The document summarizes Vince Velocci's research internship from September 2014 to April 2015 under the supervision of Dr. Walter Richardson at UTSA. The internship involved adapting existing simulation software to model electromagnetic fields and induced currents in the brain during Transcranial Magnetic Stimulation (TMS). Vince worked on installing software, reading literature, and coding examples involving fractional order derivatives to help lay the groundwork for future work modeling TMS using finite element methods.
NEURAL NETWORK FOR THE RELIABILITY ANALYSIS OF A SERIES - PARALLEL SYSTEM SUB...IAEME Publication
Artificial neural networks can achieve high computation rates by employing a massive number of simple processing elements with a high degree of connectivity between the elements. Neural networks with feedback connections provide a computing model capable of exploiting fine- grained parallelism to solve a rich class of complex problems. In this paper we discuss a complex series-parallel system subjected to finite common cause and finite human error failures and its reliability using neural network method.
Artificial Neural Network and its Applicationsshritosh kumar
Abstract
This report is an introduction to Artificial Neural
Networks. The various types of neural networks are
explained and demonstrated, applications of neural
networks like ANNs in medicine are described, and a
detailed historical background is provided. The
connection between the artificial and the real thing is
also investigated and explained. Finally, the
mathematical models involved are presented and
demonstrated.
NEURAL NETWORK FOR THE RELIABILITY ANALYSIS OF A SERIES - PARALLEL SYSTEM SUB...IAEME Publication
Artificial neural networks can achieve high computation rates by employing a massive number of simple processing elements with a high degree of connectivity between the elements. Neural networks with feedback connections provide a computing model capable of exploiting fine- grained parallelism to solve a rich class of complex problems. In this paper we discuss a complex series-parallel system subjected to finite common cause and finite human error failures and its reliability using neural network method.
Artificial Neural Network and its Applicationsshritosh kumar
Abstract
This report is an introduction to Artificial Neural
Networks. The various types of neural networks are
explained and demonstrated, applications of neural
networks like ANNs in medicine are described, and a
detailed historical background is provided. The
connection between the artificial and the real thing is
also investigated and explained. Finally, the
mathematical models involved are presented and
demonstrated.
Basics of Neural networks and its image recognition and its applications of engineering fields and medicines and how it detect those images and give the results of those images....
An Neural Network (NN) is an information processing paradigm that is inspired by the way biological nervous systems, such as the brain, process information.
It is composed of a large number of highly interconnected processing elements (neurons) working in unison to solve specific problems.
An ANN is configured for a specific application, such as pattern recognition or data classification, through a learning process.
An artificial neuron is a device with many inputs and one output. The neuron has two modes of operation; the training mode and the using mode. In the training mode, the neuron can be trained to fire (or not), for particular input patterns.
In the using mode, when a taught input pattern is detected at the input, its associated output becomes the current output.
Basic definitions, terminologies, and Working of ANN has been explained. This ppt also shows how ANN can be performed in matlab. This material contains the explanation of Feed forward back propagation algorithm in detail.
Neural Network and Artificial Intelligence.
Neural Network and Artificial Intelligence.
WHAT IS NEURAL NETWORK?
The method calculation is based on the interaction of plurality of processing elements inspired by biological nervous system called neurons.
It is a powerful technique to solve real world problem.
A neural network is composed of a number of nodes, or units[1], connected by links. Each linkhas a numeric weight[2]associated with it. .
Weights are the primary means of long-term storage in neural networks, and learning usually takes place by updating the weights.
Artificial neurons are the constitutive units in an artificial neural network.
WHY USE NEURAL NETWORKS?
It has ability to Learn from experience.
It can deal with incomplete information.
It can produce result on the basis of input, has not been taught to deal with.
It is used to extract useful pattern from given data i.e. pattern Recognition etc.
Biological Neurons
Four parts of a typical nerve cell :• DENDRITES: Accepts the inputs• SOMA : Process the inputs• AXON : Turns the processed inputs into outputs.• SYNAPSES : The electrochemical contactbetween the neurons.
ARTIFICIAL NEURONS MODEL
Inputs to the network arerepresented by the x1mathematical symbol, xn
Each of these inputs are multiplied by a connection weight , wn
sum = w1 x1 + ……+ wnxn
These products are simplysummed, fed through the transfer function, f( ) to generate a result and then output.
