Laskennallinen tekniikka - Computational Engineering

- Mathematical foundations of sequential data analysis: path-dependent functionals, iterated integrals, path signatures, and their universal approximation properties for time series. - Representation of time series through signature features and reformulation of nonlinear inference problems as linear or convex optimization problems - Signature-based modeling frameworks: signature-driven stochastic differential equations, data-driven learning of model coefficients, and construction of interpretable dynamic models consistent with financial theory. - Applications in portfolio management, derivatives markets and data-driven volatility modeling.

This course introduces how emerging quantum computers can enhance scientific computing by accelerating the numerical kernels that appear across physics, engineering, and data-driven modeling. Students study the principles behind quantum speedups and learn quantum and hybrid quantum-classical algorithms for linear algebra, optimization, sampling, and the solution of differential equations. Emphasis is placed on when quantum acceleration is actually achievable in practice.

Kurssi käy läpi topologian ja mittateorian alkeita Euklidisessa avaruudessa. Kurssilla käsiteltäviin aiheisiin kuuluvat raja-arvot, jatkuvuus, täydellisyys, kompaktius, Lebesguen mitta, mittaintegraali, klassiset konvergenssilauseet sekä Fubinin lause.

Kurssi käy läpi topologian ja mittateorian alkeita Euklidisessa avaruudessa. Kurssilla käsiteltäviin aiheisiin kuuluvat raja-arvot, jatkuvuus, täydellisyys, kompaktius, Lebesguen mitta, mittaintegraali, klassiset konvergenssilauseet sekä Fubinin lause.

Vektoriavaruudet: lineaarinen riippumattomuus, virittäjäjoukko ja kanta, aliavaruudet. Matriisilaskenta ja lineaariset kuvaukset: ydin, jälki, aste, ominaisarvot ja vektorit, determinantti, käänteismatriisi, koordinaattimuunnokset. Sisätulo, normi, ortogonaalisuus, ortogonaaliprojektiot, pienimmän neliösumman menetelmä, esimerkkejä ja sovelluksia.

Laskennallisesti älykäs agentti ja sen arkkitehtuuri, tekoäly ja sen toteutus. Automatisoitu ongelmanratkaisu ja päättely varmuudessa ja epävarmuudessa. Datapohjaisen koneoppimisen periaatteet ja paradigmat. Regressio- ja virhemittaukset, Bayes-päättely, päätöspuut, tekoälyverkot, ohjaamaton datan klusterointi sekä tekoälyn etiikka ja oikeudenmukaisuus.

Algoritminen ongelmanratkaisu. Tietorakenteet. Algoritmien suunnitteluperiaatteet. Algoritmien analysointimenetelmät. Algoritmien tehokkuus. NP-täydellisyys. Tyypilliset ongelmatyypit ja niihin sopivat tietorakenteet: järjestely-, haku- ja verkko-ongelmat sekä pinot, jonot, listat, keot, hajautustaulut ja puurakenteet. Tietorakenteiden käytännön toteutus Pythonilla. Yritysyhteistyö Ei yritysyhteistyötä

Electromagnetic radiation and light interaction with matter, sources of radiation and illumination techniques, imaging sensors and manufacturing technologies, spectroscopy, imaging optics, sensor and image acquisition modelling and characterisation, digital image encoding and characteristics, image preprocessing techniques, and image-based measurements.

Introduction to pattern recognition and machine learning based on supervised, unsupervised and reinforcement learning. Feature extraction and selection, system evaluation. Linear and non-linear classifiers based on linear models, kernels, artificial neural networks and support vector machines. Statistical pattern recognition, parameter estimation and Bayesian inference. Context-dependent and reinforcement learning. Practical pattern recognition and method-independent learning.Company cooperation: no direct cooperation.Use of AI applications: readily available tools can be used for checking the language of written reports.

Algoritminen ongelmanratkaisu: johdatus tietojenkäsittelyyn, ongelmanratkaisu, algoritmien laatiminen, algoritmien suunnittelu, algoritmien kompleksisuus, hakuongelmat ja pelien pelaaminen. Tieto ja tiedon muuntaminen: tieto ja tiedon koodaus, informaatio ja tiedon tiivistäminen, tietorakenteet, tiedon salaus, propositilogiikka ja päättely sekä kääntäminen käytännössä.

