A Note on the “Constructing” of Nonstationary Methods for Solving Nonlinear Equations with Raised Speed of Convergence

This paper is partially supported by project ISM-4 of Department for Scientific Research, “Paisii Hilendarski” University of Plovdiv.In this paper we give methodological survey of “contemporary methods” for solving the nonlinear equation f(x) = 0. The reason for this review is that many authors in present days rediscovered such classical methods. Here we develop one methodological schema for constructing nonstationary methods with a preliminary chosen speed of convergence

A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots

[EN] The aim of this paper is to introduce new high order iterative methods for multiple roots of the nonlinear scalar equation; this is a demanding task in the area of computational mathematics and numerical analysis. Specifically, we present a new Chebyshev¿Halley-type iteration function having at least sixth-order convergence and eighth-order convergence for a particular value in the case of multiple roots. With regard to computational cost, each member of our scheme needs four functional evaluations each step. Therefore, the maximum efficiency index of our scheme is 1.6818 for ¿ = 2,which corresponds to an optimal method in the sense of Kung and Traub¿s conjecture. We obtain the theoretical convergence order by using Taylor developments. Finally, we consider some real-life situations for establishing some numerical experiments to corroborate the theoretical results.This research was partially supported by Ministerio de Economia y Competitividad under Grant MTM2014-52016-C2-1-2-P and by the project of Generalitat Valenciana Prometeo/2016/089Behl, R.; Martínez Molada, E.; Cevallos-Alarcon, FA.; Alarcon-Correa, D. (2019). A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots. Mathematics. 7(4):1-12. https://doi.org/10.3390/math7040339S11274Gutiérrez, J. M., & Hernández, M. A. (1997). A family of Chebyshev-Halley type methods in Banach spaces. Bulletin of the Australian Mathematical Society, 55(1), 113-130. doi:10.1017/s0004972700030586Kanwar, V., Singh, S., & Bakshi, S. (2008). Simple geometric constructions of quadratically and cubically convergent iterative functions to solve nonlinear equations. Numerical Algorithms, 47(1), 95-107. doi:10.1007/s11075-007-9149-4Argyros, I. K., Ezquerro, J. A., Gutiérrez, J. M., Hernández, M. A., & Hilout, S. (2011). On the semilocal convergence of efficient Chebyshev–Secant-type methods. Journal of Computational and Applied Mathematics, 235(10), 3195-3206. doi:10.1016/j.cam.2011.01.005Xiaojian, Z. (2008). Modified Chebyshev–Halley methods free from second derivative. Applied Mathematics and Computation, 203(2), 824-827. doi:10.1016/j.amc.2008.05.092Amat, S., Hernández, M. A., & Romero, N. (2008). A modified Chebyshev’s iterative method with at least sixth order of convergence. Applied Mathematics and Computation, 206(1), 164-174. doi:10.1016/j.amc.2008.08.050Kou, J., & Li, Y. (2007). Modified Chebyshev–Halley methods with sixth-order convergence. Applied Mathematics and Computation, 188(1), 681-685. doi:10.1016/j.amc.2006.10.018Li, D., Liu, P., & Kou, J. (2014). An improvement of Chebyshev–Halley methods free from second derivative. Applied Mathematics and Computation, 235, 221-225. doi:10.1016/j.amc.2014.02.083Sharma, J. R. (2015). Improved Chebyshev–Halley methods with sixth and eighth order convergence. Applied Mathematics and Computation, 256, 119-124. doi:10.1016/j.amc.2015.01.002Neta, B. (2010). Extension of Murakami’s high-order non-linear solver to multiple roots. International Journal of Computer Mathematics, 87(5), 1023-1031. doi:10.1080/00207160802272263Zhou, X., Chen, X., & Song, Y. (2011). Constructing higher-order methods for obtaining the multiple roots of nonlinear equations. Journal of Computational and Applied Mathematics, 235(14), 4199-4206. doi:10.1016/j.cam.2011.03.014Hueso, J. L., Martínez, E., & Teruel, C. (2014). Determination of multiple roots of nonlinear equations and applications. Journal of Mathematical Chemistry, 53(3), 880-892. doi:10.1007/s10910-014-0460-8Behl, R., Cordero, A., Motsa, S. S., & Torregrosa, J. R. (2015). On developing fourth-order optimal families of methods for multiple roots and their dynamics. Applied Mathematics and Computation, 265, 520-532. doi:10.1016/j.amc.2015.05.004Behl, R., Cordero, A., Motsa, S. S., Torregrosa, J. R., & Kanwar, V. (2015). An optimal fourth-order family of methods for multiple roots and its dynamics. Numerical Algorithms, 71(4), 775-796. doi:10.1007/s11075-015-0023-5Geum, Y. H., Kim, Y. I., & Neta, B. (2015). A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics. Applied Mathematics and Computation, 270, 387-400. doi:10.1016/j.amc.2015.08.039Geum, Y. H., Kim, Y. I., & Neta, B. (2016). A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points. Applied Mathematics and Computation, 283, 120-140. doi:10.1016/j.amc.2016.02.029Behl, R., Alshomrani, A. S., & Motsa, S. S. (2018). An optimal scheme for multiple roots of nonlinear equations with eighth-order convergence. Journal of Mathematical Chemistry, 56(7), 2069-2084. doi:10.1007/s10910-018-0857-xMcNamee, J. M. (1998). A comparison of methods for accelerating convergence of Newton’s method for multiple polynomial roots. ACM SIGNUM Newsletter, 33(2), 17-22. doi:10.1145/290590.290592Cordero, A., & Torregrosa, J. R. (2007). Variants of Newton’s Method using fifth-order quadrature formulas. Applied Mathematics and Computation, 190(1), 686-698. doi:10.1016/j.amc.2007.01.06

