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ddemidov / AmgclC++ library for solving large sparse linear systems with algebraic multigrid method
hypre-space / HypreParallel solvers for sparse linear systems featuring multigrid methods.
JuliaLinearAlgebra / IterativeSolvers.jlIterative algorithms for solving linear systems, eigensystems, and singular value problems
lesgourg / Class PublicPublic repository of the Cosmic Linear Anisotropy Solving System (master for the most recent version of the standard code; GW_CLASS to include Cosmic Gravitational Wave Background anisotropies; classnet branch for acceleration with neutral networks; ExoCLASS branch for exotic energy injection; class_matter branch for FFTlog)
abusufyanvu / 6S191 MIT DeepLearningMIT Introduction to Deep Learning (6.S191) Instructors: Alexander Amini and Ava Soleimany Course Information Summary Prerequisites Schedule Lectures Labs, Final Projects, Grading, and Prizes Software labs Gather.Town lab + Office Hour sessions Final project Paper Review Project Proposal Presentation Project Proposal Grading Rubric Past Project Proposal Ideas Awards + Categories Important Links and Emails Course Information Summary MIT's introductory course on deep learning methods with applications to computer vision, natural language processing, biology, and more! Students will gain foundational knowledge of deep learning algorithms and get practical experience in building neural networks in TensorFlow. Course concludes with a project proposal competition with feedback from staff and a panel of industry sponsors. Prerequisites We expect basic knowledge of calculus (e.g., taking derivatives), linear algebra (e.g., matrix multiplication), and probability (e.g., Bayes theorem) -- we'll try to explain everything else along the way! Experience in Python is helpful but not necessary. This class is taught during MIT's IAP term by current MIT PhD researchers. Listeners are welcome! Schedule Monday Jan 18, 2021 Lecture: Introduction to Deep Learning and NNs Lab: Lab 1A Tensorflow and building NNs from scratch Tuesday Jan 19, 2021 Lecture: Deep Sequence Modelling Lab: Lab 1B Music Generation using RNNs Wednesday Jan 20, 2021 Lecture: Deep Computer Vision Lab: Lab 2A Image classification and detection Thursday Jan 21, 2021 Lecture: Deep Generative Modelling Lab: Lab 2B Debiasing facial recognition systems Friday Jan 22, 2021 Lecture: Deep Reinforcement Learning Lab: Lab 3 pixel-to-control planning Monday Jan 25, 2021 Lecture: Limitations and New Frontiers Lab: Lab 3 continued Tuesday Jan 26, 2021 Lecture (part 1): Evidential Deep Learning Lecture (part 2): Bias and Fairness Lab: Work on final assignments Lab competition entries due at 11:59pm ET on Canvas! Lab 1, Lab 2, and Lab 3 Wednesday Jan 27, 2021 Lecture (part 1): Nigel Duffy, Ernst & Young Lecture (part 2): Kate Saenko, Boston University and MIT-IBM Watson AI Lab Lab: Work on final assignments Assignments due: Sign up for Final Project Competition Thursday Jan 28, 2021 Lecture (part 1): Sanja Fidler, U. Toronto, Vector Institute, and NVIDIA Lecture (part 2): Katherine Chou, Google Lab: Work on final assignments Assignments due: 1 page paper review (if applicable) Friday Jan 29, 2021 Lecture: Student project pitch competition Lab: Awards ceremony and prize giveaway Assignments due: Project proposals (if applicable) Lectures Lectures will be held starting at 1:00pm ET from Jan 18 - Jan 29 2021, Monday through Friday, virtually through Zoom. Current MIT students, faculty, postdocs, researchers, staff, etc. will be able to access the lectures during this two week period, synchronously or asynchronously, via the MIT Canvas course webpage (MIT internal only). Lecture recordings will be uploaded to the Canvas as soon as possible; students are not required to attend any lectures synchronously. Please see the Canvas for details on Zoom links. The public edition of the course will only be made available after completion of the MIT course. Labs, Final Projects, Grading, and Prizes Course will be graded during MIT IAP for 6 units under P/D/F grading. Receiving a passing grade requires completion of each software lab project (through honor code, with submission required to enter lab competitions), a final project proposal/presentation or written review of a deep learning paper (submission required), and attendance/lecture viewing (through honor code). Submission of a written report or presentation of a project proposal will ensure a passing grade. MIT students will be eligible for prizes and awards as part of the class competitions. There will be two parts to the competitions: (1) software labs and (2) final projects. More information is provided