Kurssi on englanniksi.- joukko-oppi, relaatiot, verkot- logiikka ja totuustaulut- todistustekniikat (suora, kontrapositio, epäsuora, induktio)- valintojen lukumäärä (n vaihtoehtoa, valitaan k eri tavoin)- jaollisuus, Diofantoksen yhtälöt, Eukleideen algoritmi, modulaariaritmetiikka- Fermat'n pikkulause ja RSA-salausalgoritmin periaate- matriisien laskutoimitukset, Gauss-Jordanin eliminointimenetelmä

• Linear Algebra: – solving systems of linear equations (using both matrix form and symbolic math toolbox). – Singular value decomposition (SVD) with applications in, for instance, image reconstruction, solving linear equations, data compression, etc. • ODEs and DAEs: – Solving ODEs analytically and numerically (using both symbolic and numeric methods); – Solving DAEs numerically. – Applications in various dynamical systems. • Optimization: – Description of the general form of a model. – Linear least squares estimation in Matlab (using Matlab backslash) – Parameter estimation for nonlinear models using Matlab fminsearch optimizer – Specific purpose alternatives: lscurvefit and polyfit – Various applications with both dynamic models and algebraic models. • Statistics: – Basics: sample statistics – Statistics for linear models: Covariance of estimates using coefficient matrix, t-values, Rsquare value, crossvalidation. – Statistics for nonlinear models: Covariance of estimates by computing the Jacobian matrix (analytically and numerically) – Alternative ways to obtain the statistics of parameter estimates: Adding noise to data, Bootstrapping – Various applications with both dynamic models and algebraic models. The course is related to UN's Sustainable Development Goal (SDG): 4 Quality Education.The course is related to industry and employment: Research and Development.

Matlab data-rakenteiden (moniulotteiset matriisit, cell array,jne.) ja data-tyyppien käyttäminen (numeeriset, loogiset, teksti, jne.), ehdolliset rakenteet (if-else, switch-case), silmukat (for, while), Matlabin sisärakennetyt funktiot, ulkoisen datan käsittely, 2- ja 3-ulotteiset graafit, käyttäjän itsemääritellyt funktiot.

Vektoriavaruudet: lineaarinen riippumattomuus, virittäjäjoukko ja kanta, aliavaruudet. Matriisilaskenta ja lineaariset kuvaukset: ydin, jälki, aste, ominaisarvot ja vektorit, determinantti, käänteismatriisi, koordinaattimuunnokset. Sisätulo, normi, ortogonaalisuus, ortogonaaliprojektiot, pienimmän neliösumman menetelmä, esimerkkejä ja sovelluksia.

Basic concepts of discrete systems. Model-based design, basic simulation workflow, running the simulations and interpreting the results. Random numbers, discrete event generation by random numbers. Statistical and empirical distributions for event generation. Basics of stochastic differential equations. Building numerical simulation examples with MATLAB. Application examples: queuing systems, storage size optimization, stochastic dynamical systems and agent-based modeling.

Funktioiden, derivaatan, integraalin, vektorien ja matriisilaskennan perusteet sekä tieteellisen laskennan perusteet.

This is a research seminar mainly intended for research purposes. Final year MSc and PhD students can obtain credits from this course through study diary. For details on the study, you should contact the lecturers. The course is related to UN's Sustainable Development Goal (SDG): 4 Quality Education. The course is related to industry and employment: Research and Development.

Functional analysis is a classical field of mathematics, which aims to describe general vector spaces (e.g. function spaces or graphs) and mappings defined on these spaces, and aims to characterize their relationships and properties. Functional analysis offers tools for deeper understanding of many mathematical phenomena such as Fourier transform or numerical analysis. The topic of functional analysis is contemporary, since the data masses studied in modern science are often vast and high-dimensional. It is necessary to understand how different mappings between such data sets scale as the size or the dimension of the data increases. The contents of this course are mostly theoretical and exercises emphasise being able to prove mathematical statements.

This is a research level course mainly intended to final year MSc students and PhD students. The exact content is always agreed with the students. Topics include, but are not limited to, connections between deep Gaussian processes, deep neural networks and stochastic differential equations; implementation needed sampling methods with MCMC, variational Bayes or optimisation as needed; mixture of Gaussian process experts; high-performance computing and random field models for Bayesian inversion. The course is related to UN's Sustainable Development Goal (SDG): 4 Quality Education. The course is related to industry and employment: Research and Development.

Characteristics and pre-processing of data, linear and nonlinear dimensionality reduction. Logistic, multivariate statistical methods and advanced extensions of the methods. Deep neural networks, semi-supervised learning and generative models. Case-based topics on data analysis and machine learning.Company cooperation: no direct cooperation.Use of AI applications: readily available AI tools can be used for checking the language of written reports.