Numerical integration based on the hyperfunction theoryHidenoriOgata
The document discusses a numerical integration method based on the hyperfunction theory. The method represents integrals, including those with singularities, as contour integrals in the complex plane. For integrals over a finite interval, the contour integral is approximated using the trapezoidal rule. For integrals over an infinite interval, the contour is parameterized and the integral is evaluated as an infinite sum, which is accelerated using the DE transform. The method is highly accurate due to the geometric convergence of the trapezoidal rule for analytic functions.
Bregman divergences from comparative convexityFrank Nielsen
This document discusses generalized divergences and comparative convexity. It introduces Jensen divergences, Bregman divergences, and their generalizations to quasi-arithmetic and weighted means. Quasi-arithmetic Bregman divergences are defined for strictly (ρ,τ)-convex functions using two strictly monotone functions ρ and τ. Power mean Bregman divergences are obtained as a subfamily when ρ(x)=xδ1 and τ(x)=xδ2. A criterion is given to check (ρ,τ)-convexity by testing the ordinary convexity of the transformed function G=Fρ,τ.
This document discusses deep generative models including variational autoencoders (VAEs) and generational adversarial networks (GANs). It explains that generative models learn the distribution of input data and can generate new samples from that distribution. VAEs use variational inference to learn a latent space and generate new data by varying the latent variables. The document outlines the key concepts of VAEs including the evidence lower bound objective used for training and how it maximizes the likelihood of the data.
An application of the hyperfunction theory to numerical integrationHidenoriOgata
The slide of a speech in the conference "ECMI2016" (The 19th European Conference on Mathematics for Industry) held at Santiago de Compostela, Spain in June 2016.
Numerical integration based on the hyperfunction theoryHidenoriOgata
The document discusses a numerical integration method based on the hyperfunction theory. The method represents integrals, including those with singularities, as contour integrals in the complex plane. For integrals over a finite interval, the contour integral is approximated using the trapezoidal rule. For integrals over an infinite interval, the contour is parameterized and the integral is evaluated as an infinite sum, which is accelerated using the DE transform. The method is highly accurate due to the geometric convergence of the trapezoidal rule for analytic functions.
Bregman divergences from comparative convexityFrank Nielsen
This document discusses generalized divergences and comparative convexity. It introduces Jensen divergences, Bregman divergences, and their generalizations to quasi-arithmetic and weighted means. Quasi-arithmetic Bregman divergences are defined for strictly (ρ,τ)-convex functions using two strictly monotone functions ρ and τ. Power mean Bregman divergences are obtained as a subfamily when ρ(x)=xδ1 and τ(x)=xδ2. A criterion is given to check (ρ,τ)-convexity by testing the ordinary convexity of the transformed function G=Fρ,τ.
This document discusses deep generative models including variational autoencoders (VAEs) and generational adversarial networks (GANs). It explains that generative models learn the distribution of input data and can generate new samples from that distribution. VAEs use variational inference to learn a latent space and generate new data by varying the latent variables. The document outlines the key concepts of VAEs including the evidence lower bound objective used for training and how it maximizes the likelihood of the data.
An application of the hyperfunction theory to numerical integrationHidenoriOgata
The slide of a speech in the conference "ECMI2016" (The 19th European Conference on Mathematics for Industry) held at Santiago de Compostela, Spain in June 2016.
Probability formula sheet
Set theory, sample space, events, concepts of randomness and uncertainty, basic principles of probability, axioms and properties of probability, conditional probability, independent events, Baye’s formula, Bernoulli trails, sequential experiments, discrete and continuous random variable, distribution and density functions, one and two dimensional random variables, marginal and joint distributions and density functions. Expectations, probability distribution families (binomial, poisson, hyper geometric, geometric distribution, normal, uniform and exponential), mean, variance, standard deviations, moments and moment generating functions, law of large numbers, limits theorems
for more visit http://tricntip.blogspot.com/
This document discusses multilinear twisted paraproducts, which are generalizations of classical paraproduct operators to higher dimensions. It begins by reviewing classical paraproducts on the real line and their generalization to higher dimensions using dyadic squares. It then discusses complications that arise, such as twisted paraproducts. The document presents a unified framework for studying such operators using bipartite graphs and selections of vertices. It proves a main boundedness result and discusses special cases like classical dyadic paraproducts and dyadic twisted paraproducts. It introduces tools like Bellman functions and calculus of finite differences to analyze estimates for paraproduct-like operators on finite trees of dyadic squares.
Proximal Splitting and Optimal TransportGabriel Peyré
This document summarizes proximal splitting and optimal transport methods. It begins with an overview of topics including optimal transport and imaging, convex analysis, and various proximal splitting algorithms. It then discusses measure-preserving maps between distributions and defines the optimal transport problem. Finally, it presents formulations for optimal transport including the convex Benamou-Brenier formulation and discrete formulations on centered and staggered grids. Numerical examples of optimal transport between distributions on 2D domains are also shown.
This document provides a probability cheatsheet compiled by William Chen and Joe Blitzstein with contributions from others. It is licensed under CC BY-NC-SA 4.0 and contains information on topics like counting rules, probability definitions, random variables, expectations, independence, and more. The cheatsheet is designed to summarize essential concepts in probability.
The document discusses exponential decay of solutions to a second-order linear differential equation involving a self-adjoint positive operator A and an accretive damping operator D. Several theorems establish conditions under which the associated operator semigroup or pencil generates exponential decay. If D is accretive and satisfies certain positivity conditions, the semigroup will decay exponentially. Explicit bounds on the rate of decay and estimates of the spectrum are provided depending on properties of A and D.
1) The document discusses probit transformation for nonparametric kernel estimation of copulas. It introduces a standard kernel estimator for copulas that is inconsistent on boundaries.
