Hybrid Quantum-Classical Solver for the Helmholtz Equation

We present a versatile and efficient quantum algorithm based on the Lattice Boltzmann method (LBM) approximate solution of the linear advection-diffusion equation (ADE). We emphasize that the LBM approximation modifies the diffusion term of the underlying exact ADE and leads to a modified equation (mADE). Due to its versatility in terms of operator splitting, the proposed quantum LBM algorithm for the mADE provides a building block for future quantum algorithms to solve the linearized Navier-Stokes equation on quantum computers. We split the algorithm into four operations: initialization, collision, streaming, and calculation of the macroscopic quantities. We propose general quantum building blocks for each operator, which adapt intrinsically from the general three-dimensional case to smaller dimensions and apply to arbitrary lattice-velocity sets. Based on (sub-linear) amplitude data encoding, we propose improved initialization and collision operations with reduced complexity and efficient sampling-based simulation. Quantum streaming algorithms are based on previous developments. The proposed quantum algorithm allows for the computation of successive time steps, requiring full state measurement and reinitialization after every time step. It is validated by comparison with a digital implementation and based on analytical solutions in one and two dimensions. Furthermore, we demonstrate the versatility of the quantum algorithm for two cases with non-uniform advection velocities in two and three dimensions. Various velocity sets are considered to further highlight the flexibility of the algorithm. We benchmark our optimized quantum algorithm against previous methods employed in sampling-based quantum simulators. We demonstrate sampling efficiency, with sampling accelerated convergence requiring fewer shots.

This thesis is concerned with the solution of the shallow water wave equations. To solve the equations for a real life domain of flow a numerical solution is required. Analytical solutions are useful for testing numerical solutions. Previous research in this area is examined. Existing analytical solutions of the nonlinear shallow water equations involving moving boundaries do not include bottom friction. Existing analytical solutions of the nonlinear shallow water equations involving fixed boundaries can be modifed. The SLM (Selective Lumped Mass) scheme for numerically solving the shallow water equations has been used to accurately model tides both in bays with fixed boundaries and bays with moving boundaries. The SLM scheme for domains with moving boundaries has not been fully explained in published papers. Analytical solutions are established for moving boundary shallow water equations involving nonlinear continuity for unforced frictional flow in parabolic canals, circular paraboloids and elliptical paraboloids, and both for forced frictional flow and unforced frictionless flow in a parabolic canal. Analytical solutions are established for one dimensional nonlinear frictionless shallow water wave flow in a basin of constant depth, with a fixed boundary and with a sinusoidal input at the open boundary. The SLM scheme is coded in Visual C ++ and validated against an analytical solution, a convergence study is carried out for the SLM scheme and a computer program is written in Visual C ++ to generate finite element meshes. The SLM scheme is used to model one dimensional nonlinear shallow water flow in a basin of constant depth, with fixed boundary and with sinusoidal input at the open boundary; results obtained were close to those in the analytical solution. The SLM scheme when applied to a moving boundary domain has been modifed, with the resultant scheme used to model forced frictionless flow and forced frictional flow in a parabolic canal, with results close to those of the analytical solutions developed. The SLM scheme is applied to model accurately the existing tidal heights and currents in Port Phillip Bay, Victoria, Australia. The SLM scheme is applied to model the effect of proposed channel deepening on the tides in Port Phillip Bay; the effect is small, increasing the tidal heights by at most seven millimetres. The conclusion of this thesis is that successful analytical and numerical solutions of the shallow water equations are developed both for domains with fixed boundaries and domains with moving boundaries.

Seismic forward modeling is a computationally and data-intensive stage in the seismic processing workflow. By profiling the kernels of seismic forward modeling algorithms, it was observed that they need to access a wide variety of memory locations, in addition to the computational cost of performing floating-point operations for the numerical solution of wave equations. In this context, the Roofline model was used to analyze six representative computing kernels in seismic modeling on GPU environment to indicate bottlenecks in the performance and suggest improvements of these wave equation propagators. Based on this, six viscoacoustic equations were implemented using the Devito tool. Experimental data have shown that optimizations in increasing data reuse and decreasing off-chip memory traffic can significantly improve performance.