Solving the nearest rotation matrix problem in three and four dimensions with applications in robotics

Aplicat embargament des de la data de defensa fins ei 31/5/2022Since the map from quaternions to rotation matrices is a 2-to-1 covering map, this map cannot be smoothly inverted. As a consequence, it is sometimes erroneously assumed that all inversions should necessarily contain singularities that arise in the form of quotients where the divisor can be arbitrarily small. This misconception was clarified when we found a new division-free conversion method. This result triggered the research work presented in this thesis. At first glance, the matrix to quaternion conversion does not seem to be a relevant problem. Actually, most researchers consider it as a well-solved problem whose revision is not likely to provide any new insight in any area of practical interest. Nevertheless, we show in this thesis how solving the nearest rotation matrix problem in Frobenius norm can be reduced to a matrix to quaternion conversion. Many problems, such as hand-eye calibration, camera pose estimation, location recognition, image stitching etc. require finding the nearest proper orthogonal matrix to a given matrix. Thus, the matrix to quaternion conversion becomes of paramount importance. While a rotation in 3D can be represented using a quaternion, a rotation in 4D can be represented using a double quaternion. As a consequence, the computation of the nearest rotation matrix in 4D, using our approach, essentially follow the same steps as in the 3D case. Although the 4D case might seem of theoretical interest only, we show in this thesis its practical relevance thanks to a little known mapping between 3D displacements and 4D rotations. In this thesis we focus our attention in obtaining closed-form solutions, in particular those that only require the four basic arithmetic operations because they can easily be implemented on microcomputers with limited computational resources. Moreover, closed-form methods are preferable for at least two reasons: they provide the most meaningful answer because they permit analyzing the influence of each variable on the result; and their computational cost, in terms of arithmetic operations, is fixed and assessable beforehand. We have actually derived closed-form methods specifically tailored for solving the hand-eye calibration and the pointcloud registration problems which outperform all previous approaches.Dado que la función que aplica a cada cuaternión su matrix de rotación correspondiente es 2 a 1, la inversa de esta función no es diferenciable en todo su dominio. Por consiguiente, a veces se asume erróneamente que todas las inversiones deben contener necesariamente singularidades que surgen en forma de cocientes donde el divisor puede ser arbitrariamente pequeño. Esta idea errónea se aclaró cuando encontramos un nuevo método de conversión sin división. Este resultado desencadenó el trabajo de investigación presentado en esta tesis. A primera vista, la conversión de matriz a cuaternión no parece un problema relevante. En realidad, la mayoría de los investigadores lo consideran un problema bien resuelto cuya revisión no es probable que proporcione nuevos resultados en ningún área de interés práctico. Sin embargo, mostramos en esta tesis cómo la resolución del problema de la matriz de rotación más cercana según la norma de Frobenius se puede reducir a una conversión de matriz a cuaternión. Muchos problemas, como el de la calibración mano-cámara, el de la estimación de la pose de una cámara, el de la identificación de una ubicación, el del solapamiento de imágenes, etc. requieren encontrar la matriz de rotación más cercana a una matriz dada. Por lo tanto, la conversión de matriz a cuaternión se vuelve de suma importancia. Mientras que una rotación en 3D se puede representar mediante un cuaternión, una rotación en 4D se puede representar mediante un cuaternión doble. Como consecuencia, el cálculo de la matriz de rotación más cercana en 4D, utilizando nuestro enfoque, sigue esencialmente los mismos pasos que en el caso 3D. Aunque el caso 4D pueda parecer de interés teórico únicamente, mostramos en esta tesis su relevancia práctica gracias a una función poco conocida que relaciona desplazamientos en 3D con rotaciones en 4D. En esta tesis nos centramos en la obtención de soluciones de forma cerrada, en particular aquellas que solo requieren las cuatro operaciones aritméticas básicas porque se pueden implementar fácilmente en microcomputadores con recursos computacionales limitados. Además, los métodos de forma cerrada son preferibles por al menos dos razones: proporcionan la respuesta más significativa porque permiten analizar la influencia de cada variable en el resultado; y su costo computacional, en términos de operaciones aritméticas, es fijo y evaluable de antemano. De hecho, hemos derivado nuevos métodos de forma cerrada diseñados específicamente para resolver el problema de la calibración mano-cámara y el del registro de nubes de puntos cuya eficiencia supera la de todos los métodos anteriores.Postprint (published version