2) It then presents a "naive" probit transformation kernel copula density estimator that transforms data to standard normal using the probit function to address boundary issues.
3) It further improves upon this by introducing local log-linear and log-quadratic approximations for the transformed density, yielding two new estimators with better asymptotic properties.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
This document discusses using the Wasserstein distance for inference in generative models. It begins with an overview of approximate Bayesian computation (ABC) and how distances between samples are used. It then introduces the Wasserstein distance as an alternative distance that can have lower variance than the Euclidean distance. Computational aspects and asymptotics of using the Wasserstein distance are discussed. The document also covers how transport distances can handle time series data.
Classification with mixtures of curved Mahalanobis metricsFrank Nielsen
This document discusses curved Mahalanobis distances in Cayley-Klein geometries and their application to classification. Specifically:
1. It introduces Mahalanobis distances and generalizes them to curved distances in Cayley-Klein geometries, which can model both elliptic and hyperbolic geometries.
2. It describes how to learn these curved Mahalanobis metrics using an adaptation of Large Margin Nearest Neighbors (LMNN) to the elliptic and hyperbolic cases.
3. Experimental results on several datasets show that curved Mahalanobis distances can achieve comparable or better classification accuracy than standard Mahalanobis distances.
The document describes Approximate Bayesian Computation (ABC), a technique for performing Bayesian inference when the likelihood function is intractable or impossible to evaluate directly. ABC works by simulating data under different parameter values, and accepting simulations that are close to the observed data according to a distance measure and tolerance level. ABC provides an approximation to the posterior distribution that improves as the tolerance level decreases and more informative summary statistics are used. The document discusses the ABC algorithm, properties of the exact ABC posterior distribution, and challenges in selecting appropriate summary statistics.
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operatorsVjekoslavKovac1
This document summarizes a talk given by Vjekoslav Kovač at a joint mathematics conference. The talk concerned establishing T(1)-type theorems for entangled multilinear Calderón-Zygmund operators. Specifically, Kovač discussed studying multilinear singular integral forms where the functions partially share variables, known as an "entangled structure." He outlined establishing generalized modulation invariance and Lp estimates for such operators. The talk motivated further studying related problems involving bilinear ergodic averages and forms with more complex graph structures. Kovač specialized his techniques to bipartite graphs, multilinear Calderón-Zygmund kernels, and "perfect" dyadic models.
The document discusses probability distributions and their natural parameters. It provides examples of several common distributions including the Bernoulli, multinomial, Gaussian, and gamma distributions. For each distribution, it derives the natural parameter representation and shows how to write the distribution in the form p(x|η) = h(x)g(η)exp{η^T μ(x)}. Maximum likelihood estimation for these distributions is also briefly discussed.
Multiple estimators for Monte Carlo approximationsChristian Robert
This document discusses multiple estimators that can be used to approximate integrals using Monte Carlo simulations. It begins by introducing concepts like multiple importance sampling, Rao-Blackwellisation, and delayed acceptance that allow combining multiple estimators to improve accuracy. It then discusses approaches like mixtures as proposals, global adaptation, and nonparametric maximum likelihood estimation (NPMLE) that frame Monte Carlo estimation as a statistical estimation problem. The document notes various advantages of the statistical formulation, like the ability to directly estimate simulation error from the Fisher information. Overall, the document presents an overview of different techniques for combining Monte Carlo simulations to obtain more accurate integral approximations.
The dual geometry of Shannon informationFrank Nielsen
The document discusses the dual geometry of Shannon information. It covers:
1. Shannon entropy and related concepts like maximum entropy principle and exponential families.
2. The properties of Kullback-Leibler divergence including its interpretation as a statistical distance and relation to maximum entropy.
3. How maximum likelihood estimation for exponential families can be viewed as minimizing Kullback-Leibler divergence between the empirical distribution and model distribution.
Murphy: Machine learning A probabilistic perspective: Ch.9Daisuke Yoneoka
This document summarizes key concepts about the exponential family and generalized linear models (GLMs). It defines the exponential family and provides examples like the Bernoulli, multinomial, and Gaussian distributions. The exponential family has important properties like finite sufficient statistics, existence of conjugate priors, and convexity. Maximum likelihood estimation for the exponential family involves matching sample moments to population moments. Conjugate priors allow tractable Bayesian inference for the exponential family. The document outlines maximum entropy derivation of the exponential family and how GLMs can generate classifiers.
Poster for Bayesian Statistics in the Big Data Era conferenceChristian Robert
The document proposes a new version of Hamiltonian Monte Carlo (HMC) sampling that is essentially calibration-free. It achieves this by learning the optimal leapfrog scale from the distribution of integration times using the No-U-Turn Sampler algorithm. Compared to the original NUTS algorithm on benchmark models, this new enhanced HMC (eHMC) exhibits significantly improved efficiency with no hand-tuning of parameters required. The document tests eHMC on a Susceptible-Infected-Recovered model of disease transmission.
Maximum likelihood estimation of regularisation parameters in inverse problem...Valentin De Bortoli
This document discusses an empirical Bayesian approach for estimating regularization parameters in inverse problems using maximum likelihood estimation. It proposes the Stochastic Optimization with Unadjusted Langevin (SOUL) algorithm, which uses Markov chain sampling to approximate gradients in a stochastic projected gradient descent scheme for optimizing the regularization parameter. The algorithm is shown to converge to the maximum likelihood estimate under certain conditions on the log-likelihood and prior distributions.
We start with motivation, few examples of uncertainties. Then we discretize elliptic PDE with uncertain coefficients, apply TT format for permeability, the stochastic operator and for the solution. We compare sparse multi-index set approach with full multi-index+TT.
Tensor Train format allows us to keep the whole multi-index set, without any multi-index set truncation.