Longest common substring (LCS) is an important text processing problem, which has recently been investigated in the quantum query model. The decision version of this problem, LCS with threshold \(d\) , asks whether two length- \(n\) input strings have a common substring of length \(d\) . The two extreme cases, \(d=1\) and \(d=n\) , correspond, respectively to Element Distinctness and Unstructured Search, two fundamental problems in quantum query complexity. However, the intermediate case \(1\ll d\ll n\) was not fully understood. We show that the complexity of LCS with threshold \(d\) smoothly interpolates between the two extreme cases up to \(n^{o(1)}\) factors: — LCS with threshold \(d\) has a quantum algorithm in \(n^{2/3+o(1)}/d^{1/6}\) query complexity and time complexity, and requires at least \(\Omega(n^{2/3}/d^{1/6})\) quantum query complexity. Our result improves upon previous upper bounds \(\widetilde{O}(\min\{n/d^{1/2},n^{2/3}\})\) (Le Gall and Seddighin ITCS 2022, Akmal and Jin SODA 2022), and answers an open question of Akmal and Jin. Our main technical contribution is a quantum speed-up of the powerful String Synchronizing Set technique introduced by Kempa and Kociumaka (STOC 2019). It consistently samples \(n/\tau^{1-o(1)}\) synchronizing positions in the string depending on their length- \(\Theta(\tau)\) contexts, and each synchronizing position can be reported by a quantum algorithm in \(\widetilde{O}(\tau^{1/2+o(1)})\) time. Our quantum string synchronizing set also yields a near-optimal LCE data structure in the quantum setting. As another application of our quantum string synchronizing set, we study the \(k\) -mismatch Matching problem, which asks if the pattern has an occurrence in the text with at most \(k\) Hamming mismatches. Using a structural result of Charalampopoulos et al. (FOCS 2020), we obtain: — \(k\) -mismatch matching has a quantum algorithm with \(k^{3/4}n^{1/2+o(1)}\) query complexity and \(\widetilde{O}(kn^{1/2})\) time complexity. We also observe a non-matching quantum query lower bound of \(\Omega(\sqrt{kn})\) .

At present, the theory of fractional calculus is widely used in many fields of science for modeling various processes. Differential equations with fractional derivatives are used to model the migration of pollutants in porous inhomogeneous media and allow a more correct description of the behavior of pollutants at large distances from the source. The analytical solution of differential equations with fractional order derivatives is often very complicated or even impossible. There has been proposed a numerical method for solving fractional differential equations in partial derivatives with respect to time to describe the migration of pollutants in groundwater. An implicit difference scheme is developed for the numerical solution of a non-stationary fractional differential equation, which is an analogue of the well-known implicit Crank-Nicholson difference scheme. The system of difference equations is presented in matrix form. The solution of the problem is reduced to the multiple solution of a tridiagonal system of linear algebraic equations by the tridiagonal matrix algorithm. The results of evaluating the spread of pollutant in groundwater based on the numerical method for model examples are presented. The concentrations of the substance obtained on the basis of the analytical and numerical solutions of the unsteady one-dimensional fractional differential equation are compared. The results obtained using the proposed method and on the basis of the well-known analytical solution of the fractional differential equation are in fairly good agreement with each other. The relative error is on average 9%. In contrast to the well-known analytical solution, the developed numerical method can be used to model the spread of pollutants in groundwater, taking into account their biodegradation.

The two-site Heisenberg model has an extraordinarily simple analytical solution and is traditionally used as a benchmark against numerical methods, such as exact diagonalization and Monte Carlo methods. In the same spirit, we benchmark three quantum algorithms that are implemented in a quantum computer against the analytical solution of this model. In particular, this presentation includes a description of the standard and iterative quantum phase estimation algorithms, as well as the variational quantum eigensolver. These quantum algorithms are introduced in a pedagogical fashion allowing newcomers to the subject, familiar with only the most basic quantum mechanical systems, to easily reproduce the presented results and apply the methods to other problems, thus building a seemingly under-appreciated path towards useful quantum algorithms through the lens of simulating and computing properties of physical quantum systems.

An implementation of a second order unsplit perfectly matched layer (PML) absorbing boundary condition is presented. The formulation is given in the context of seismic modelling and simple models are used to showcase the increased performance. The formulation is presented as a correction procedure to the unchanged governing equations. This approach makes the implementation especially suited for existing FDTD modelling codes. Although, a second order PML might perform better in some cases than the classic or CFS-PML the increased computational requirements and the difficulty in optimising a greater number of PML parameters must be taken into account in deciding the most appropriate PML for a given seismic modelling problem.

