Please see the attached image/problem[Interface coupled problem statement] below. It has two domains with two PDEs coupled at the interface. I have no problem solving two coupled PDEs for a single geometry, but for two different geometry, I am not sure how can I add two interfaces BCs at the interface of two domains in COMSOL. Can you give me an idea/useful links/example code of how to solve two PDEs in two domains in COMSOL coupled at the interface?
I need to find the variation of two variables(u1 and u2) along the x-axis(1D problem). If you need to know more about the problem,please let me know.
Regards,
Anadi
Interface coupled problem statement
In COMSOL, you can use the Multiphysics interface to couple two different domains with different physics. To set up a simulation with two domains that are coupled at the interface, you can create a new model in COMSOL and add two physics interfaces for the two different domains. For example, if you are solving two PDEs, you can add a "PDE" interface for each domain.Then, in each physics interface, define the PDE that you want to solve. later,create a new boundary condition at the interface between the two domains. In this boundary condition, you can specify the coupling between the two domains. For example, you can specify that the values of the variables u1 and u2 are equal at the interface.
Set up the mesh for the model, taking into account the interface between the two domains. You may need to use a finer mesh at the interface to ensure that the solution is accurately captured.Finally, solve the model and post-process the results to visualize the variation of u1 and u2 along the x-axis.
Here is a link to the COMSOL documentation on coupling different physics interfaces.
Related
I am learning how to build a thermo-fluid model with Modelica, I notice that in order to reduce the nonlinearity, it is recommended to use two different models: flow model and volume model, here is the explanation I found in a commercial library, but I am not sure how to arrange the equation in the components to realize this idea.
So I am looking for some papers and examples about how to use this idea in Modelica coding.
The flow port which indicates that a flow model is directly coupled to
it on the inside of the component, i.e. pressure from this port is
further processed to compute e.g. mass flow rate. It is recommended to
connect this port with a corresponding VolumePort in order to obtain
an alternating flow model-volume system model structure. If two models
are connected with their ports being both of this class, non-linear
equation systems for the algebraic pressure in the ports may be
created and require an initial guess value.
I want to create a deep classification net, but my classes aren't mutually exclusive (that is what sofmaxlayer do).
Is it possible to define a non mutually exclusive classification layer (i.e., a data can be in more than one class)?
One way to do it, it would be with a logsig function in the classification layer, instead of a softmax, but I have no idea how to acomplish that....
In CNN you can have multiple class in last layer as you know. But if I understand correctly your need in last layer an out put with that is in a range of numbers instead of 1 or 0 for each class. Its mean you need regression. If your labels support this task it's OK and you can do it with regression just like what happen in bounding box regression for localization. And you don't need soft-max in last layer. just use other activation functions that produce sufficient out put for your task.
I want to use pre-trained model for the face identification. I try to use Siamese architecture which requires a few number of images. Could you give me any trained model which I can change for the Siamese architecture? How can I change the network model which I can put two images to find their similarities (I do not want to create image based on the tutorial here)? I only want to use the system for real time application. Do you have any recommendations?
I suppose you can use this model, described in Xiang Wu, Ran He, Zhenan Sun, Tieniu Tan A Light CNN for Deep Face Representation with Noisy Labels (arXiv 2015) as a a strating point for your experiments.
As for the Siamese network, what you are trying to earn is a mapping from a face image into some high dimensional vector space, in which distances between points reflects (dis)similarity between faces.
To do so, you only need one network that gets a face as an input and produce a high-dim vector as an output.
However, to train this single network using the Siamese approach, you are going to duplicate it: creating two instances of the same net (you need to explicitly link the weights of the two copies). During training you are going to provide pairs of faces to the nets: one to each copy, then the single loss layer on top of the two copies can compare the high-dimensional vectors representing the two faces and compute a loss according to a "same/not same" label associated with this pair.
Hence, you only need the duplication for the training. In test time ('deploy') you are going to have a single net providing you with a semantically meaningful high dimensional representation of faces.
For a more advance Siamese architecture and loss see this thread.