Probability formula sheet
Set theory, sample space, events, concepts of randomness and uncertainty, basic principles of probability, axioms and properties of probability, conditional probability, independent events, Baye’s formula, Bernoulli trails, sequential experiments, discrete and continuous random variable, distribution and density functions, one and two dimensional random variables, marginal and joint distributions and density functions. Expectations, probability distribution families (binomial, poisson, hyper geometric, geometric distribution, normal, uniform and exponential), mean, variance, standard deviations, moments and moment generating functions, law of large numbers, limits theorems
for more visit http://tricntip.blogspot.com/
This document discusses multilinear twisted paraproducts, which are generalizations of classical paraproduct operators to higher dimensions. It begins by reviewing classical paraproducts on the real line and their generalization to higher dimensions using dyadic squares. It then discusses complications that arise, such as twisted paraproducts. The document presents a unified framework for studying such operators using bipartite graphs and selections of vertices. It proves a main boundedness result and discusses special cases like classical dyadic paraproducts and dyadic twisted paraproducts. It introduces tools like Bellman functions and calculus of finite differences to analyze estimates for paraproduct-like operators on finite trees of dyadic squares.
Proximal Splitting and Optimal TransportGabriel Peyré
This document summarizes proximal splitting and optimal transport methods. It begins with an overview of topics including optimal transport and imaging, convex analysis, and various proximal splitting algorithms. It then discusses measure-preserving maps between distributions and defines the optimal transport problem. Finally, it presents formulations for optimal transport including the convex Benamou-Brenier formulation and discrete formulations on centered and staggered grids. Numerical examples of optimal transport between distributions on 2D domains are also shown.
This document provides a probability cheatsheet compiled by William Chen and Joe Blitzstein with contributions from others. It is licensed under CC BY-NC-SA 4.0 and contains information on topics like counting rules, probability definitions, random variables, expectations, independence, and more. The cheatsheet is designed to summarize essential concepts in probability.
The document discusses exponential decay of solutions to a second-order linear differential equation involving a self-adjoint positive operator A and an accretive damping operator D. Several theorems establish conditions under which the associated operator semigroup or pencil generates exponential decay. If D is accretive and satisfies certain positivity conditions, the semigroup will decay exponentially. Explicit bounds on the rate of decay and estimates of the spectrum are provided depending on properties of A and D.
1) The document discusses probit transformation for nonparametric kernel estimation of copulas. It introduces a standard kernel estimator for copulas that is inconsistent on boundaries.
2) It then presents a "naive" probit transformation kernel copula density estimator that transforms data to standard normal using the probit function to address boundary issues.
3) It further improves upon this by introducing local log-linear and log-quadratic approximations for the transformed density, yielding two new estimators with better asymptotic properties.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
This document discusses using the Wasserstein distance for inference in generative models. It begins with an overview of approximate Bayesian computation (ABC) and how distances between samples are used. It then introduces the Wasserstein distance as an alternative distance that can have lower variance than the Euclidean distance. Computational aspects and asymptotics of using the Wasserstein distance are discussed. The document also covers how transport distances can handle time series data.
Classification with mixtures of curved Mahalanobis metricsFrank Nielsen
This document discusses curved Mahalanobis distances in Cayley-Klein geometries and their application to classification. Specifically:
1. It introduces Mahalanobis distances and generalizes them to curved distances in Cayley-Klein geometries, which can model both elliptic and hyperbolic geometries.
2. It describes how to learn these curved Mahalanobis metrics using an adaptation of Large Margin Nearest Neighbors (LMNN) to the elliptic and hyperbolic cases.
3. Experimental results on several datasets show that curved Mahalanobis distances can achieve comparable or better classification accuracy than standard Mahalanobis distances.
The document describes Approximate Bayesian Computation (ABC), a technique for performing Bayesian inference when the likelihood function is intractable or impossible to evaluate directly. ABC works by simulating data under different parameter values, and accepting simulations that are close to the observed data according to a distance measure and tolerance level. ABC provides an approximation to the posterior distribution that improves as the tolerance level decreases and more informative summary statistics are used. The document discusses the ABC algorithm, properties of the exact ABC posterior distribution, and challenges in selecting appropriate summary statistics.
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operatorsVjekoslavKovac1
This document summarizes a talk given by Vjekoslav Kovač at a joint mathematics conference. The talk concerned establishing T(1)-type theorems for entangled multilinear Calderón-Zygmund operators. Specifically, Kovač discussed studying multilinear singular integral forms where the functions partially share variables, known as an "entangled structure." He outlined establishing generalized modulation invariance and Lp estimates for such operators. The talk motivated further studying related problems involving bilinear ergodic averages and forms with more complex graph structures. Kovač specialized his techniques to bipartite graphs, multilinear Calderón-Zygmund kernels, and "perfect" dyadic models.
The document discusses probability distributions and their natural parameters. It provides examples of several common distributions including the Bernoulli, multinomial, Gaussian, and gamma distributions. For each distribution, it derives the natural parameter representation and shows how to write the distribution in the form p(x|η) = h(x)g(η)exp{η^T μ(x)}. Maximum likelihood estimation for these distributions is also briefly discussed.
Multiple estimators for Monte Carlo approximationsChristian Robert
This document discusses multiple estimators that can be used to approximate integrals using Monte Carlo simulations. It begins by introducing concepts like multiple importance sampling, Rao-Blackwellisation, and delayed acceptance that allow combining multiple estimators to improve accuracy. It then discusses approaches like mixtures as proposals, global adaptation, and nonparametric maximum likelihood estimation (NPMLE) that frame Monte Carlo estimation as a statistical estimation problem. The document notes various advantages of the statistical formulation, like the ability to directly estimate simulation error from the Fisher information. Overall, the document presents an overview of different techniques for combining Monte Carlo simulations to obtain more accurate integral approximations.