NEURON MODEL
Neuron Consist of:
Inputs (Synapses): inputsignal.Weights (Dendrites):determines the importance ofincoming value.Output (Axon): output toother neuron or of NN .
Calculating voltage magnitudes and voltage phase angles of real electrical ne...IJECEIAES
In the field of electrical network, it is necessary, under different conditions, to learn about the behavior of the system. Power Flow Analysis is the tool per excellent that allow as to make a deep study and define all quantities of each bus of the system. To determine power flow analysis there is a lot of methods, we have either numerical or intelligent techniques. Lately, researchers always work on finding intelligent methods that allow them to solve their complex problems. The goal of this article is to compare two intelligent methods that are capable of predicting quantities; artificial neural network and adaptive neuro-fuzzy inference system using real electrical networks. To do that we used few significant discrepancies. These methods are characterized by giving results in real time. To make this comparison successful, we implemented these two methods, to predict the voltage magnitudes and the voltage phase angles, on two Moroccan electrical networks. The results of the comparison show that the method of adaptive neuro-fuzzy inference system have more advantages than the method of artificial neural network.
Basics of Neural networks and its image recognition and its applications of engineering fields and medicines and how it detect those images and give the results of those images....
An Neural Network (NN) is an information processing paradigm that is inspired by the way biological nervous systems, such as the brain, process information.
It is composed of a large number of highly interconnected processing elements (neurons) working in unison to solve specific problems.
An ANN is configured for a specific application, such as pattern recognition or data classification, through a learning process.
An artificial neuron is a device with many inputs and one output. The neuron has two modes of operation; the training mode and the using mode. In the training mode, the neuron can be trained to fire (or not), for particular input patterns.
In the using mode, when a taught input pattern is detected at the input, its associated output becomes the current output.
Basic definitions, terminologies, and Working of ANN has been explained. This ppt also shows how ANN can be performed in matlab. This material contains the explanation of Feed forward back propagation algorithm in detail.
Neural Network and Artificial Intelligence.
Neural Network and Artificial Intelligence.
WHAT IS NEURAL NETWORK?
The method calculation is based on the interaction of plurality of processing elements inspired by biological nervous system called neurons.
It is a powerful technique to solve real world problem.
A neural network is composed of a number of nodes, or units[1], connected by links. Each linkhas a numeric weight[2]associated with it. .
Weights are the primary means of long-term storage in neural networks, and learning usually takes place by updating the weights.
Artificial neurons are the constitutive units in an artificial neural network.
WHY USE NEURAL NETWORKS?
It has ability to Learn from experience.
It can deal with incomplete information.
It can produce result on the basis of input, has not been taught to deal with.
It is used to extract useful pattern from given data i.e. pattern Recognition etc.
Biological Neurons
Four parts of a typical nerve cell :• DENDRITES: Accepts the inputs• SOMA : Process the inputs• AXON : Turns the processed inputs into outputs.• SYNAPSES : The electrochemical contactbetween the neurons.
ARTIFICIAL NEURONS MODEL
Inputs to the network arerepresented by the x1mathematical symbol, xn
Each of these inputs are multiplied by a connection weight , wn
sum = w1 x1 + ……+ wnxn
These products are simplysummed, fed through the transfer function, f( ) to generate a result and then output.
NEURON MODEL
Neuron Consist of:
Inputs (Synapses): inputsignal.Weights (Dendrites):determines the importance ofincoming value.Output (Axon): output toother neuron or of NN .