On the other hand, you might want to consider the approach described in Oren Tadmor, Yonatan Wexler, Tal Rosenwein, Shai Shalev-Shwartz, Amnon Shashua Learning a Metric Embedding for Face Recognition using the Multibatch Method (arXiv 2016). This approach is more efficient and easy to implement than pair-wise losses over image pairs.
I am trying to detect the faces using the Matlab built-in viola jones face detection. Is there anyway that I can combine two classification models like "FrontalFaceCART" and "ProfileFace" into one in order to get a better result?
Thank you.
You can't combine models. That's a non-sense in any classification task since every classifier is different (works differently, i.e. different algorithm behind it, and maybe is also trained differently).
According to the classification model(s) help (which can be found here), your two classifiers work as follows:
FrontalFaceCART is a model composed of weak classifiers, based on classification and regression tree analysis
ProfileFace is composed of weak classifiers, based on a decision stump
More infos can be found in the link provided but you can easily see that their inner behaviour is rather different, so you can't mix them or combine them.
It's like (in Machine Learning) mixing a Support Vector Machine with a K-Nearest Neighbour: the first one uses separating hyperplanes whereas the latter is simply based on distance(s).
You can, however, train several models in parallel (e.g. independently) and choose the model that better suits you (e.g. smaller error rate/higher accuracy): so you basically create as many different classifiers as you like, give them the same training set, evaluate each accuracy (and/or other parameters) and choose the best model.
One option is to make a hierarchical classifier. So in a first step you use the frontal face classifier (assuming that most pictures are frontal faces). If the classifier fails, you try with the profile classifier.
I did that with a dataset of faces and it improved my overall classification accuracy. Furthermore, if you have some a priori information, you can use it. In my case the faces were usually in the middle up part of the picture.
To further improve your performance, without using the two classifiers in MATLAB you are using, you would need to change your technique (and probably your programming language). This is the best method so far: Facenet.
I have networks of roughly 10K to 100K nodes which are all connected. These nodes are typically grouped into clusters of communities which are strongly connected with many edges between them and there are hubs etc. Between the communities there are nodes with a few edges bridging / connecting the communities together. These datasets are in adjacency matrices
I have tried spectral clustering (Ding et al 2001) but it is really slow on large data sets and seems to stop working when there is a lot of ambiguity (bridges which are not the only bridge route to another cluster- other communities can act as alternative proxy routes).
I have tried some of the methods from martelot such as the Newman algorithm for modularity optimisation but have not incorporated the stability optimisation functions in that effort (could that be crucial?). On synthetic data sets where the clusters are created by random graphs (ER graphs) the methods work but on real ones where there is nested hierarchy the results are scattered. Using a standalone visualization application/tool the bridges are evident though.
What methods would you recommend/advise to try? I am using MATLAB.
What do you want to do, exactly? Detect communities, or bridges between them? Those are two different problems. Once you have the communities, it's straightforward enough identifying the edges connecting nodes from two distinct communities. So, I guess you want to detect communities.
There are actually thousands methods for this purpose, some of them implemented in Matlab, such as the one you cite, or the generalized Louvain algorithm (also based on modularity optimization). However, most of them are rather available as C or C++ programs, such as InfoMap (based on a data compression paradigm), WalkTrap (clustering using a random walk-based distance), Markov Cluster (simulates some propagation mechanism), and the list goes on...
Those tools formalize the notion of community structure more or less differently, potentially leading to different (estimated) community structures, when applied on the same network. And of course, different communities means different bridges, too. So the question is rather to know how to pick the appropriate method for your data. You seem to have a priori knowledge regarding the networks you are studying, so you should use that to make your choice (rather than the programming language). For instance, even if you don't state it explicitly, you seem to be looking for a hierarchical community structure: not all tools are able to detect this kind of structure. Similarly, if you think one node can belong to several communities at the same time, then you should consider looking for overlapping communities, for instance using CFinder (based on clique percolation).
I'd advise you to have a look at this excellent review of community detection, you might find some interesting information allowing you to pick a method: Community Detection in Graphs. Also, from a programming point of view, I'd advise you to play with the igraph library (available for C, R and Python): it contains several standard community detection tools. You can try them on your data and see what you get.