The dual geometry of Shannon informationFrank Nielsen
The document discusses the dual geometry of Shannon information. It covers:
1. Shannon entropy and related concepts like maximum entropy principle and exponential families.
2. The properties of Kullback-Leibler divergence including its interpretation as a statistical distance and relation to maximum entropy.
3. How maximum likelihood estimation for exponential families can be viewed as minimizing Kullback-Leibler divergence between the empirical distribution and model distribution.
Murphy: Machine learning A probabilistic perspective: Ch.9Daisuke Yoneoka
This document summarizes key concepts about the exponential family and generalized linear models (GLMs). It defines the exponential family and provides examples like the Bernoulli, multinomial, and Gaussian distributions. The exponential family has important properties like finite sufficient statistics, existence of conjugate priors, and convexity. Maximum likelihood estimation for the exponential family involves matching sample moments to population moments. Conjugate priors allow tractable Bayesian inference for the exponential family. The document outlines maximum entropy derivation of the exponential family and how GLMs can generate classifiers.
Poster for Bayesian Statistics in the Big Data Era conferenceChristian Robert
The document proposes a new version of Hamiltonian Monte Carlo (HMC) sampling that is essentially calibration-free. It achieves this by learning the optimal leapfrog scale from the distribution of integration times using the No-U-Turn Sampler algorithm. Compared to the original NUTS algorithm on benchmark models, this new enhanced HMC (eHMC) exhibits significantly improved efficiency with no hand-tuning of parameters required. The document tests eHMC on a Susceptible-Infected-Recovered model of disease transmission.
Maximum likelihood estimation of regularisation parameters in inverse problem...Valentin De Bortoli
This document discusses an empirical Bayesian approach for estimating regularization parameters in inverse problems using maximum likelihood estimation. It proposes the Stochastic Optimization with Unadjusted Langevin (SOUL) algorithm, which uses Markov chain sampling to approximate gradients in a stochastic projected gradient descent scheme for optimizing the regularization parameter. The algorithm is shown to converge to the maximum likelihood estimate under certain conditions on the log-likelihood and prior distributions.
We start with motivation, few examples of uncertainties. Then we discretize elliptic PDE with uncertain coefficients, apply TT format for permeability, the stochastic operator and for the solution. We compare sparse multi-index set approach with full multi-index+TT.
Tensor Train format allows us to keep the whole multi-index set, without any multi-index set truncation.
First principle, power rule, derivative of constant term, product rule, quotient rule, chain rule, derivatives of trigonometric functions and their inverses, derivatives of exponential functions and natural logarithmic functions, implicit differentiation, parametric differentiation, L'Hopital's rule
Image sciences, image processing, image restoration, photo manipulation. Image and videos representation. Digital versus analog imagery. Quantization and sampling. Sources and models of noises in digital CCD imagery: photon, thermal and readout noises. Sources and models of blurs. Convolutions and point spread functions. Overview of other standard models, problems and tasks: salt-and-pepper and impulse noises, half toning, inpainting, super-resolution, compressed sensing, high dynamic range imagery, demosaicing. Short introduction to other types of imagery: SAR, Sonar, ultrasound, CT and MRI. Linear and ill-posed restoration problems.
This document discusses antiderivatives and indefinite integrals. It begins by introducing the concept of an antiderivative, which is a function whose derivative is a known function. It then defines the indefinite integral as representing the set of all antiderivatives. Several properties of antiderivatives and indefinite integrals are presented, including: the constant of integration; basic integration rules like power, exponential, and logarithmic rules; and notation used to represent indefinite integrals. Examples are provided to illustrate key concepts and properties.
Here the interest is mainly to compute characterisations like the entropy,
the Kullback-Leibler divergence, more general $f$-divergences, or other such characteristics based on
the probability density. The density is often not available directly,
and it is a computational challenge to just represent it in a numerically
feasible fashion in case the dimension is even moderately large. It
is an even stronger numerical challenge to then actually compute said characteristics
in the high-dimensional case.
The task considered here was the numerical computation of characterising statistics of
high-dimensional pdfs, as well as their divergences and distances,
where the pdf in the numerical implementation was assumed discretised on some regular grid.
We have demonstrated that high-dimensional pdfs,
pcfs, and some functions of them
can be approximated and represented in a low-rank tensor data format.
Utilisation of low-rank tensor techniques helps to reduce the computational complexity
and the storage cost from exponential $\C{O}(n^d)$ to linear in the dimension $d$, e.g.\
$O(d n r^2 )$ for the TT format. Here $n$ is the number of discretisation
points in one direction, $r<<n$ is the maximal tensor rank, and $d$ the problem dimension.
This document proposes a linear programming (LP) based approach for solving maximum a posteriori (MAP) estimation problems on factor graphs that contain multiple-degree non-indicator functions. It presents an existing LP method for problems with single-degree functions, then introduces a transformation to handle multiple-degree functions by introducing auxiliary variables. This allows applying the existing LP method. As an example, it applies this to maximum likelihood decoding for the Gaussian multiple access channel. Simulation results demonstrate the LP approach decodes correctly with polynomial complexity.
This document discusses probability density functions (pdfs) and how they relate to probability distribution functions. It provides examples of common pdfs like the uniform and Gaussian distributions. The Gaussian or normal distribution is described in more detail. The document also discusses how to determine the pdf of a random variable that is a function of another random variable, whether the function is monotonic or non-monotonic. Key aspects like changing of variables in integrals and combining probabilities for multiple values are addressed.
The document discusses various types of integrals and rules for finding antiderivatives. It defines definite and indefinite integrals. It then lists and explains the main antiderivative rules for powers, chain rule, product rule, quotient rule, scalar multiples, sums and differences, trigonometric functions, and inverse trigonometric functions. Examples are provided to illustrate each rule.