below. Winners will be announced on the last day of class, with thousands of dollars of prizes being given away! Software labs There are three TensorFlow software lab exercises for the course, designed as iPython notebooks hosted in Google Colab. Software labs can be found on GitHub: https://github.com/aamini/introtodeeplearning. These are self-paced exercises and are designed to help you gain practical experience implementing neural networks in TensorFlow. For registered MIT students, submission of lab materials is not necessary to get credit for the course or to pass the course. At the end of each software lab there will be task-associated materials to submit (along with instructions) for entry into the competitions, open to MIT students and affiliates during the IAP offering. This includes MIT students/affiliates who are taking the class as listeners -- you are eligible! These instructions are provided at the end of each of the labs. Completing these tasks and submitting your materials to Canvas will enter you into a per-lab competition. MIT students and affiliates will be eligible for prizes during the IAP offering; at the end of the course, prize-winners will be awarded with their prizes. All competition submissions are due on January 26 at 11:59pm ET to Canvas. For the software lab competitions, submissions will be judged on the basis of the following criteria: Strength and quality of final results (lab dependent) Soundness of implementation and approach Thoroughness and quality of provided descriptions and figures Gather.Town lab + Office Hour sessions After each day’s lecture, there will be open Office Hours in the class GatherTown, up until 3pm ET. An MIT email is required to log in and join the GatherTown. During these sessions, there will not be a walk through or dictation of the labs; the labs are designed to be self-paced and to be worked on on your own time. The GatherTown sessions will be hosted by course staff and are held so you can: Ask questions on course lectures, labs, logistics, project, or anything else; Work on the labs in the presence of classmates/TAs/instructors; Meet classmates to find groups for the final project; Group work time for the final project; Bring the class community together. Final project To satisfy the final project requirement for this course, students will have two options: (1) write a 1 page paper review (single-spaced) on a recent deep learning paper of your choice or (2) participate and present in the project proposal pitch competition. The 1 page paper review option is straightforward, we propose some papers within this document to help you get started, and you can satisfy a passing grade with this option -- you will not be eligible for the grand prizes. On the other hand, participation in the project proposal pitch competition will equivalently satisfy your course requirements but additionally make you eligible for the grand prizes. See the section below for more details and requirements for each of these options. Paper Review Students may satisfy the final project requirement by reading and reviewing a recent deep learning paper of their choosing. In the written review, students should provide both: 1) a description of the problem, technical approach, and results of the paper; 2) critical analysis and exposition of the limitations of the work and opportunities for future work. Reviews should be submitted on Canvas by Thursday Jan 28, 2021, 11:59:59pm Eastern Time (ET). Just a few paper options to consider... https://papers.nips.cc/paper/2017/file/3f5ee243547dee91fbd053c1c4a845aa-Paper.pdf https://papers.nips.cc/paper/2018/file/69386f6bb1dfed68692a24c8686939b9-Paper.pdf https://papers.nips.cc/paper/2020/file/1457c0d6bfcb4967418bfb8ac142f64a-Paper.pdf https://science.sciencemag.org/content/362/6419/1140 https://papers.nips.cc/paper/2018/file/0e64a7b00c83e3d22ce6b3acf2c582b6-Paper.pdf https://arxiv.org/pdf/1906.11829.pdf https://www.nature.com/articles/s42256-020-00237-3 https://pubmed.ncbi.nlm.nih.gov/32084340/ Project Proposal Presentation Keyword: proposal This is a 2 week course so we do not require results or working implementations! However, to win the top prizes, nice, clear results and implementations will demonstrate feasibility of your proposal which is something we look for! Logistics -- please read! You must sign up to present before 11:59:59pm Eastern Time (ET) on Wednesday Jan 27, 2021 Slides must be in a Google Slide before 11:59:59pm Eastern Time (ET) on Thursday Jan 28, 2021 Project groups can be between 1 and 5 people Listeners welcome To be eligible for a prize you must have at least 1 registered MIT student in your group Each participant will only be allowed to be in one group and present one project pitch Synchronous attendance on 1/29/21 is required to make the project pitch! 