Inverse Problems and Dynamical Systems in Tomography and Optics

The dissertation concerns inverse problems and dynamical systems. Inverse problems, as a subfield of mathematics, studies the mathematical theory of indirect measurements. It is an active area of research with extensive mathematical theory and numerous applications. Many inverse problems are concerned with physical systems that evolve in time. A mathematical model that describes how a quantity evolves in time is called a dynamical system. X-ray computed tomography is a technique where the inner structure of an object is computed from a number of its X-ray images. In the first publication of the dissertation we consider X-ray computed tomography in a setting where the orientations in which the object was imaged are unknown. This problem is called tomography with unknown view angles; such a problem arises e.g. in cryogenic electron microscopy of viral particles. We show that under general assumptions it is possible to reconstruct the structure of the object in tomography with unknown view angles. Diffusion is the flow of a substance from areas of high concentration to areas of low concentration. In the second publication we consider an inverse problem for the space–time fractional diffusion equation. This equation models diffusion and anomalous diffusion processes, such as those sometimes observed in fractured geological formations. We show that the geometry of the underlying space can be determined by observing the evolution of a solution of the equation in a subset of the space. In the third and fourth publications we consider a dynamical system that models injection locking in a laser. Injection locking is a technique where light from one laser is injected into another laser's cavity (the part of a laser where the emitted light is created) with the intention of altering the laser's properties. The idea is to consider injection locking as a process that can provide the basis for optical computing devices. In the third publication we derive an approximation for the nonlinear relationship between the injected light and the injection-locked emitted light, and we show that it is possible to construct an optical logic gate based on this relationship. In the fourth publication we do a detailed analysis of the dynamical system that models injection locking in lasers, and based on this analysis, we propose a design for an optical neural network.Väitöskirja käsittelee käänteisten ongelmien ja dynaamisten järjestelmien matemaattista teoriaa. Käänteisissä ongelmissa pyritään ymmärtämään ilmiön syy sen seurauksia. Esimerkki käänteisestä ongelmasta on röntgenkuvaus, jossa kappaleen läpäisseiden säteiden vaimenemisesta (seuraus) halutaan päätellä vaimenemisen syy (kappaleen sisäinen rakenne). Usein käänteisen ongelman liittyvä tarkasteltava suure on dynaaminen, eli se muuttuu ajan mukana. Tällaisen ilmiön matemaattista mallia kutsutaan dynaamiseksi järjestelmäksi. Väitöskirjan ensimmäisessä julkaisussa tutkitaan tietokonekerroskuvauksen matemaattista teoriaa. Tietokonekerroskuvauksessa tarkasteltavasta kohteesta kuvataan röntgenkuvia useista eri suunnista, ja näistä kuvista pyritään laskemaan kohteen sisäinen rakenne. Usein suunnat joista kuvat on otettu tunnetaan, mutta tietyissä tilanteissa nämä kuvaussuunnat ovat tuntemattomia. Tällainen tilanne on esimerkiksi virusten kuvantamisessa käytetyssä kryogeenisessa elektronimikroskopiassa. Julkaisussa osoitetaan, että kohteen rakenteen määrittäminen on yleensä mahdollista vaikka kuvaussuunnat olisivatkin tuntemattomia. Lämmön johtumisen matemaattista mallia kutsutaan lämpöyhtälöksi. Lämmön johtuminen aineen sisällä riippuu sekä aineen kyvystä johtaa lämpöä (ns. lämmönjohtavuuskertoimesta), että tarkasteltavan kappaleen muodosta. Väitöskirjan toisessa julkaisussa tutkitaan käänteistä ongelmaa lämpöyhtälölle. Julkaisussa osoitetaan, että tarkastelemalla kappaleen lämpötilan käyttäytymistä osassa kappaletta voidaan päätellä sekä koko kappaleen muoto, että sen lämmönjohtavuuskerroin. Kolmannessa ja neljännessä julkaisussa tutkitaan optiseen laskentaan liittyvän dynaamisen järjestelmän matemaattista teoriaa. Nykyisten puolijohdekomponentteihin perustuvien tietokoneiden toiminta perustuu sähköön, optisessa laskennassa tavoite on rakentaa tietokone jonka toiminta perustuu sähkön sijaan valoon. Julkaisuissa tutkitaan lasereiden synkronoitumiseen liittyvän ilmiön (ns. injektiolukituksen) matemaattisia ominaisuuksia. Julkaisuissa osoitetaan että ominaisuuksiensa puolesta injektiolukitusta voi olla mahdollista hyödyntää optisten transistoreiden tai optisten neuroverkkojen rakentamisessa