Modifikasi Metode Iterasi Dua Langkah Menggunakan Kombinasi Linear Tiga Parameter Real

Makalah ini membahas modifikasi  metode iterasi dua langkah dengan menggunakan kombinasi linier tiga parameter dan tiga metode iterasi berorde konvergensi tiga yang masing-masing dihasilkan dari penjumlahan metode Potra-Ptak dan metode varian Newton, modifikasi metode varian Newton  rata-rata kontra harmonik, dan Metode Newton-Steffensen. Berdasarkan hasil kajian diperoleh bahwa metode iterasi baru memiliki orde konvergensi empat untuk q 1 = -2, q 2 = 3 - q 3 dan q3 ÎÂ yang melibatkan tiga evaluasi fungsi dengan indeks efisiensi sebesar 41/3 » 1,5874. Simulasi numerik diberikan untuk menunjukkan performa metode iterasi baru dibandingkan dengan metode Newton, metode Potra-Ptak, dan metode Chebyshe

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Modelado matemático y simulación numérica de disipadores de calor para luminarias LED. Aplicaciones a alumbrado público

[ES] En esta tesis se plasma un ejemplo paradigmático de Matemática Industrial: se define un problema real de enorme interés actual, se presenta un modelo matemático del mismo, se resuelve numéricamente mediante métodos de elementos Finitos, se realiza diferentes prototipos y se verifican experimentalmente las predicciones teóricas; además, en este caso particular, los prototipos aquí analizados se llevaron al mercado, cerrando un ciclo que se inicia con el modelado matemático y se termina con la transferencia a la sociedad de una solución competitiva a un problema real. El problema que se aborda en esta tesis se enmarca en el desarrollo de soluciones de iluminación basadas en tecnología de diodos emisores de luz (LED, por su abreviación en inglés) de alta potencia. De hecho, el problema que se afronta es el desarrollo de disipadores pasivos de calor que garanticen la correcta evacuación del calor producido en el dispositivo LED y aseguren su adecuado funcionamiento. Para ello, se modela el problema de transferencia de calor (incluyendo conducción, radiación y convección) en diferentes prototipos, se resuelve con técnicas de Elementos Finitos y se optimizan los diseños propuestos, garantizando siempre que la temperatura de operación del chip LED sea correcta. Una vez realizado este análisis teórico, se construyen los prototipos y se verifican experimentalmente las predicciones realizadas. Por último, en los anexos se recoge una serie de aportaciones complementarias: una sobre el gas de van der Waals y la Geometría de Contacto y otras dos sobre la convergencia de métodos iterativos.[CA] En aquesta tesi es plasma un exemple paradigmàtic de Matemàtica Industrial: es defineix un problema real d'enorme interès actual, es presenta un model matemàtic del mateix, es resol numèricament mitjançant mètodes d'Elements Finits, es realitza diferents prototips i es verifiquen experimentalment les prediccions teòriques; a més, en aquest cas particular, els prototips aquí analitzats es van dur a mercat, tancant un cicle que s'inicia amb el modelatge matemàtic i s'acaba amb la transferència a la societat d'una solució competitiva a un problema real. El problema que s'aborda en aquesta tesi s'emmarca en el desenvolupament de solucions d'il·luminació basades en tecnologia LED d'alta potència. De fet, el problema que s'afronta és el desenvolupament de dissipadors passius de calor que garanteixin la correcta evacuació de la calor produïda da en el dispositiu LED i assegurin la seva adequat funcionament. Per a això, es modela el problema de transferència de calor (incloent conducció, radiació i convecció) en diferents prototips, es resol amb tècniques d'Elements Finits i s'optimitzen els dissenys proposats, garantint sempre que la temperatura d'operació de l'xip LED sigui correcta. Un cop realitzat aquest anàlisi teòrica, es construeixen els prototips i es verifiquen experimentalment les prediccions realitzades. Finalment, en els annexos es recull una sèrie d'aportacions complementàries: una sobre el gas de van der Waals i la Geometria de Contacte i dues sobre la convergència de mètodes iteratius.[EN] In this thesis, a paradigmatic example of Industrial Mathematics is captured: a real problem of enormous current interest is defined, a mathematical model of it is presented, it is solved numerically using Finite Element methods, different prototypes are made and the theoretical predictions are experimentally verified; Furthermore, in this particular case, the prototypes analyzed here were brought to the market, closing a cycle that begins with mathematical modeling and ends with the transfer to society of a competitive solution to a real problem. The problem addressed in this thesis is part of the development of lighting solutions based on high-power LED technology. In fact, the problem being faced is the development of passive heat sinks that guarantee the correct evacuation of the heat produced in the LED device and ensure its proper operation. For this, the heat transfer problem (including conduction, radiation and convection) is modeled in different prototypes, it is solved with Finite Element techniques and the proposed designs are optimized, always guaranteeing that the operating temperature of the LED chip is correct. Once this theoretical analysis has been carried out, the prototypes are built and the predictions made are experimentally verified. Finally, the annexes contain a series of complementary contributions: one on van der Waals gas and Contact Geometry and two others on the convergence of iterative methods.A la Secretarıa de Educación Superior, Ciencia,Tecnología e Innovación (SENESCYT) por el apoyo económico para poder realizar mis estudios en el extranjero con el fin de fortalecer el talento humano en el Ecuador.Alarcón Correa, DF. (2020). Modelado matemático y simulación numérica de disipadores de calor para luminarias LED. Aplicaciones a alumbrado público [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/155989TESI