3 min presentation on your idea (we will be very strict with the time limits) Prizes! (see below) Sign up to Present here: by 11:59pm ET on Wednesday Jan 27 Once you sign up, make your slide in the following Google Slides; submit by midnight on Thursday Jan 28. Please specify the project group # on your slides!!! Things to Consider This doesn’t have to be a new deep learning method. It can just be an interesting application that you apply some existing deep learning method to. What problem are you solving? Are there use cases/applications? Why do you think deep learning methods might be suited to this task? How have people done it before? Is it a new task? If so, what are similar tasks that people have worked on? In what aspects have they succeeded or failed? What is your method of solving this problem? What type of model + architecture would you use? Why? What is the data for this task? Do you need to make a dataset or is there one publicly available? What are the characteristics of the data? Is it sparse, messy, imbalanced? How would you deal with that? Project Proposal Grading Rubric Project proposals will be evaluated by a panel of judges on the basis of the following three criteria: 1) novelty and impact; 2) technical soundness, feasibility, and organization, including quality of any presented results; 3) clarity and presentation. Each judge will award a score from 1 (lowest) to 5 (highest) for each of the criteria; the average score from each judge across these criteria will then be averaged with that of the other judges to provide the final score. The proposals with the highest final scores will be selected for prizes. Here are the guidelines for the criteria: Novelty and impact: encompasses the potential impact of the project idea, its novelty with respect to existing approaches. Why does the proposed work matter? What problem(s) does it solve? Why are these problems important? Technical soundness, feasibility, and organization: encompasses all technical aspects of the proposal. Do the proposed methodology and architecture make sense? Is the architecture the best suited for the proposed problem? Is deep learning the best approach for the problem? How realistic is it to implement the idea? Was there any implementation of the method? If results and data are presented, we will evaluate the strength of the results/data. Clarity and presentation: encompasses the delivery and quality of the presentation itself. Is the talk well organized? Are the slides aesthetically compelling? Is there a clear, well-delivered narrative? Are the problem and proposed method clearly presented? Past Project Proposal Ideas Recipe Generation with RNNs Can we compress videos with CNN + RNN? Music Generation with RNNs Style Transfer Applied to X GAN’s on a new modality Summarizing text/news articles Combining news articles about similar events Code or spec generation Multimodal speech → handwriting Generate handwriting based on keywords (i.e. cursive, slanted, neat) Predicting stock market trends Show language learners articles or videos at their level Transfer of writing style Chemical Synthesis with Recurrent Neural networks Transfer learning to learn something in a domain for which it’s hard or risky to gather data or do training RNNs to model some type of time series data Computer vision to coach sports players Computer vision system for safety brakes or warnings Use IBM Watson API to get the sentiment of your Facebook newsfeed Deep learning webcam to give wifi-access to friends or improve video chat in some way Domain-specific chatbot to help you perform a specific task Detect whether a signature is fraudulent Awards + Categories Final Project Awards: 1x NVIDIA RTX 3080 4x Google Home Max 3x Display Monitors Software Lab Awards: Bose headphones (Lab 1) Display monitor (Lab 2) Bebop drone (Lab 3) Important Links and Emails Course website: http://introtodeeplearning.com Course staff: introtodeeplearning-staff@mit.edu Piazza forum (MIT only): https://piazza.com/mit/spring2021/6s191 Canvas (MIT only): https://canvas.mit.edu/courses/8291 Software lab repository: https://github.com/aamini/introtodeeplearning Lab/office hour sessions (MIT only): https://gather.town/app/56toTnlBrsKCyFgj/MITDeepLearning
curiousily / Machine Learning From ScratchSuccinct Machine Learning algorithm implementations from scratch in Python, solving real-world problems (Notebooks and Book). Examples of Logistic Regression, Linear Regression, Decision Trees, K-means clustering, Sentiment Analysis, Recommender Systems, Neural Networks and Reinforcement Learning.