Calculating voltage magnitudes and voltage phase angles of real electrical ne...IJECEIAES
In the field of electrical network, it is necessary, under different conditions, to learn about the behavior of the system. Power Flow Analysis is the tool per excellent that allow as to make a deep study and define all quantities of each bus of the system. To determine power flow analysis there is a lot of methods, we have either numerical or intelligent techniques. Lately, researchers always work on finding intelligent methods that allow them to solve their complex problems. The goal of this article is to compare two intelligent methods that are capable of predicting quantities; artificial neural network and adaptive neuro-fuzzy inference system using real electrical networks. To do that we used few significant discrepancies. These methods are characterized by giving results in real time. To make this comparison successful, we implemented these two methods, to predict the voltage magnitudes and the voltage phase angles, on two Moroccan electrical networks. The results of the comparison show that the method of adaptive neuro-fuzzy inference system have more advantages than the method of artificial neural network.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Model of Differential Equation for Genetic Algorithm with Neural Network (GAN...Sarvesh Kumar
The work is carried on the application of differential equation (DE) and its computational technique of genetic algorithm and neural (GANN) in C#, which is frequently used in globalised world by human wings. Diagrammatical and flow chart presentation is the major concerned for easy undertaking of these two concepts with indication of its present and future application is the new initiative taken in this paper along with computational approaches in C#. Little observation has been also pointed during working, functioning and development process of above algorithm in C# under given boundary value condition of DE for genetic and neural. Operations of fitness function and Genetic operations were completed for behavioural transmission of chromosome.
USING SINGULAR VALUE DECOMPOSITION IN A CONVOLUTIONAL NEURAL NETWORK TO IMPRO...ijcsit
A brain tumor consists of cells showing abnormal brain growth. The area of the brain tumor significantly
affects choosing the type of treatment and following the course of the disease during the treatment. At the
same time, pictures of Brain MRIs are accompanied by noise. Eliminating existing noises can significantly
impact the better segmentation and diagnosis of brain tumors. In this work, we have tried using the
analysis of eigenvalues. We have used the MSVD algorithm, reducing the image noise and then using the
deep neural network to segment the tumor in the images. The proposed method's accuracy was increased
by 2.4% compared to using the original images. With Using the MSVD method, convergence speed has
also increased, showing the proposed method's effectiveness.
A brain tumor consists of cells showing abnormal brain growth. The area of the brain tumor significantly
affects choosing the type of treatment and following the course of the disease during the treatment. At the
same time, pictures of Brain MRIs are accompanied by noise. Eliminating existing noises can significantly
impact the better segmentation and diagnosis of brain tumors. In this work, we have tried using the
analysis of eigenvalues. We have used the MSVD algorithm, reducing the image noise and then using the
deep neural network to segment the tumor in the images. The proposed method's accuracy was increased
by 2.4% compared to using the original images. With Using the MSVD method, convergence speed has
also increased, showing the proposed method's effectiveness
The Dawn of the Age of Artificially Intelligent NeuroprostheticsSagar Hingal
A summary or an overview of the existing technologies that encapsulate the concepts of NeuroScience and Bio-Technology using the enhanced methods of Artificial-intelligence.
In this review paper, there are several case studies and methodologies of implementations of neuroprosthetics as well as how A.I (Artificial Intelligence) is evolved over the period of time and what is next on the future.....
The aim of this research is to find accurate solution for the Troesch’s problem by using high performance technique based on parallel processing implementation.
Design/methodology/approach – Feed forward neural network is designed to solve important type of differential equations that arises in many applied sciences and engineering applications. The suitable designed based on choosing suitable learning rate, transfer function, and training algorithm. The authors used back propagation with new implement of Levenberg - Marquardt training algorithm. Also, the authors depend new idea for choosing the weights. The effectiveness of the suggested design for the network is shown by using it for solving Troesch problem in many cases.
Findings – New idea for choosing the weights of the neural network, new implement of Levenberg - Marquardt training algorithm which assist to speeding the convergence and the implementation of the suggested design demonstrates the usefulness in finding exact solutions.
Deep learning for pose-invariant face detection in unconstrained environmentIJECEIAES
In the recent past, convolutional neural networks (CNNs) have seen resurgence and have performed extremely well on vision tasks. Visually the model resembles a series of layers each of which is processed by a function to form a next layer. It is argued that CNN first models the low level features such as edges and joints and then expresses higher level features as a composition of these low level features. The aim of this paper is to detect multi-view faces using deep convolutional neural network (DCNN). Implementation, detection and retrieval of faces will be obtained with the help of direct visual matching technology. Further, the probabilistic measure of the similarity of the face images will be done using Bayesian analysis. Experiment detects faces with ±90 degree out of plane rotations. Fine tuned AlexNet is used to detect pose invariant faces. For this work, we extracted examples of training from AFLW (Annotated Facial Landmarks in the Wild) dataset that involve 21K images with 24K annotations of the face.