The document discusses functions and evaluating functions. It provides examples of determining if a given equation is a function using the vertical line test and evaluating functions by substituting values into the function equation. It also includes examples of evaluating composite functions using flow diagrams to illustrate the steps of evaluating each individual function.
The document discusses convex functions and related concepts. It defines convex functions and provides examples of convex and concave functions on R and Rn, including norms, logarithms, and powers. It describes properties that preserve convexity, such as positive weighted sums and composition with affine functions. The conjugate function and quasiconvex functions are also introduced. Key concepts are illustrated with examples throughout.
This document summarizes some statistical models used for calibrating imperfect mathematical models. It discusses three main approaches:
1. Gaussian stochastic process (GaSP) calibration, which models bias as a Gaussian process. This is commonly used but can produce inconsistent parameter estimates.
2. L2 calibration, which estimates reality separately from the model before estimating parameters. However, it does not use model information.
3. Scaled Gaussian stochastic process (S-GaSP) calibration, which constrains the GaSP to have a fixed L2 norm. This satisfies predicting reality and calibrated parameters. The S-GaSP is equivalent to penalized kernel ridge regression.
The document analyzes the nonparametric regression setting
This document discusses backpropagation in convolutional neural networks. It begins by explaining backpropagation for single neurons and multi-layer neural networks. It then discusses the specific operations involved in convolutional and pooling layers, and how backpropagation is applied to convolutional neural networks as a composite function with multiple differentiable operations. The key steps are decomposing the network into differentiable operations, propagating error signals backward using derivatives, and computing gradients to update weights.
SOLVING BVPs OF SINGULARLY PERTURBED DISCRETE SYSTEMSTahia ZERIZER
In this article, we study boundary value problems of a large
class of non-linear discrete systems at two-time-scales. Algorithms are given to implement asymptotic solutions for any order of approximation.
S1. Fixed point iteration is a numerical method for solving equations of the form x = g(x) by making an initial guess x0 and repeatedly substituting xn into the right side to obtain xn+1.
S2. The method converges if |g'(α)| < 1, where α is the root and g' is the derivative of g. This ensures the error decreases at each iteration.
S3. Examples show the method can converge rapidly, as in Newton's method, or diverge, depending on the properties of g near the root. Aitken extrapolation can provide a better estimate of the root than the current iterate xn.
S1. Fixed point iteration is a numerical method for solving equations of the form x = g(x) by making an initial guess x0 and repeatedly substituting xn into the right side to obtain xn+1.
S2. The method converges if g(x) is continuous and λ, the maximum absolute value of the derivative of g(x), is less than 1.
S3. Examples show that fixed point iteration can converge slowly if the derivative of g(x) at the root is close to 1, and Aitken's method can be used to accelerate convergence by extrapolating the iterates.
This document discusses derivatives of various functions including:
- Exponential functions like ex and ax where the derivative of ex is ex and the derivative of ax is axln(a)
- Inverse functions where the derivative of the inverse is the reciprocal of the derivative of the original function
- Logarithmic functions like ln(x), loga(x) where the derivatives are 1/x and 1/(xln(a))
- Using logarithmic differentiation to find derivatives of functions like f(x)g(x)
It also provides practice problems finding derivatives of various functions and solving related equations.
A Szemeredi-type theorem for subsets of the unit cubeVjekoslavKovac1
This document summarizes a talk on gaps between arithmetic progressions in subsets of the unit cube. It presents three key propositions:
1) For subsets A of positive measure, structured progressions contribute a lower bound depending on the measure of A and the best known bounds for Szemerédi's theorem.
2) Estimating errors by pigeonholing scales, the difference between smooth and sharp progressions over various scales is bounded above by a sublinear function of scales.
3) For sufficiently nice subsets, the difference between measure and smoothed measure is arbitrarily small by choosing a small smoothing parameter.
Combining these propositions shows that for sufficiently nice subsets, gaps between progressions contain an interval
The document discusses the chain rule for derivatives. It begins by defining function composition and provides examples of composing linear functions. It then states the chain rule theorem, which says that the derivative of a composition is the product of the individual function derivatives evaluated at the same point. Several examples are worked out applying the chain rule to find the derivative of various compositions of functions.
Similar to Hyperfunction method for numerical integration and Fredholm integral equations of the second kind (20)
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...University of Maribor
Slides from talk presenting:
Aleš Zamuda: Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapter and Networking.
Presentation at IcETRAN 2024 session:
"Inter-Society Networking Panel GRSS/MTT-S/CIS
Panel Session: Promoting Connection and Cooperation"
IEEE Slovenia GRSS
IEEE Serbia and Montenegro MTT-S
IEEE Slovenia CIS
11TH INTERNATIONAL CONFERENCE ON ELECTRICAL, ELECTRONIC AND COMPUTING ENGINEERING
3-6 June 2024, Niš, Serbia
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
Electric vehicle and photovoltaic advanced roles in enhancing the financial p...IJECEIAES
Climate change's impact on the planet forced the United Nations and governments to promote green energies and electric transportation. The deployments of photovoltaic (PV) and electric vehicle (EV) systems gained stronger momentum due to their numerous advantages over fossil fuel types. The advantages go beyond sustainability to reach financial support and stability. The work in this paper introduces the hybrid system between PV and EV to support industrial and commercial plants. This paper covers the theoretical framework of the proposed hybrid system including the required equation to complete the cost analysis when PV and EV are present. In addition, the proposed design diagram which sets the priorities and requirements of the system is presented. The proposed approach allows setup to advance their power stability, especially during power outages. The presented information supports researchers and plant owners to complete the necessary analysis while promoting the deployment of clean energy. The result of a case study that represents a dairy milk farmer supports the theoretical works and highlights its advanced benefits to existing plants. The short return on investment of the proposed approach supports the paper's novelty approach for the sustainable electrical system. In addition, the proposed system allows for an isolated power setup without the need for a transmission line which enhances the safety of the electrical network
Introduction- e - waste – definition - sources of e-waste– hazardous substances in e-waste - effects of e-waste on environment and human health- need for e-waste management– e-waste handling rules - waste minimization techniques for managing e-waste – recycling of e-waste - disposal treatment methods of e- waste – mechanism of extraction of precious metal from leaching solution-global Scenario of E-waste – E-waste in India- case studies.