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haasad / PyPardisoPython interface to the Intel MKL Pardiso library to solve large sparse linear systems of equations
SAFRAN-LAB / HODLRA fast, accurate direct solver and determinant computation for dense linear systems
anishida / LisLis (Library of Iterative Solvers for linear systems, pronounced [lis]) is a parallel software library to solve discretized linear equations and eigenvalue problems that arise from the numerical solution of partial differential equations using iterative methods.
linbox-team / LinboxLinBox - C++ library for exact, high-performance linear algebra
SCIInstitute / SCI Solver FEMSCI-Solver_FEM is a C++/CUDA library written to solve an FEM linear system. It is designed to solve the FEM system quickly by using GPU hardware.
maple3142 / Gf2bvSolving linear systems over GF(2) by manipulating bitvectors
Aryia-Behroziuan / NeuronsAn ANN is a model based on a collection of connected units or nodes called "artificial neurons", which loosely model the neurons in a biological brain. Each connection, like the synapses in a biological brain, can transmit information, a "signal", from one artificial neuron to another. An artificial neuron that receives a signal can process it and then signal additional artificial neurons connected to it. In common ANN implementations, the signal at a connection between artificial neurons is a real number, and the output of each artificial neuron is computed by some non-linear function of the sum of its inputs. The connections between artificial neurons are called "edges". Artificial neurons and edges typically have a weight that adjusts as learning proceeds. The weight increases or decreases the strength of the signal at a connection. Artificial neurons may have a threshold such that the signal is only sent if the aggregate signal crosses that threshold. Typically, artificial neurons are aggregated into layers. Different layers may perform different kinds of transformations on their inputs. Signals travel from the first layer (the input layer) to the last layer (the output layer), possibly after traversing the layers multiple times. The original goal of the ANN approach was to solve problems in the same way that a human brain would. However, over time, attention moved to performing specific tasks, leading to deviations from biology. Artificial neural networks have been used on a variety of tasks, including computer vision, speech recognition, machine translation, social network filtering, playing board and video games and medical diagnosis. Deep learning consists of multiple hidden layers in an artificial neural network. This approach tries to model the way the human brain processes light and sound into vision and hearing. Some successful applications of deep learning are computer vision and speech recognition.[68] Decision trees Main article: Decision tree learning Decision tree learning uses a decision tree as a predictive model to go from observations about an item (represented in the branches) to conclusions about the item's target value (represented in the leaves). It is one of the predictive modeling approaches used in statistics, data mining, and machine learning. Tree models where the target variable can take a discrete set of values are called classification trees; in these tree structures, leaves represent class labels and branches represent conjunctions of features that lead to those class labels. Decision trees where the target variable can take continuous values (typically real numbers) are called regression trees. In decision analysis, a decision tree can be used to visually and explicitly represent decisions and decision making. In data mining, a decision tree describes data, but the resulting classification tree can be an input for decision making. Support vector machines Main article: Support vector machines Support vector machines (SVMs), also known as support vector networks, are a set of related supervised learning methods used for classification and regression. Given a set of training examples, each marked as belonging to one of two categories, an SVM training algorithm builds a model that predicts whether a new example falls into one category or the other.