Smart Brain Wave Sensor for Paralyzed- A Real Time ImplementationSiraj Ahmed
ABSTRACT
As the title of the paper indicates about brainwaves and its uses for various applications based on their frequencies and different parameters which can be implemented as real time application with the title a smart brain wave sensor system for paralyzed patients. Brain wave sensing is to detect a person's mental status. The purpose of brain wave sensing is to give exact treatment to paralyzed patients. The data or signal is obtained from the brainwaves sensing band. This data are converted as object files using Visual Basics. The processed data is further sent to Arduino which has the human's behavioral aspects like emotions, sensations, feelings, and desires. The proposed device can sense human brainwaves and detect the percentage of paralysis that the person is suffering. The advantage of this paper is to give a real-time smart sensor device for paralyzed patients with paralysis percentage for their exact treatment.
Keywords:-Brainwave sensor, BMI, Brain scan, EEG, MCH.
1. Vince Velocci, Research Student – AIM 6943 Final Report (Spring 2015) – April 27, 2015
Internship start date: Monday September 8, 2014
Internship end date: Thursday April 30, 2015
Supervisor: Dr. Walter Richardson, walter.richardson@utsa.edu, 210-458-4760
Department of Mathematics, The University of Texas at San Antonio
One UTSA Circle
San Antonio, TX 78249 (www.math.utsa.edu)
Duties
I am a research student working with Dr. Richardson on a project whose ultimate goal is
the numerical simulation of electromagnetic fields and induced currents in the brain as a result
of a medical procedure known as Transcranial Magnetic Stimulation (TMS). My duties included
trying out a couple of toy numerical problems, and helping the project get off the ground, to be
continued at a future time by another student.
Description of Work and Research Methodology
Transcranial Magnetic Stimulation is a treatment tool that has been used to treat certain
neurological or psychiatric disorders, such as Parkinson’s disease and depression. TMS is also
used as a tool to study the brain, its function, and it structure. In this procedure, a coil,
sometimes in the shape of a figure-eight, is placed a short distance above the head. A highly
oscillating current is run through the coil. A current through a wire causes a magnetic field, B,
to surround the coil. The highly oscillating current, however, causes the formation of a rapidly
varying magnetic pulse that travels through the brain and induces (via Faraday’s Law of
Induction) weak circular currents (eddy currents) in certain targeted brain regions, thereby
activating those brain regions. Our goal is to adapt existing simulation software to simulate
these currents and the electromagnetic fields in the brain as a result of this procedure. This
problem involves solving Maxwell’s Equations (which, in this case, take the form of a wave
equation due to the highly varying nature of the fields involved) in three dimensions on the
human brain.
Open source software known as EIDORS (Electrical Impedance Tomography and Diffuse
Optical Tomography Reconstruction Software), initially developed for Matlab, is a set of Matlab
programs for image reconstruction in electrical impedance tomography. This involves obtaining
a map of the conductivity and permittivity structure of the brain from the surface
measurements of electrodes attached to the brain. Mathematically, it involves solving the
inverse problem of determining functions σ(x) and ε(x) from surface measurements of the
2. electric field E and magnetic field H, where these four quantities are related by Maxwell’s
Equations which, in this system, take the form
curl(E) = jωμH and curl(H) = (σ – jωε)E
Assuming a small magnetic permeability μ, and field frequency ω, one can express E = grad(φ),
where φ is the complex-valued electric potential, and the two Maxwell’s Equations become the
elliptic partial differential equation, Div[(σ – jωε)*grad(φ)] = 0. The measured boundary data
are the potential φ, and the component of grad(φ) normal to the head. In the lab, one applies
known currents to the head (boundary) and makes measurements of potentials. By solving this
inverse problem, EIDORS obtains a map of the brain from the numerically computed values σ
and ε. Of course, what I have explained is a simplification of the full version of the process,
which involves solving both the forward problem and the inverse problem. The forward
problem solves the elliptic PDE given an initial guess for σ and ε and one set of boundary data.