Gluing small black holes along timelike geodesics I: formal solution

Given a smooth globally hyperbolic $(3+1)$-dimensional spacetime satisfying the Einstein vacuum equations (possibly with cosmological constant) and an inextendible timelike geodesic, we construct a family of metrics depending on a small parameter $\epsilon>0$ with the following properties. (1) They solve the Einstein vacuum equations modulo $\mathcal{O}(\epsilon^\infty)$. (2) Away from the geodesic they tend to the original metric as $\epsilon\to 0$. (3) Their $\epsilon^{-1}$-rescalings near every point of the geodesic tend to a fixed subextremal Kerr metric (assuming a condition on the mode stability at zero frequency which we verify in the very slowly rotating case). Our results apply on spacetimes which do not admit nontrivial Killing vector fields in a neighborhood of a point on the geodesic. They also apply in a neighborhood of the domain of outer communications of subextremal Kerr and Kerr-de Sitter spacetimes, in which case our metrics model extreme mass ratio inspirals if we choose the timelike geodesic to cross the event horizon. The metrics which we construct here depend on $\epsilon$ and the (rescaled) coordinates on the original spacetime in a log-smooth fashion. This in particular justifies the formal perturbation theoretic setup in work of Gralla-Wald on gravitational self-force in the case of small black holes.Comment: 165 pages, 14 figure

Research in computerized structural analysis and synthesis

Computer applications in dynamic structural analysis and structural design modeling are discussed