[69] An SVM training algorithm is a non-probabilistic, binary, linear classifier, although methods such as Platt scaling exist to use SVM in a probabilistic classification setting. In addition to performing linear classification, SVMs can efficiently perform a non-linear classification using what is called the kernel trick, implicitly mapping their inputs into high-dimensional feature spaces. Illustration of linear regression on a data set. Regression analysis Main article: Regression analysis Regression analysis encompasses a large variety of statistical methods to estimate the relationship between input variables and their associated features. Its most common form is linear regression, where a single line is drawn to best fit the given data according to a mathematical criterion such as ordinary least squares. The latter is often extended by regularization (mathematics) methods to mitigate overfitting and bias, as in ridge regression. When dealing with non-linear problems, go-to models include polynomial regression (for example, used for trendline fitting in Microsoft Excel[70]), logistic regression (often used in statistical classification) or even kernel regression, which introduces non-linearity by taking advantage of the kernel trick to implicitly map input variables to higher-dimensional space. Bayesian networks Main article: Bayesian network A simple Bayesian network. Rain influences whether the sprinkler is activated, and both rain and the sprinkler influence whether the grass is wet. A Bayesian network, belief network, or directed acyclic graphical model is a probabilistic graphical model that represents a set of random variables and their conditional independence with a directed acyclic graph (DAG). For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases. Efficient algorithms exist that perform inference and learning. Bayesian networks that model sequences of variables, like speech signals or protein sequences, are called dynamic Bayesian networks. Generalizations of Bayesian networks that can represent and solve decision problems under uncertainty are called influence diagrams. Genetic algorithms Main article: Genetic algorithm A genetic algorithm (GA) is a search algorithm and heuristic technique that mimics the process of natural selection, using methods such as mutation and crossover to generate new genotypes in the hope of finding good solutions to a given problem. In machine learning, genetic algorithms were used in the 1980s and 1990s.[71][72] Conversely, machine learning techniques have been used to improve the performance of genetic and evolutionary algorithms.[73] Training models Usually, machine learning models require a lot of data in order for them to perform well. Usually, when training a machine learning model, one needs to collect a large, representative sample of data from a training set. Data from the training set can be as varied as a corpus of text, a collection of images, and data collected from individual users of a service. Overfitting is something to watch out for when training a machine learning model. Federated learning Main article: Federated learning Federated learning is an adapted form of distributed artificial intelligence to training machine learning models that decentralizes the training process, allowing for users' privacy to be maintained by not needing to send their data to a centralized server. This also increases efficiency by decentralizing the training process to many devices. For example, Gboard uses federated machine learning to train search query prediction models on users' mobile phones without having to send individual searches back to Google.[74] Applications There are many applications for machine learning, including: Agriculture Anatomy Adaptive websites Affective computing Banking Bioinformatics Brain–machine interfaces Cheminformatics Citizen science Computer networks Computer vision Credit-card fraud detection Data quality DNA sequence classification Economics Financial market analysis[75] General game playing Handwriting recognition Information retrieval Insurance Internet fraud detection Linguistics Machine learning control Machine perception Machine translation Marketing Medical diagnosis Natural language processing Natural language understanding Online advertising Optimization Recommender systems Robot locomotion Search engines Sentiment analysis Sequence mining Software engineering Speech recognition Structural health monitoring Syntactic pattern recognition Telecommunication Theorem proving Time series forecasting User behavior analytics In 2006, the media-services provider Netflix held the first "Netflix Prize" competition to find a program to better predict user preferences and improve the accuracy of its existing Cinematch movie recommendation algorithm by at least 10%. A joint team made up of researchers from AT&T Labs-Research in collaboration with the teams Big Chaos and Pragmatic Theory built an ensemble model to win the Grand Prize in 2009 for $1 million.[76] Shortly after the prize was awarded, Netflix realized that viewers' ratings were not the best indicators of their viewing patterns ("everything is a recommendation") and they changed their recommendation engine accordingly.[77] In 2010 The Wall Street Journal wrote about the firm Rebellion Research and their use of machine learning to predict the financial crisis.[78] In 2012, co-founder of Sun Microsystems, Vinod Khosla, predicted that 80% of medical doctors' jobs would be lost in the next two decades to automated machine learning medical diagnostic software.