The forward problem is solved by modeling the brain (domain) as a 3D finite element mesh of
tetrahedra. (Software exists within EIDORS to set up the finite element mesh.) One linearizes
the equations resulting from this elliptic PDE about the initial guesses for the constants σ and ε,
and EIDORS solves this linear system to update these constants. This is done iteratively until
the numerical model agrees with the measured data to the desired precision.
The goal of our research is to adapt the EIDORS software to TMS in order to determine
the induced currents in the brain as a result of this medical procedure. An important
component of this work is to obtain high quality MRI images of the human brain that can be
converted into a finite element mesh to serve as the domain for the solution of Maxwell’s
Equations in the brain. Software exists to accomplish this, and the software treats the brain as
being composed of five different tissue types, each with its own set of properties: skull, skin,
white matter, grey matter, and cerebrospinal fluid.
Solving Maxwell’s equations on the human brain via the finite element method is one
avenue of approach to the problem of modelling the induced currents in TMS. Another avenue
of approach involves the interesting theory of fractional order derivatives. It has been recently
established that the dielectric behavior of systems (eg. the brain) may be well mathematically
modeled by a particularly successful model of viscoelasticity involving differential equations
that employ fractional order derivatives. Numerically investigating these fractional order
derivatives as well as toy problems associated with such derivatives (eg. the fractional order
wave-diffusion equation) was done as a start to this avenue.
Accomplishments Thus Far
Initially, I spent a bit of time installing the EIDORS software as well as it myriad of
components onto my computer, dealing with intricacies concerning MyApps (as that is the way
I am accessing Matlab), as well as playing around with the software, by trying various included
examples and tutorials, and investigating the EIDORS code. I also worked on trying to install the
program FreeSurfer (freesurfer.net), which is a free, open source program that runs on Linux
3. and allows you to analyze and visualize MRI images of the brain. More importantly, Freesurfer
allows you to segment the different tissue types from each other before the model is converted
into a 3D finite element mesh that may be used to simulate processes on the brain. I
successfully installed the program and tried it out on various different toy systems.
One particular program that has been used to simulate different forms of brain
stimulation is a package called SimNIBS (Simulation of Non-Invasive Brain Stimulation). The
program uses Freesurfer and a routine called mri2mesh to form the finite element mesh of the
brain. SimNIBS allows one to calculate the electric field induced by transcranial magnetic
stimulation. SimNIBS performs the electric field calculations using Matlab, so having Matlab
installed on your machine is a requirement. One cannot, for example, use myapps to access
Matlab for this purpose.
I contacted the person behind SimNIBS (Axel Thielscher) and, apparently, one can use
the meshes generated by SimNIBS in EIDORS if you want to perform simulations in EIDORS.
This is because EIDORS supports meshes created by gmsh, and gmsh is also used by SimNIBS for
this purpose. Using the EIDORS function gmsh_read_mesh, one should be able to read in the
head meshes created by SimNIBS.
To familiarize myself with this field, I have read numerous papers:
Windhoff, Mirko; Opitz, Alexander; Thielscher, Axel. “Electric Field Calculations in Brain
Stimulation Based on Finite Elements: An Optimized Processing Pipeline for the Generation and
Usage of Accurate Individual Head Models,” Human Brain Mapping, 34: 923-935 (2013).
Elloian, Jeffrey M.; Noetscher, Gregory M.; Makarov, Sergey N.; Pascual-Leone, Alvaro.
“Continuous Wave Simulations on the Propagation of Electromagnetic Fields Through the
Human Head,” IEEE Transactions on Biomedical Engineering, Vol. 61, No. 6, June 2014.
Lionheart, W.R.B.; Arridge, S.R.; Schweiger, M.; Vauhkonen, M.; Kaipio, J.P. “Electrical
Impedance and Diffuse Optical Tomography Reconstruction Software,” 1st
World Congress on
Industrial Process Tomography, Buxton, Greater Manchester, April 14-17, 1999.
Mugler, D.H.; Scott, R.A. “Fast Fourier Transform Method For Partial Differential Equations,
Case Study: The 2-D Diffusion Equation,” Comput. Math. Applic. Vol. 16, No. 3, 221-228, 1988.
