t)}{\partial t} = A - BU(x,t)$$
with finite difference methods?
I ask as I am trying to solve the ...
2
votes
2answers
204 views
von Neumann stability analysis for spatial variable flux
Can we use the von Neumann stability analysis to investigate the stability of the discrete form of the following problem?
$$u_t+\frac{\partial(x^2u)}{\partial x}=S(u,x)\ .$$
Please give some hint ...
6
votes
1answer
98 views
Stable computation of ratio of sums of large numbers
I have two sets of large positive numbers $a_1,\ldots,a_n$ and $b_1,\ldots,b_n$. By 'large' I mean of the order of $10^{10}$. I want to calculate the ratio $$R = \frac{a_1 - a_2 + \cdots +(-1)^{n+1}...
3
votes
1answer
136 views
Stability of PDE Discretizations with Multistep Time Discretizations
Let's pretend we have a spatially discretized PDE of the following form:
\begin{align}
\frac{\partial^2 \boldsymbol{u}^k}{\partial t^2} = D\boldsymbol{u}^k
\end{align}
where $D$ can be any form for ...
4
votes
1answer
687 views
Von Neumann stability analysis with a constant term
I have a question concerning the von Neumann stability analysis of finite difference approximations of PDEs. There seem to be a wealth of online source explaining the application of this stability ...
9
votes
2answers
505 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 ...
0
votes
0answers
344 views
Stability of nonlinear partial differential equation
I want to find an expression for the stability of the nonlinear Poisson equation. I know about von Neumann stability analysis which applies to linear equations as far as I know. Any suggestion how to ...
1
vote
0answers
83 views
Kahan Summation for Three-Term Recurrences
Kahan summation applies to summation problems, but not to three-term recurrence relations. However, a three-term recurrence shares many of the features of a summation-albeit with a rescaling step at ...
5
votes
1answer
1k views
What is the origin of the spurious oscillations in the Crank-Nicolson scheme?
I was reading about the Crank-Nicolson method, and it is often said that it can produce "spurious oscillations" or that this method is prone to "ringing", especially for large time step and stiff ...
1
vote
1answer
848 views
Improve numeric stability of subtraction in C++ [closed]
I'm writing a matrix library in c++. After some debugging I found that a simple double difference is not zero for two "equals" numbers. This is due how double are represented in a computer of course. ...
7
votes
1answer
401 views
CFL condition in polar coordinates
In this question, I suggested that the Couran-Friedrichs-Lewy (CFL) condition for the wave equation in polar coordinates reads
$$C = 2c\frac{\Delta t}{\Delta r \Delta \phi} \leq C_\max \enspace ,$$
...
3
votes
1answer
81 views
Finding Numerical Stability of Simple System with Integral Term Due to Low Pass Filter
As a toy problem, I was looking at a classical second order system of the following form:
\begin{align}
\ddot{x}(t) + c \dot{x}(t) + k x(t) = 0
\end{align}
Instead of the basic system above, I want ...
3
votes
2answers
119 views
In numerical methods, eg, finite differencing approaches, does there exist convergent schemes that are not both consistent and stable?
In a book that our course is following this semester, the theorem given is only in one direction: if the scheme is both consistent and stable, then the scheme is convergent.
However, since this ...