[79] In 2014, it was reported that a machine learning algorithm had been applied in the field of art history to study fine art paintings and that it may have revealed previously unrecognized influences among artists.[80] In 2019 Springer Nature published the first research book created using machine learning.[81] Limitations Although machine learning has been transformative in some fields, machine-learning programs often fail to deliver expected results.[82][83][84] Reasons for this are numerous: lack of (suitable) data, lack of access to the data, data bias, privacy problems, badly chosen tasks and algorithms, wrong tools and people, lack of resources, and evaluation problems.[85] In 2018, a self-driving car from Uber failed to detect a pedestrian, who was killed after a collision.[86] Attempts to use machine learning in healthcare with the IBM Watson system failed to deliver even after years of time and billions of dollars invested.[87][88] Bias Main article: Algorithmic bias Machine learning approaches in particular can suffer from different data biases. A machine learning system trained on current customers only may not be able to predict the needs of new customer groups that are not represented in the training data. When trained on man-made data, machine learning is likely to pick up the same constitutional and unconscious biases already present in society.[89] Language models learned from data have been shown to contain human-like biases.[90][91] Machine learning systems used for criminal risk assessment have been found to be biased against black people.[92][93] In 2015, Google photos would often tag black people as gorillas,[94] and in 2018 this still was not well resolved, but Google reportedly was still using the workaround to remove all gorillas from the training data, and thus was not able to recognize real gorillas at all.[95] Similar issues with recognizing non-white people have been found in many other systems.[96] In 2016, Microsoft tested a chatbot that learned from Twitter, and it quickly picked up racist and sexist language.[97] Because of such challenges, the effective use of machine learning may take longer to be adopted in other domains.[98] Concern for fairness in machine learning, that is, reducing bias in machine learning and propelling its use for human good is increasingly expressed by artificial intelligence scientists, including Fei-Fei Li, who reminds engineers that "There’s nothing artificial about AI...It’s inspired by people, it’s created by people, and—most importantly—it impacts people. It is a powerful tool we are only just beginning to understand, and that is a profound responsibility.”[99] Model assessments Classification of machine learning models can be validated by accuracy estimation techniques like the holdout method, which splits the data in a training and test set (conventionally 2/3 training set and 1/3 test set designation) and evaluates the performance of the training model on the test set. In comparison, the K-fold-cross-validation method randomly partitions the data into K subsets and then K experiments are performed each respectively considering 1 subset for evaluation and the remaining K-1 subsets for training the model. In addition to the holdout and cross-validation methods, bootstrap, which samples n instances with replacement from the dataset, can be used to assess model accuracy.[100] In addition to overall accuracy, investigators frequently report sensitivity and specificity meaning True Positive Rate (TPR) and True Negative Rate (TNR) respectively. Similarly, investigators sometimes report the false positive rate (FPR) as well as the false negative rate (FNR). However, these rates are ratios that fail to reveal their numerators and denominators. The total operating characteristic (TOC) is an effective method to express a model's diagnostic ability. TOC shows the numerators and denominators of the previously mentioned rates, thus TOC provides more information than the commonly used receiver operating characteristic (ROC) and ROC's associated area under the curve (AUC).[101] Ethics Machine learning poses a host of ethical questions. Systems which are trained on datasets collected with biases may exhibit these biases upon use (algorithmic bias), thus digitizing cultural prejudices.[102] For example, using job hiring data from a firm with racist hiring policies may lead to a machine learning system duplicating the bias by scoring job applicants against similarity to previous successful applicants.[103][104] Responsible collection of data and documentation of algorithmic rules used by a system thus is a critical part of machine learning. Because human languages contain biases, machines trained on language corpora will necessarily also learn these biases.