An analysis for some methods and algorithms of quantum chemistry

In der theoretischen Berechnung der Eigenschaften von Atomen, Molekülen und Festkörpern spielt die Lösung der elektronischen Schrödingergleichung, einer Operatoreigenwertgleichung für den Hamiltonoperator H des jeweiligen Systems, eine zentrale Rolle. Besondere Bedeutung kommt hierbei dem kleinsten Eigenwert von H zu, der die Grundzustandsenergie des Systems angibt. Um den unterschiedlichen Anforderungen in der Fülle von Anwendungsgebieten der elektronischen Schrödingergleichung gerecht zu werden, wurden in den letzten Jahrzehnten verschiedenste Näherungsverfahren entwickelt, um die Lösung dieses extrem hochdimensionalen Minimierungsproblems zu approximieren. Das Ziel der vorliegenden Arbeit ist es, eine (mathematische) Analysis für Aspekte einiger der verwendeten Methoden der Quantenchemie zu liefern. Zu diesem Zweck gliedert sich die Arbeit in vier Teile: Der erste Teil gibt eine Einführung in den mathematischen, hauptsächlich der Funktionalanalysis zuzuschreibenden Hintergrund, der bei der Behandlung der elektronischen Schrödingergleichung als Operatoreigenwertgleichung notwendig ist, und stellt viele der in den späteren Kapiteln benötigten Handwerkszeuge zur Verfügung. Der zweite Teil beschäftigt sich mit einem Gradientenalgorithmus mit Orthogonalitätsnebenbedingungen, der zur der Lösung der in der Beschreibung größerer Systeme wichtigen Hartree-Fock- und Kohn-Sham-Gleichungen, aber auch zur algorithmischen Behandlung der CI-Methode und außerhalb der Elektronenstrukturberechnung in der Berechnung invarianter Unterräume verwendet wird. Wir formulieren den Algorithmus allgemeiner als Verfahren zur Behandlung von Minimierungsproblemen auf der sogenannten Grassmann-Mannigfaltigkeit [1] und beweisen mit Hilfe dieses Formalismus unter anderem lineare Konvergenz des Algorithmus und die quadratische Konvergenz der zugeh√∂rigen Energien. Im dritten Teil der Arbeit wird die in der Praxis für hochgenaue Rechnungen bedeutsame Coupled-Cluster-Methode, traditionell ein Ansatz zur Approximation der Galerkinlösung der Schrödingergleichung innerhalb einer gegebenen Diskretisierung [2], als Verfahren im unendlichdimensionalen, undiskretisierten Raum formuliert. Zu diesem Zweck wird zunächst die Stetigkeit des Clusteroperators T als Operator vom Sobolevraum H1 in sich bewiesen: hieraus lässt sich dann die (unendlichdimensionale Verallgemeinerung der bekannten) Nullstellengleichung für die Coupled-Cluster-Funktion formulieren. Wir zeigen die lokale starke Monotonie der CC-Funktion, mit deren Hilfe wir Existenz- und Eindeutigkeitsaussagen und einen zielorientierten Fehlerschätzer nach [3] beweisen. Schließlich diskutieren wir die algorithmische Behandlung der oben genannten Nullstellengleichung. Teil 4 beschäftigt sich mit der DIIS-Methode, einem im Rahmen der Quantenchemie standardmäßig verwendeten Verfahren zur Konvergenzbeschleuningung iterativer Algorithmen. Wir identifizieren DIIS mit einer Variante des projezierten Broyden-Verfahrens [4] und zeigen, dass sich das Verfahren, angewandt auf lineare Probleme, als Variante des GMRES-Verfahrens auffassen lässt. Für den allgemeinen Fall beweisen wir schließlich zwei lokale Konvergenzaussagen und diskutieren die Umstände, unter denen DIIS superlineares Konvergenzverhalten zeigen kann. [1] T. A. Arias, A. Edelman, S. T. Smith, SIAM J. Matrix Anal. and Appl. 20, 2, 1999. [2] R. Schneider, Num. Math. 113, 3, 2009. [3] R. Becker, R. Rannacher, Acta Numerica 2000 (A. Iserlet, ed.), Cambridge University Press, 2001. [4] D. M. Gay, R. B. Schnabel, Nonlinear Programming 3, Academic Press, 1978.In the field of ab-initio calculation of the properties of atoms, molecules and solids, the solution of the electronic Schrödinger equation, an operator eigenvalue equation for the Hamiltonian of the system, plays a major role. Of utmost significance is the lowest eigenvalue of H, representing the ground state energy of the system. To meet the requirements of the multitude of possible applications of the elctronic Schrödinger equation, the last decades have seen the development of a variety of different methods designed to approximate the solution of this extremely high-dimensional minimization problem. The aim of the present work is to deliver a (mathematical) analysis for some aspects of some of these methods used in the context of quantum chemistry. The work consists of four parts: The first part gives an introduction to the mathematical background, mainly belonging to the field of functional analysis, that is needed for the rigogous treatment of the electronic Schrödinger equation as an operator eigenvalue equation, and provides many of the technical means needed in the later chapters. The second part is concerned with a gradient algorithm with orthogonality constraints, which is used for the solution Hartree-Fock and Kohn-Sham equations playing an important role in the description of larger systems and which also serves for the algorithmic treatment of the CI method and - outside of the field of electronic structure calculation - for the calculation of invariant subspaces. The algorithm is formulated as an abstract method for the treatment of minimization problems on the so-called Grassmann manifold [1]; with the help of this formalism, linear convergence of the algorithm and quadratic convergence of the corresponding eigenvalues is proven. The third part of the work is concerned with the Coupled Cluster method, being of high practical significance in calculations where high accuracy is demanded. We lift the method, usually formulated as an ansatz for the approximation of the Galerkin solution in a finite dimensional, discretised subspace [2] to the continuous, undiscretised space, resulting in what we will call the continuous Coupled Cluster method. To define the continuous method, we first prove the continuity of the cluster operator T as an operator mapping the Sobolev space H1 to itself; with the help of this result, the infinite dimensional globalization of the canonical) Coupled Cluster equations can be formulated. Afterwards, we prove local strong monotonicity of the CC function, from which we derive existence and (local) uniqueness statements and a goal-oriented a-posteriori error estimator in the fashion of [3]. Finally, we discuss the algorithmic treatment of the CC root equation. The last part of this work features an analysis for the acceleration technique DIIS that is commonly used in quantum chemistry codes. We identify DIIS with a variant of a projected Broyden's method [4] and show that when applied to linear systems, the method can be interpreted as a variant of the well-known GMRES method. For the global nonlinear case, we finally prove two local convergence results and discuss the circumstances under which DIIS can show superlinear convergence. [1] T. A. Arias, A. Edelman, S. T. Smith, SIAM J. Matrix Anal. and Appl. 20, 2, 1999. [2] R. Schneider, Num. Math. 113, 3, 2009. [3] R. Becker, R. Rannacher, Acta Numerica 2000 (A. Iserlet, ed.), Cambridge University Press, 2001. [4] D. M. Gay, R. B. Schnabel, Nonlinear Programming 3, Academic Press, 1978