Hyperfunction method for numerical integration and Fredholm integral equations of the second kind
1. 1 / 23
Hyperfunction Method for Numerical Integration
and Fredholm Integral Equations of the Second Kind
Hidenori Ogata
The University of Electro-Communications, Japan
13 July, 2017
2. Aim of this study
2 / 23
Hyperfunction theory (M. Sato, 1958)✓ ✏
• A theory of generalized functions based on complex function theory.
• A “hyperfunction” is expressed in terms of complex analytic functions.
hyperfunctions
= functions with singularities
pole
discontinuity
delta impluse, ...
←−
complex analytic function
easy to treat
numerically
✒ ✑
In this talk, we propose hyperfunction methods for
• numerical integration
• Fredholm integral equations of the second kind.
3. Contents
3 / 23
1. Hyperfunction thoery
2. Hyperfunction method for numerical integration
3. Hyperfunction method for Fredholm integral equations
4. Summary
4. Contents
4 / 23
1. Hyperfunction thoery
2. Hyperfunction method for numerical integration
3. Hyperfunction method for Fredholm integral equations
4. Summary
5. 1. Hyperfunction theory
5 / 23
Hyperfunction theory (M. Sato, 1958)✓ ✏
• hyperfunction on an interval I
. . . the difference between the values of a complex analytic funtion F(z) on I
f(x) = [F(z)] ≡ F(x + i0) − F(x − i0).
F(z) : defining function of the hyperfunction f(x)
analytic in D I, where D is a complex neighborhood of I
✒ ✑
D
I
F(z)
=Re z
m z
7. 1. Hyperfunctions: examples
6 / 23
Dirac’s delta function
δ(x) = −
1
2πi
1
x + i0
−
1
x − i0
.
O
D
a b
C
+ǫ
−ǫ
Suppose that φ(z) is analytic in D. By Cauchy’s integral formula,
φ(0) =
b
a
φ(x)δ(x)dx = −
1
2πi
b
a
φ(x)
1
x + i0
−
1
x − i0
dx.
8. 1. Hyperfunctions: examples
6 / 23
Dirac’s delta function
δ(x) = −
1
2πi
1
x + i0
−
1
x − i0
.
O
D
a b
C
+ǫ
−ǫ
Suppose that φ(z) is analytic in D. By Cauchy’s integral formula,
φ(0) =
b
a
φ(x)δ(x)dx = −
1
2πi
b
a
φ(x)
1
x + i0
−
1
x − i0
dx.
9. 1. Hyperfunction: examples
7 / 23
Heaviside step function
H(x) =
1 ( x > 0 )
0 ( x < 0 )
= F(x + i0) − F(x − i0), F(z) = −
1
2πi
log(−z).
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1Re z
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Im z
-1
-0.5
0
0.5
1
Re F(z)
The real part of F(z) = −
1
2πi
log(−z).
10. 1. Hyperfunction theory: integral
8 / 23
integral of a hyperfunction✓ ✏
f(x) = F(x + i0) − F(x − i0) : hyperfunction on an interval I
I
f(x)dx ≡ −
C
F(z)dz,
C : closed path encircling I in the positive sense and included in D
(F(z) is analytic in D I)
✒ ✑
D
C
I
11. 1. Hyperfunction theory: integral
8 / 23
integral of a hyperfunction✓ ✏
f(x) = F(x + i0) − F(x − i0) : hyperfunction on an interval I
I
f(x)dx ≡ −
C
F(z)dz,
C : closed path encircling I in the positive sense and included in D
(F(z) is analytic in D I)
✒ ✑
D
C
I
I
f(x)dx =
I
[F(x + i0) − F(x − i0)] dx.
12. Contents
9 / 23
1. Hyperfunction thoery
2. Hyperfunction method for numerical integration
3. Hyperfunction method for Fredholm integral equations
4. Summary
13. 2. Hyperfunction method for numerical integration
10 / 23
We consider an integral of the form
I
f(x)w(x)dx,
f(x) : analytic in D (I ⊂ D ⊂ C, )
w(x) : weight function.
D
I
14. 2. Hyperfunction method for numerical integration
10 / 23
We consider an integral of the form
I
f(x)w(x)dx,
f(x) : analytic in D (I ⊂ D ⊂ C, )
w(x) : weight function.
D
I
We can regard the integrand as a hyperfunction.
✓ ✏
f(x)w(x)χI(x) = −
1
2πi
{f(x + i0)Ψ(x + i0) − f(x − i0)Ψ(x − i0)}
with χI(x) =
1 (x ∈ I)
0 (x ∈ I)
, Ψ(z) =
I
w(x)
z − x
dx.
✒ ✑
15. 2. Hyperfunction method for numerical integration
10 / 23
We consider an integral of the form
I
f(x)w(x)dx,
f(x) : analytic in D (I ⊂ D ⊂ C, )
w(x) : weight function.
D
C : z = ϕ(u)
I
We can regard the integrand as a hyperfunction.
✓ ✏
I
f(x)w(x)dx =
1
2πi C
f(z)Ψ(z)dz
=
1
2πi
τperiod
0
f(ϕ(τ))Ψ(ϕ(τ))ϕ′
(τ)dτ,
C : z = ϕ(τ) ( 0 ≦ τ ≦ τperiod ) periodic function (of period τperiod)
✒ ✑
Approximating the complex integral by the trapezoidal rule, we have ...