[105][106] Other forms of ethical challenges, not related to personal biases, are more seen in health care. There are concerns among health care professionals that these systems might not be designed in the public's interest but as income-generating machines. This is especially true in the United States where there is a long-standing ethical dilemma of improving health care, but also increasing profits. For example, the algorithms could be designed to provide patients with unnecessary tests or medication in which the algorithm's proprietary owners hold stakes. There is huge potential for machine learning in health care to provide professionals a great tool to diagnose, medicate, and even plan recovery paths for patients, but this will not happen until the personal biases mentioned previously, and these "greed" biases are addressed.[107] Hardware Since the 2010s, advances in both machine learning algorithms and computer hardware have led to more efficient methods for training deep neural networks (a particular narrow subdomain of machine learning) that contain many layers of non-linear hidden units.[108] By 2019, graphic processing units (GPUs), often with AI-specific enhancements, had displaced CPUs as the dominant method of training large-scale commercial cloud AI.[109] OpenAI estimated the hardware compute used in the largest deep learning projects from AlexNet (2012) to AlphaZero (2017), and found a 300,000-fold increase in the amount of compute required, with a doubling-time trendline of 3.4 months.[110][111] Software Software suites containing a variety of machine learning algorithms include the following: Free and open-source so
attaoveisi / AFISMCa new observer-based adaptive fuzzy integral sliding mode controller (AFISMC) is proposed based on the Lyapunov stability theorem. The plant under study is subjected to a square-integrable disturbance and is assumed to have mismatch uncertainties both in state- and input-matrices. In addition, a norm-bounded time varying term is introduced to address the possible existence of un-modelled/nonlinear dynamics. Based on the classical sliding mode controller (SMC), the equivalent control effort is obtained to satisfy the sufficient requirement of SMC and then the control law is modified to guarantee the reachability of the system trajectory to the sliding manifold. The sliding surface is compensated based on the observed states in the form of linear matrix inequality (LMI). In order to relax the norm-bounded constrains on the control law and solve the chattering problem of SMC, a fuzzy logic (FL) inference mechanism is combined with the controller. An adaptive law is then introduced to tune the parameters of the fuzzy system on-line. Finally, by aiming at evaluating the validity of the controller and the robust performance of the closed-loop system, the proposed regulator is implemented on a real-time mechanical vibrating system.
wo80 / CSparse.NETA concise library for solving sparse linear systems with direct methods.
ShelvanLee / XFEM# XFEM_Fracture2D ### Description This is a Matlab program that can be used to solve fracture problems involving arbitrary multiple crack propagations in a 2D linear-elastic solid based on the principle of minimum potential energy. The extended finite element method is used to discretise the solid continuum considering cracks as discontinuities in the displacement field. To this end, a strong discontinuity enrichment and a square-root singular crack tip enrichment are used to describe each crack. Several crack growth criteria are available to determine the evolution of cracks over time; apart from the classic maximum tension (or hoop-stress) criterion, the minimum total energy criterion and the local symmetry criterion are implemented implicitly with respect to the discrete time-stepping. ### Key features * *Fast:* The stiffness matrix and the force vector (i.e. the equations' system) and the enrichment tracking data structures are updated at each time step only with respect to the changes in the fracture topology. This ultimately results in the major part of the computational expense in the solution to the linear system of equations rather than in the post-processing of the solution or in the assembly and updating of the equations. As Matlab offers fast and robust direct solvers, the computational times are reasonably fast. * *Robust.* Suitable for multiple crack propagations with intersections. Furthermore, the stress intensity factors are computed robustly via the interaction integral approach (with the inclusion of the terms to account for crack surface pressure, residual stresses or strains). The minimum total energy criterion and the principle of local symmetry are implemented implicitly in time. The energy release rates are computed based on the stiffness derivative approach using algebraic differentiation (rather than finite differencing of the potential energy). On the other hand, the crack growth direction based on the local symmetry criterion is determined such that the local mode-II stress intensity factor vanishes; the change in a crack tip kink angle is approximated using the ratio of the crack tip stress intensity factors. * *Easy to run.