Intuitively, when sampling Δt is too infrequent (i.e. v(x,t) is big) we blowup (diverge from true state). You can think of faster velocity as corresponding to high frequency/low period.
SKwon1220
Does there exist some sort of optimization that we can perform to pick the optimal sampling rate for Δt to minimize the error of this method? Or is this something that is usually found through trial and error and observation?
rcorona
@SKwon1220 this reminds of adaptive learning rate optimization methods in machine learning such as Adam.
During gradient descent optimization, if the learning rate is set too high, the model can diverge because it ends up taking steps of increasing magnitude, similarly to the examples on this slide.
What methods like Adam do is adapt the step size by scaling it down proportionally to the spread of the gradients.
With that in mind, I wonder if one could adapt the step size in the Forward Euler method by scaling it down proportionally to the spread of updates to the position variables?
aishikbhattacharyya
I wonder if there's an algorithm that's able to control the inaccuracy, or error, as the step size increases. For example, if we compute with varying step sizes to figure out what size would be most optimal.
sparky-ed
I find it very interesting to see the importance of precision in computational simulations. While the Forward Euler method is straightforward and perhaps easy to implement, it clearly has limitations that could possibly lead to such results. But I believe understanding why and how this might lead to a wrong result is essential to understand and balance the trade-off between computational simplicity and accuracy of results. I was just wondering, how many more do we need to address this instability and accuracy?
Zzz212zzZ
To address the inaccuracies issue caused by delta t, we can adopt an adaptive stepping. Especially in those movements (like fast or complex movements), we could dynamically decrease the step size to collect more information.
Intuitively, when sampling Δt is too infrequent (i.e. v(x,t) is big) we blowup (diverge from true state). You can think of faster velocity as corresponding to high frequency/low period.
Does there exist some sort of optimization that we can perform to pick the optimal sampling rate for Δt to minimize the error of this method? Or is this something that is usually found through trial and error and observation?
@SKwon1220 this reminds of adaptive learning rate optimization methods in machine learning such as Adam. During gradient descent optimization, if the learning rate is set too high, the model can diverge because it ends up taking steps of increasing magnitude, similarly to the examples on this slide.
What methods like Adam do is adapt the step size by scaling it down proportionally to the spread of the gradients.
With that in mind, I wonder if one could adapt the step size in the Forward Euler method by scaling it down proportionally to the spread of updates to the position variables?
I wonder if there's an algorithm that's able to control the inaccuracy, or error, as the step size increases. For example, if we compute with varying step sizes to figure out what size would be most optimal.
I find it very interesting to see the importance of precision in computational simulations. While the Forward Euler method is straightforward and perhaps easy to implement, it clearly has limitations that could possibly lead to such results. But I believe understanding why and how this might lead to a wrong result is essential to understand and balance the trade-off between computational simplicity and accuracy of results. I was just wondering, how many more do we need to address this instability and accuracy?
To address the inaccuracies issue caused by delta t, we can adopt an adaptive stepping. Especially in those movements (like fast or complex movements), we could dynamically decrease the step size to collect more information.