Ziegler, et al. “A finite-element reciprocity solution for EEG forward modeling with realistic
individual head models,” NeuroImage, 103 (2014) 542-551.
Wharmby, Andrew W.; Bagley, Ronald L. “Modifying Maxwell’s equations for dielectric
materials based on techniques from viscoelasticity and concepts from fractional calculus,”
International Journal of Engineering Science, 79 (2014) 59-80.
Wagner, Tim A.; Zahn, Markus; Grodzinsky, Alan J.; Pascual-Leone, Alvara. “Three-Dimensional
Head Model Simulation of Transcranial Magnetic Stimulation,” IEEE Transactions on Biomedical
Engineering, Vol. 51, No. 9, September 2004.
4. I also got in touch with medical researchers at The University of Toronto and McGill
University to inquire about a high resolution image set made for the brain, known as the
BigBrain dataset: see https://bigbrain.loris.ca/main.php and
http://en.wikipedia.org/wiki/BigBrain and http://www.ncbi.nlm.nih.gov/pubmed/23788795
This dataset may prove useful for setting up finite element meshes and determining whether
the software can deal with such a large dataset.
The other tasks I accomplished involved using Matlab to code up examples involving
fractional order derivatives.
Code 1: A code that computes finite difference approximations to fractional order derivatives,
by using the spectral decomposition of the matrix corresponding to the FD approximation to
the 2nd
derivative. These orders are then illustrated in a special plot, showing full matrices for
fractional orders gradually becoming the tri-diagonal for order 2.
N=20;
M=21;
x=linspace(0,1,N+2);
noboundary=zeros(N,1);
deltax=1/(N+1);
r=1/deltax;
y=zeros(N+2,1);
for i=1:N+2
y(i)=x(i)*(1-x(i))*cos(4*pi*x(i));
end;
v=zeros(N,1);
for i=1:N
v(i)=y(i+1);
noboundary(i)=x(i+1);
end;
A=zeros(N,N);
A(1,1)=2;
A(1,2)=-1;
A(N,N)=2;
A(N,N-1)=-1;
for i=2:N-1
A(i,i-1)=-1;
A(i,i)=2;
A(i,i+1)=-1;
end;
[V,D]=eig(A);
alpha=linspace(0,2,M);
for i=1:M
pcolor(V*(D^(alpha(i)))*V'), drawnow;
pause(0.5);
end;
for i=1:M
plot(noboundary,[r*V*(D^(alpha(i)))*V']*v), drawnow;
pause(0.05);
end;
5. Code 2: A code which numerically computes and plots the solution to the equation
0Dt
3/2
y(t) + y(t) = f(t) (t > 0)
y(0) = y’(0) = 0
using four different choices for the “forcing function”, f(t).
for forcing = 1:4
if forcing==1
%------------------------------------------------------------
n=501;
h=0.1;
alpha=3/2;
L=5;
t=linspace(0,50,n);
y=zeros(1,n);
f=ones(1,n);
for i=3:n
sum=0;
w=1;
k=min([i L/h]);
for j=1:k-1
w=(1-(alpha+1)/j)*w;
sum=sum+w*y(i-j);
end;
y(i)=((h^(alpha))*f(i)-sum)/(1+h^(alpha));
end;
plot(t,y)
elseif forcing==2
%----------------------------------------------------------
n=501;
h=0.1;
alpha=3/2;
L=5;
t=linspace(0,50,n);
y=zeros(1,n);
f=zeros(1,n);
for i=1:n
f(i)=t(i)*exp(-t(i));
end;
for i=3:n
sum=0;
w=1;
k=min([i L/h]);
for j=1:k-1
w=(1-(alpha+1)/j)*w;
sum=sum+w*y(i-j);
end;
y(i)=((h^(alpha))*f(i)-sum)/(1+h^(alpha));
end;
plot(t,y)
7. Code 3: A code that solves the fractional order wave-diffusion equation
0Dt
α
u(x,t) = uxx(x,t) for 0 ≤ x ≤ 1, 0 ≤ t ≤ 10
u(x,0) = sin(πx)
for different values of the order of the time derivative, α.