16. 2. Hyperfunction method for numerical integration
11 / 23
Hyperfunction method✓ ✏
I
f(x)w(x)dx ≃
h
2πi
N−1
k=0
f(ϕ(kh))Ψ(ϕ(kh))ϕ′
(kh),
with Ψ(z) =
b
a
w(x)
z − x
dx and h =
τperiod
N
.
✒ ✑
D
C : z = ϕ(τ), 0 ≦ τ ≦ τperiod
I
17. 2. Hyperfunction method for numerical integration
11 / 23
Hyperfunction method✓ ✏
I
f(x)w(x)dx ≃
h
2πi
N−1
k=0
f(ϕ(kh))Ψ(ϕ(kh))ϕ′
(kh),
with Ψ(z) =
b
a
w(x)
z − x
dx and h =
τperiod
N
.
✒ ✑
Ψ(z) for some typical weight functions w(x)
I w(x) Ψ(z)
(a, b) 1 log
z − a
z − b
∗
(0, 1) xα−1
(1 − x)β−1
B(α, β)z−1
F(α, 1; α + β; z−1
)∗∗
( α, β > 0 )
∗ log z is the branch s.t. −π ≦ arg z < π.
∗∗ F(α, 1; α + β; z−1
) can be easily evaluated using a continued fraction.
18. 2. Hyperfunction method for numerical integration
11 / 23
Hyperfunction method✓ ✏
I
f(x)w(x)dx ≃
h
2πi
N−1
k=0
f(ϕ(kh))Ψ(ϕ(kh))ϕ′
(kh),
with Ψ(z) =
b
a
w(x)
z − x
dx and h =
τperiod
N
.
✒ ✑
If f(z) is real-valued on R, we can reduce the number of sampling points N by half
using the reflection principle.
19. 2. Numerical integration: theoretical error estimate
12 / 23
theoretical error estimate✓ ✏
If f(ϕ(w)) and ϕ(w) are analytic in | Im w| < d0,
|error| ≦
τperiod
π
max
Im w=±d
|f(ϕ(w))Ψ(ϕ(w))ϕ′
(w)|
×
exp(−(4πd/τperiod)N)
1 − exp(−(4πd/uperiod)N)
( 0 < ∀d < d0 ).
. . . geometric convergence.
✒ ✑
20. 2. Numerical integration: example
13 / 23
✓ ✏
1
0
ex
xα−1
(1 − x)β−1
dx = B(α, β)F(α; α + β; 1) ( α, β > 0 ).
✒ ✑
We computed this integral by
• hyperfunction method (with N reduction),
• DE formula (efficient for integrals with end-point singularities)
• Gauss-Jacobi formula
• C++ program, double precision
• complex integral path for the hyperfunction method (an ellipse)
z = ϕ(τ) =
1
2
+
1
4
ρ +
1
ρ
cos τ +
i
4
ρ −
1
ρ
sin τ ( ρ = 10 )
= 0.5 + 2.575 cos τ + i2.425 sin τ.
21. 2. Numerical integration: example
14 / 23
-16
-14
-12
-10
-8
-6
-4
-2
0
0 10 20 30 40 50 60
log10(error)
N
hyperfunction
hyperfunction
Gauss-Jacobi
Gauss-Jacobi
DE
DE
-16
-14
-12
-10
-8
-6
-4
-2
0
0 20 40 60 80 100 120
log10(error)
N
hyperfunction
hyperfunction
Gauss-Jacobi
DE
DE
α = β = 0.5 α = β = 10−4
(very strong singularities)
The errors of the hyperfunction method, Gauss-Jacobi formula and the DE formula
hyperfunction Gauss-Jacobi DE
α = β = 0.5 O(0.025N
) O((8.2 × 10−4
)N
) O(0.36N
)
α = β = 10−4
O(0.029N
) — O(0.70N
)
22. 2. Numerical integration: example
14 / 23
-16
-14
-12
-10
-8
-6
-4
-2
0
0 10 20 30 40 50 60
log10(error)
N
hyperfunction
hyperfunction
Gauss-Jacobi
Gauss-Jacobi
DE
DE
-16
-14
-12
-10
-8
-6
-4
-2
0
0 20 40 60 80 100 120
log10(error)
N
hyperfunction
hyperfunction
Gauss-Jacobi
DE
DE
α = β = 0.5 α = β = 10−4
(very strong singularities)
The hyperfunction method converges geometricaly,
and its performance is not affected by the end-point singularities.
23. Contents
15 / 23
1. Hyperfunction thoery
2. Hyperfunction method for numerical integration
3. Hyperfunction method for Fredholm integral equations
4. Summary
24. 3. Hyperfunction method for integral equations
16 / 23
Fredholm integral equation for unknown u(x)✓ ✏
λu(x) −
b
a
K(x, ξ)u(ξ)w(ξ)dξ = g(x),
w(ξ) : weight function, K(x, ξ), g(x), λ(= 0) : given.
✒ ✑
We apply the hyperfunction method to this integral equation.
25. 3. Hyperfunction method for integral equations
17 / 23
λu(x) −
b
a
K(x, ξ)u(ξ)w(ξ)dξ = g(x).
(Assumption)
• g(z) : analytic in D except for
a finite number of poles at a1, . . . , aK
• K(z, ζ) : analytic function in D w.r.t. z and ζ D
a b
ak
26. 3. Hyperfunction method for integral equations
17 / 23
λu(x) −
b
a
K(x, ξ)u(ξ)w(ξ)dξ = g(x).
(Assumption)
• g(z) : analytic in D except for
a finite number of poles at a1, . . . , aK
• K(z, ζ) : analytic function in D w.r.t. z and ζ D
a b
ak
ua(z) ≡ u(z) − λ−1
g(z) is analytic in D.
ua(x) satisfies the integral equation
✓ ✏
λua(x) −
b
a
K(x, ξ)ua(ξ)w(ξ)dξ =
1
λ
b
a
K(x, ξ)g(ξ)w(ξ)dξ.