Quantum computing algorithms for inverse problems on graphs and an NP-complete inverse problem

We consider an inverse problem for a finite graph $(X,E)$ where we are given a subset of vertices $B\subset X$ and the distances $d_{(X,E)}(b_1,b_2)$ of all vertices $b_1,b_2\in B$. The distance of points $x_1,x_2\in X$ is defined as the minimal number of edges needed to connect two vertices, so all edges have length 1. The inverse problem is a discrete version of the boundary rigidity problem in Riemannian geometry or the inverse travel time problem in geophysics. We will show that this problem has unique solution under certain conditions and develop quantum computing methods to solve it. We prove the following uniqueness result: when $(X,E)$ is a tree and $B$ is the set of leaves of the tree, the graph $(X,E)$ can be uniquely determined in the class of all graphs having a fixed number of vertices. We present a quantum computing algorithm which produces a graph $(X,E)$, or one of those, which has a given number of vertices and the required distances between vertices in $B$. To this end we develop an algorithm that takes in a qubit representation of a graph and combine it with Grover's search algorithm. The algorithm can be implemented using only $O(|X|^2)$ qubits, the same order as the number of elements in the adjacency matrix of $(X,E)$. It also has a quadratic improvement in computational cost compared to standard classical algorithms. Finally, we consider applications in theory of computation, and show that a slight modification of the above inverse problem is NP-complete: all NP-problems can be reduced to a discrete inverse problem we consider

Inverse problems for heat equation and space-time fractional diffusion equation with one measurement

Given a connected compact Riemannian manifold (M, g) without boundary, dim M >= 2, we consider a space-time fractional diffusion equation with an interior source that is supported on an open subset Vof the manifold. The time-fractional part of the equation is given by the Caputo derivative of order alpha is an element of(0, 1], and the space fractional part by (-Delta(g))(beta), where beta is an element of(0, 1] and Delta(g) is the Laplace-Beltrami operator on the manifold. The case alpha= beta= 1, which corresponds to the standard heat equation on the manifold, is an important special case. We construct a specific source such that measuring the evolution of the corresponding solution on Vdetermines the manifold up to a Riemannian isometry. (c) 2020 Published by Elsevier Inc.Peer reviewe

Inverse problems for heat equation and space-time fractional diffusion equation with one measurement

Given a connected compact Riemannian manifold (M, g) without boundary, dim M >= 2, we consider a space-time fractional diffusion equation with an interior source that is supported on an open subset Vof the manifold. The time-fractional part of the equation is given by the Caputo derivative of order alpha is an element of(0, 1], and the space fractional part by (-Delta(g))(beta), where beta is an element of(0, 1] and Delta(g) is the Laplace-Beltrami operator on the manifold. The case alpha= beta= 1, which corresponds to the standard heat equation on the manifold, is an important special case. We construct a specific source such that measuring the evolution of the corresponding solution on Vdetermines the manifold up to a Riemannian isometry. (c) 2020 Published by Elsevier Inc.Peer reviewe

Analysis of a Dynamical System Modeling Lasers and Applications for Optical Neural Networks

An analytical study of dynamical properties of a semiconductor laser with optical injection of arbitrary polarization is presented. It is shown that if the injected field is sufficiently weak, then the laser has nine equilibrium points; however, only one of them is stable. Even if the injected field is linearly polarized, six of the equilibrium points have a state of polarization that is elliptical. Dependence of the equilibrium points on the injected field is described, and it is shown that as the intensity of the injected field increases, the number of equilibrium points decreases, with only a single equilibrium point remaining for strong enough injected fields. As an application, a complex-valued optical neural network with working principle based on injection locking is proposed.Peer reviewe

Analysis of a Dynamical System Modeling Lasers and Applications for Optical Neural Networks

An analytical study of dynamical properties of a semiconductor laser with optical injection of arbitrary polarization is presented. It is shown that if the injected field is sufficiently weak, then the laser has nine equilibrium points; however, only one of them is stable. Even if the injected field is linearly polarized, six of the equilibrium points have a state of polarization that is elliptical. Dependence of the equilibrium points on the injected field is described, and it is shown that as the intensity of the injected field increases, the number of equilibrium points decreases, with only a single equilibrium point remaining for strong enough injected fields. As an application, a complex-valued optical neural network with working principle based on injection locking is proposed.Peer reviewe