T=10;
X=1;
h=0.1;
Nt=(T/h)+1;
alpha=2;
t=linspace(0,T,Nt);
deltax=0.1;
Nx=(X/deltax)+1;
x=linspace(0,X,Nx);
s=(h^alpha)/((deltax)^2);
u=zeros(Nx,Nt);
for i=1:Nx
u(i,1)=sin(pi*x(i));
end;
for i=1:Nt
u(1,i)=0;
u(Nx,i)=0;
end;
A=zeros(Nx-2,Nx-2);
A(1,1)=-2;
A(1,2)=1;
A(Nx-2,Nx-3)=1;
A(Nx-2,Nx-2)=-2;
for i=2:Nx-3
A(i,i-1)=1;
A(i,i)=-2;
A(i,i+1)=1;
end;
M=inv(s*A-eye(Nx-2));
c=zeros(Nx-2,Nt);
for i=1:Nx-2
c(i,1)=sin(pi*x(i+1));
end;
for i=2:Nt
old=zeros(Nx-2,1);
w=1;
for j=1:i-1
w=(1-(alpha+1)/j)*w;
old=old+w*c(:,i-j);
end;
c(:,i)=M*old;
end;
for j=2:Nt
for i=2:Nx-1
u(i,j)=c(i-1,j);
end;
end;
figure;
8. for j=1:Nt
plot(x,u(:,j)),axis([0 1 -2 2]), drawnow;
pause(0.5);
end;
In code 1, A is the matrix which represents the finite-difference approximation to minus
the 2nd
derivative: -u” ≈ [-u(xi-1) + 2u(xi) –u(xi+1)]/(∆x)2
. The A matrix just contains the
coefficients -1, 2, -1. The matrix is symmetric and positive-definite. Thus, we can decompose
the matrix using the spectral theorem and take fractional powers of it. This is what is done in
code 1.
In codes 2 and 3, we are using the approximation that the αth
order derivative of f(t)
with t ≥ a, which is denoted by a 𝐷𝑡
𝛼
, is given by:
a 𝐷𝑡
𝛼
f(t) ≈ ∑ (−1) 𝑗
( 𝛼
𝑗
)
[
𝑡−𝑎
ℎ
]
𝑗=0
𝑓(𝑡 − 𝑗ℎ)
This just comes from the definition of this derivative as
lim
ℎ→0
𝑛ℎ=𝑡−𝑎
ℎ−𝛼 ∑(−1) 𝑗 (
𝛼
𝑗
)
𝑛
𝑗=0
𝑓(𝑡 − 𝑗ℎ)
We denote 𝑤𝑗
(𝛼)
= (-1)j
( 𝛼
𝑗
), (j = 0, 1, 2, …) and it turns out that these coefficients obey the
following recurrence relationships:
𝑤0
(𝛼)
= 1; 𝑤𝑘
(𝛼)
= (1 −
𝛼+1
𝑘
) 𝑤𝑘−1
(𝛼)
, 𝑘 = 1, 2, 3, …
Note that if α is an integer or a fraction, we define ( 𝛼
𝑗
) =
𝛼(𝛼−1)…(𝛼−𝑗+1)
𝑗!
Also note that h is the time-step size and, in the upper limit for the summation in the
approximation to a 𝐷𝑡
𝛼
f(t), [ 𝐴] denotes the largest integer not greater than A.
Internship Benefits
Learning how to use Matlab to simulate complex systems as well as numerically solving
PDEs is an obvious benefit. This kind of experience can prove valuable in industry. Being
involved with a research project from start to finish (though, the ultimate goal is nowhere near
complete) is very much unlike any homework problem whose solution is already known. One
has to construct the path where one does not (yet) exist, and doing so is an important skill to
learn, one that industry values. Jumping into a new research project also involves familiarizing
oneself with the literature just enough to get started, and to learn what you need to learn as
you go along. This is clearly a benefit, for one can never learn everything at the very start of a
project. Being able jump into a field in which I am not an expert, and teaching myself what I
9. need to know, ultimately producing something of value, is an important thing to show off to
potential employers. And this is true for both industry and academia.
Acknowledgements
Thanks to Dr. Richardson for his guidance, and to the researchers Marc-Etienne Rousseau, D.
Louis Collins, Alan Evans, and Alain Dagher for referring me to the BigBrain dataset.