✒ ✑
1. We discretize the integral equation for ua(x) by the hyperfunction method.
2. We solve the discretized equation by the collocation method.
27. 3. Integral equations: Collocation equation
18 / 23
h
2πi
N
k=1
λ
ϕ(kh) − zi
− K(zi, ϕ(kh))Ψ(ϕ(kh)) ϕ′
(ϕ(kh))ua(ϕ(kh))
=
1
2πiλ C
K(zi, ζ)g(ζ)Ψ(ζ)dζ−
1
λ
N
k=1
Res(K(zi, ·)Ψg, ak) (i = 1, . . . , N),
where
C : z = ϕ(τ) ( 0 ≦ τ ≦ τperiod ) closed path encircling [a, b],
periodic function (period τperiod)
z1, . . . , zN : the collocation points inside C, h = τperiod/N.
The collocation equation
... a system of linear equations for ua(ϕ(kh))
( k = 1, . . . , N ). a b
ak
C : z = ϕ(τ)
D
zi
28. 3. Integral equations: Collocation equation
18 / 23
h
2πi
N
k=1
λ
ϕ(kh) − zi
− K(zi, ϕ(kh))Ψ(ϕ(kh)) ϕ′
(ϕ(kh))ua(ϕ(kh))
=
1
2πiλ C
K(zi, ζ)g(ζ)Ψ(ζ)dζ−
1
λ
N
k=1
Res(K(zi, ·)Ψg, ak) (i = 1, . . . , N),
where
C : z = ϕ(τ) ( 0 ≦ τ ≦ τperiod ) closed path encircling [a, b],
periodic function (period τperiod)
z1, . . . , zN : the collocation points inside C, h = τperiod/N.
The approximate solution u(z) is given by
u(z) =
1
2πi C
ua(ζ)
ζ − z
dζ + g(z)
≃
h
2πi
N
j=1
ua(ϕ(kh))
ϕ(kh) − z
ϕ′
(kh) + g(z).
a b
ak
C : z = ϕ(τ)
D
zi
29. 3. Integral equations: example
19 / 23
✓ ✏
u(x) +
1
0
(x − ξ)u(ξ)ξα−1
(1 − ξ)β−1
dξ = g(x),
g(x) =
1
1 + x2
+ B(α, β) Re{F(α, 1; α + β; i)}x
− B(α + 1, β) Re{F(α + 1, 1; α + β + 1; i)} ( α = β = 0.5, 10−4
).
✒ ✑
We solved the integral equation by the hyperfunction method, DE-Nystr¨om method and
Gauss-Jacobi-Nystr¨om method.
• complex integral path
C : z = ϕ(τ) =
1
2
+
1
4
ρ +
1
ρ
cos τ +
i
4
ρ −
1
ρ
sin τ ( ρ = 200 )
• collocation points zi = ϕcol
2π(i − 1)
N
( i = 1, . . . , N )
ϕc(τ) =
1
2
+
1
4
ρc +
1
ρc
cos τ +
i
4
ρc −
1
ρc
sin τ ( 1 < ρc < ρ ).
33. 3. Integral equations: example (α = β = 10−4
)
21 / 23
-60
-50
-40
-30
-20
-10
0
0 20 40 60 80 100
log10(error)
N
rhoc=1.2
rhoc=2.0
rhoc=4.0
rhoc=6.0
rhoc=8.0
DE
Gauss-Jacobi
0
20
40
60
80
100
0 20 40 60 80 100
log10(cond)
N
rhoc=1.2
rhoc=2.0
rhoc=4.0
rhoc=6.0
rhoc=8.0
DE
Gauss-Jacobi
error ǫN condition number κN of
the collocation equation
(rhoc = ρc)
ρcol/ρ 0.006 0.01 0.02 0.03 0.04
ǫN O(0.0058N
) O(0.010N
) O(0.020N
) O(0.030N
) O(0.040N
)
κN O(160N
) O(97N
) O(48N
) O(32N
) O(24N
)
• error ǫN = O[(ρcol/ρ)N
], cond. number κN = O[(ρ/ρcol)N
].
• The DE-Nystr¨om method does not work if the end-point singularities are
very strong.
34. Contents
22 / 23
1. Hyperfunction thoery
2. Hyperfunction method for numerical integration
3. Hyperfunction method for Fredholm integral equations
4. Summary
35. 4. Summary
23 / 23
• We applied hyperfunction theory to numerical integration and Fredholm integral
equations of the second kind.
◦ Hyperfunction theory: a generalized function theory where a “hyperfunction” is
expressed in terms of complex analytic functions.
◦ A hyperfunction integral is given by a complex loop integral, which is evaluated
numerically in the hyperfunction method.
• Hyperfunction method
◦ (Theoretical error estimate) geometric convergence
◦ (Numerical examples) efficiency for problems with strong end-point singularities
◦ Integral equation: The linear system of the collocation equation is
very ill-conditioned.
• Problems for future study
◦ Volterra integral equations.
◦ theoretical error estimate.
36. 4. Summary
23 / 23
• We applied hyperfunction theory to numerical integration and Fredholm integral
equations of the second kind.
◦ Hyperfunction theory: a generalized function theory where a “hyperfunction” is
expressed in terms of complex analytic functions.
◦ A hyperfunction integral is given by a complex loop integral, which is evaluated
numerically in the hyperfunction method.
• Hyperfunction method
◦ (Theoretical error estimate) geometric convergence
◦ (Numerical examples) efficiency for problems with strong end-point singularities
◦ Integral equation: The linear system of the collocation equation is
very ill-conditioned.
• Problems for future study
◦ Volterra integral equations.
◦ theoretical error estimate.
Thank you!