Diminished brain responses to second-language words are linked with native-language literacy skills in dyslexia

Dyslexia is characterized by poor reading skills, yet often also difficulties in second-language learning. The differences between native- and second-language speech processing and the establishment of new brain representations for spoken second language in dyslexia are not, however, well understood. We used recordings of the mismatch negativity component of event-related potential to determine possible differences between the activation of long-term memory representations for spoken native- and second-language word forms in Finnish-speaking 9-11-year-old children with or without dyslexia, studying English as their second language in school. In addition, we sought to investigate whether the bottleneck of dyslexic readers' second-language learning lies at the level of word representations or smaller units and whether the amplitude of mismatch negativity is correlated with native-language literacy and related skills. We found that the activation of brain representations for familiar second-language words, but not for second-language speech sounds or native-language words, was weaker in children with dyslexia than in typical readers. Source localization revealed that dyslexia was associated with weak activation of the right temporal cortex, which has been previously linked with word-form learning. Importantly, the amplitude of the mismatch negativity for familiar second-language words correlated with native-language literacy and rapid naming scores, suggesting a close link between second-language processing and these skills.Peer reviewe

Diminished brain responses to second-language words are linked with native-language literacy skills in dyslexia

Dyslexia is characterized by poor reading skills, yet often also difficulties in second-language learning. The differences between native- and second-language speech processing and the establishment of new brain representations for spoken second language in dyslexia are not, however, well understood. We used recordings of the mismatch negativity component of event-related potential to determine possible differences between the activation of long-term memory representations for spoken native- and second-language word forms in Finnish-speaking 9–11-year-old children with or without dyslexia, studying English as their second language in school. In addition, we sought to investigate whether the bottleneck of dyslexic readers' second-language learning lies at the level of word representations or smaller units and whether the amplitude of mismatch negativity is correlated with native-language literacy and related skills. We found that the activation of brain representations for familiar second-language words, but not for second-language speech sounds or native-language words, was weaker in children with dyslexia than in typical readers. Source localization revealed that dyslexia was associated with weak activation of the right temporal cortex, which has been previously linked with word-form learning. Importantly, the amplitude of the mismatch negativity for familiar second-language words correlated with native-language literacy and rapid naming scores, suggesting a close link between second-language processing and these skills.</p

All-optical majority gate based on an injection-locked laser

An all-optical computer has remained an elusive concept. To construct a practical computing primitive equivalent to an electronic Boolean logic, one should utilize nonlinearity that overcomes weaknesses that plague many optical processing schemes. An advantageous nonlinearity provides a complete set of logic operations and allows cascaded operations without changes in wavelength or in signal encoding format. Here we demonstrate an all-optical majority gate based on a vertical-cavity surface-emitting laser (VCSEL). Using emulated signal coupling, the arrangement provides Bit Error Ratio (BER) of 10(-6) at the rate of 1 GHz without changes in the wavelength or in the signal encoding format. Cascaded operation of the injection-locked laser majority gate is simulated on a full adder and a 3-bit ripple-carry adder circuits. Finally, utilizing the spin-flip model semiconductor laser rate equations, we prove that injection-locked lasers may perform normalization operations in the steady-state with an arbitrary linear state of polarization.Peer reviewe

All-optical majority gate based on an injection-locked laser

An all-optical computer has remained an elusive concept. To construct a practical computing primitive equivalent to an electronic Boolean logic, one should utilize nonlinearity that overcomes weaknesses that plague many optical processing schemes. An advantageous nonlinearity provides a complete set of logic operations and allows cascaded operations without changes in wavelength or in signal encoding format. Here we demonstrate an all-optical majority gate based on a vertical-cavity surface-emitting laser (VCSEL). Using emulated signal coupling, the arrangement provides Bit Error Ratio (BER) of 10⁻⁶ at the rate of 1 GHz without changes in the wavelength or in the signal encoding format. Cascaded operation of the injection-locked laser majority gate is simulated on a full adder and a 3-bit ripple-carry adder circuits. Finally, utilizing the spin-flip model semiconductor laser rate equations, we prove that injection-locked lasers may perform normalization operations in the steady-state with an arbitrary linear state of polarization