Numerical Geometric Acoustics

Sound propagation in air is accurately described by a small perturbation of the ambient pressure away from a quiescent state. This is the realm of linear acoustics, where the propagation of a time-harmonic wave can be modeled using the Helmholtz equation. When the wavelength is small relative to the size of a scattering obstacle, techniques from geometric optics are applicable. Geometric methods such as raytracing are often used for computational room acoustics simulations in situations where the geometry of the built environment is sufficiently complicated. At the same time, the high-frequency approximation of the Helmholtz equation is described by two partial differential equations: the eikonal equation, whose solution gives the first arrival time of a geometric acoustics/optics wavefront as a field; and a transport equation, the solution of which describes the amplitude of that wavefield. Phenomena related to high-frequency acoustic diffraction are frequently omitted from these models because of their complexity. These phenomena can be modeled using a high-frequency diffraction theory, such as the uniform theory of diffraction. Despite their shortcomings, geometric methods for room acoustics provide a useful trade-off between realism and computational efficiency. Motivated by the limitations of geometric methods, we approach the problem of geometric acoustics using numerical methods for solving partial differential equations. Our focus is offline sound propagation in a high-frequency regime where directly solving the wave or Helmholtz equations is infeasible. To this end, we conduct a broad-based survey of semi-Lagrangian solvers for the eikonal equation, which make the local ray information of the solution explicit. We develop efficient, first-order solvers for the eikonal equation in 3D, called ordered line integral methods (OLIMs). The OLIMs provide intuition about how to design work-efficient semi-Lagrangian eikonal solvers, but their first order accuracy is not sufficient to compute the amplitude consistently. Motivated by the requirements of sound propagation simulations, we develop higher-order semi-Lagrangian eikonal solvers which we term jet marching methods (JMMs). JMMs augment the efficiency of OLIMs by additionally transporting higher-order derivative information of the eikonal in a causal fashion, which allows for high-order solution of the eikonal equation using compact stencils. We use the information made available locally by our JMMs to use paraxial raytracing to simultaneously solve the transport equation yielding the amplitude. We initially develop a JMM which handles a smoothly varying speed of sound on a regular grid in 2D. Motivated by the requirements of room acoustics applications, we develop a second-order JMM for solving the eikonal equation on a tetrahedron mesh for a constant speed of sound as a special case. As before, we use paraxial raytracing to compute the amplitude. Additionally, we compute multiple arrivals by reinitializing the eikonal equation on reflecting walls and diffracting edges. To compute these scattered fields, we devise algorithms which allow us to apply reflection and diffraction boundary conditions for the eikonal and amplitude. For the amplitude, we construct algorithms that allow us to apply the uniform theory of diffraction in a semi-Lagrangian setting efficiently