* Each job has its own input files which are independent form those of all other jobs. The code especially lends itself to running parametric studies. Various results can be saved relating to the fracture geometry, fracture mechanics parameters, and the elastic fields in the solid domain. Extensive visualisation library is available for plotting results. ### Instructions 1. Get started by running the demo to showcase some of the capabilities of the program and to determine if it can be useful for you. At the Matlab's command line enter: ```Matlab >> RUN_JOBS.m ``` This will execute a series of jobs located inside the *jobs directory* `./JOBS_LIBRARY/`. These jobs do not take very long to execute (around 5 minutes in total). 2. Subsequently, you can pick one of the jobs inside `./JOBS_LIBRARY/` by defining the job title: ```Matlab >> job_title = 'several_cracks/edge/vertical_tension' ``` 3. Then you can open all the relevant scripts for this job as follows: ```Matlab >> open_job ``` The following input scripts for the *job* will be open in the Matlab's editor: 1. `JOB_MAIN.m`: This is the job's main script. It is called when executing `RUN_JOB` (or `RUN_JOBS`) and acts like a wrapper. Notably, it can serve as a convenient interface to run parametric studies and to save intermediate simulation results. 2. `Input_Scope.m`: This defines the scope of the simulation. From which crack growth criteria to use, to what to compute and what results to show via plots and/or movies. To put it simply, the script is a bunch of "switches" that tell the program what the user wants to be done. 3. `Input_Material.m`: Defines the material's elastic properties in different regions or layers (called "phases") of the computational domain. Moreover, it defines the fracture toughness of the material (assumed to be constant in all material phases). 4. `Input_Crack.m`: Defines the initial crack geometry. 5. `Input_BC.m`: Defines boundary conditions, such as displacements, tractions, crack surface pressure (assumed to be constant in all cracks), body loads (e.g. gravity, pre-stress or pre-strain). 6. `Mesh_make.m`: In-house structured mesh generator for rectangular domains using either linear triangle or bilinear quadrilateral elements. It is possible to mesh horizontal layers using different mesh sizes. 7. `Mesh_read.m`: Gmsh based mesh reader for version-1 mesh files. Of course you can use your own mesh reader provided the output variables are of the correct format (see later). 8. `Mesh_file.m`: Specifies the mesh input file (.msh). At the moment, only Gmsh mesh files of version-1 are allowed. ### Mesh_file.m A mesh file needs to be able to output the following data or variables: * `mNdCrd`: Node coordinates, size = `[nNdStd, 2]` * `mLNodS`: Element connectivities, size = `[nElemn,nLNodS]` * `vElPhz`: Element material phase (or region) ID's, size = `[nElemn,1]` * `cBCNod`: cell of boundary nodes, cell size = `{nBound,1}`, cell element size = `[nBnNod,2]` Example mesh files are located in `./JOBS_LIBRARY/`. Gmsh version-1 file format is described [here](http://www.manpagez.com/info/gmsh/gmsh-2.4.0/gmsh_60.php). ### Additional notes * global variables are defined in `.\Routines_AuxInput\Declare_Global.m` * External libraries are `.\Other_Libs\distmesh` and `.\Other_Libs\mesh2d` ### References Two external meshing libraries are used for the local mesh refinement and remeshing at the crack tip during crack propagation or prior to a crack intersection with another crack or with a boundary of the domain. Specifically, these libraries, which are located in `.\Other_Libs\`, are the following: * [*mesh2d*](https://people.sc.fsu.edu/~jburkardt/m_src/mesh2d/mesh2d.html) by Darren Engwirda * [*distmesh*](http://persson.berkeley.edu/distmesh/) by Per-Olof Persson and Gilbert Strang. ### Issues and Support For support or questions please email [sutula.danas@gmail.com](mailto:sutula.danas@gmail.com). ### Authors Danas Sutula, University of Luxembourg, Luxembourg. If you find this code useful, we kindly ask that you consider citing us. * [Minimum energy multiple crack propagation](http://hdl.handle.net/10993/29414)
bjodah / PyneqsysSolve symbolically defined systems of non-linear equations numerically.
datamole-ai / GomezFramework and implementation for mathematical optimization and solving non-linear systems of equations.
shayneobrien / Numerical MethodsMethods in numerical analysis. Includes: Lagrange interpolation, Chebyshev polynomials for optimal node spacing, iterative techniques to solve linear systems (Gauss-Seidel, Jacobi, SOR), SVD, PCA, and more.