A unified analytical solution of the steady-state atmospheric diffusion equation

Trace formulas are derived for the coefficients of a matrix differential equation representing wave propagation in a medium which supports N types of waves. In general these waves are coupled to each other but are assumed to become uncoupled as lzl -> ?o. The two coefficients of the ordinary differential equation, N x N matrices, are assumed known for z 0. A point source is located in the region z 0. The unique features of this work are a derivation of the trace formula in which the Jost functions are not needed, an alternate measurement method based on an impedance concept and a trace equation based on the impedance concept. Introduction. Motivated by the work of Trubowitz (1), trace formula methods were introduced by Deift and Trubowitz (2) to study one-dimensional inverse quantum mechanical scattering problems on the line. The trace formula expresses the potential as a functional of the reflection coefficient. Greene (3) used the same techniques to study the one-dimensional acoustical wave equation with the angular frequency playing the role of the quantum mechanical energy. Stickler and Deift (4) examined the acoustic problem of a point source in an inhomogeneous half space bounded by a pressure release surface. The sound speed was recovered from a measurement of the normal component of the acoustic velocity at the pressure release surface. The role of the quantum mechanical energy was played by a vertical wave number, which does not correspond to either a spatial or temporal derivative. Using again the analogy between vertical wave number (the wave number in the direction of the coefficients' variation) and the quantum mechanical energy, Stickler (5) showed how the trace formula methods could be used to recover the sound speed when it approaches different values at plus and minus infinity. In a quantum mechanical context this means that the potential approaches zero, say at minus infinity, but some positive constant, not necessarily known, at plus infinity. (In the Deift-Trubowitz analysis the potential approached zero in both limits.) Stickler (6) also discussed this same problem in an electromagnetic context. Tomei (7) has used the trace methods to recover the coefficients of a linear, third order equation associated with the Boussinesq equation. In the present paper the trace method is extended to a one-dimensional vector Helmholtz equation which supports N types of waves and for which the waves are coupled to each other except at infinity. The trace method yields a functional relation- ship between the coefficients of the Helmholtz equation, a reflection coefficient described below and assumed known and a specific solution of the vector Helmholtz equation with known and prescribed initial conditions. These relationships are called the trace formulas and are derived here. The coefficients of the vector Helmholtz differential equation can be replaced by these trace formulas and this results in a system of coupled, nonlinear ordinary differential equations with known initial condi- tions. Once this solution is determined, the coefficients can be recovered. The existence of a solution of this coupled system is not given.

Analytical solutions of the linearized Boussinesq equation and a fully implicit finite-difference numerical solution of the nonlinear Boussinesq equation were obtained to study transient and steady-state water table rise in a homogeneous, isotropic, and incompressible unconfined sloping aquifer. The rise was due to seepage from two canals located at different elevations above the sloping impermeable barrier and constant recharge from the land surface. Proposed analytical solutions were verified with existing analytical solutions for a horizontal aquifer and were found in close agreement. The effect of recharge and slope of the impermeable barrier on water table rise predicted by both analytical and numerical solutions was studied by considering a numerical example. The effect of the linearization of the Boussinesq equation on water table rise was also studied by comparing the water table heights predicted by the numerical solution with those computed from the analytical solution. The analytical solution overestimates water table elevations compared to those obtained from the numerical solution, and the difference in water table in the middle region decreases with increase in time.

Simplified solution to one-dimensional consolidation with threshold gradient

The aim of this paper is first to briefly describes a mathematical approach for the direct coupling of analytical formal solution of Maxwell's equations (AM) and magnetic equivalent circuits (MEC) (reluctance networks or permeance networks) methods for the electromagnetic modeling of electric machines. This coupling can help solve the problem of air-gap modeling in MEC method, and the consideration of the local magnetic saturation in modeling approaches involving analytical formal solution of Maxwell's equations. The formal solution of Maxwell's equations is obtained using separation of variable method along with Fourier series analysis; it is used to model constant permeability regions (magnetic air-gap). The MEC method is used to model regions which include ferromagnetic parts. MEC model is generated based on a mesh of the studied domain. The aim of this contribution is mainly to study the effect of mesh density, used to generate the MEC model, on results quality issued from this new model. A permanent magnet linear synchronous machine (PMLSM) is used as a case study. Results issued from a finite element analysis (FEA) are used as reference for assessing the results quality.

Hybrid finite analytic solution (HFAS), Galerkin's method based finite element solution (FES) and fully implicit finite difference solution (FIFDS) of one dimensional nonlinear Boussinesq equation and Analytical solution of Boussinesq equation linearized by Baumann's transformation (analytical solution I, AS I) as well as linearized by Werner's transformation (analytical solution II, AS II) were employed to obtain water table rise in a horizontal unconfined aquifer lying between two canals located at finite distance having different elevations and subjected to various patterns of recharge, i.e. zero recharge, constant recharge, as well as time varying recharge. Considering HFAS as benchmark solution, water table in mid region as obtained from FES followed by FIFDS was observed quite close to that obtained from HFAS and as per L2 and Tchebycheff norms computation, it was ranked at first and second place, respectively. Both AS I and AS II predicted higher water table at t = 5 days but at t = 10 days, AS I predicted lower and AS II predicted higher water table at all distances due to linearization effect. So, analytical solutions of linearized Boussinesq equation were rated lower than numerical solutions of nonlinear Boussinesq equation.

In this study, an exact analytical solution for the convective heat transfer equation from a semi-spherical fin was presented. To obtain a mathematical model, the system was assumed to be a lump in the vertical direction and the governing equation in the Cartesian coordinate was transferred to the Mathieu equation. The exact solution was compared with numerical results such as the finite difference method and midpoint method with Richardson extrapolation (Midrich). Not surprisingly, the exact solution prevailed over the numerical solutions in terms of accuracy and ease of use. Furthermore, the effect of Biot number on the heat transfer of the fin and the fine performance was investigated. The relative error of the results obtained from the analytical and numerical solutions at the base, center, and tip of the fin was 0, 7.72, and 40.25 percent, respectively. The results showed that the relative error between the analytical and numerical solutions depends on the Biot number and varies as a function of the fin length. The obtained analytical solution could be encouraging from different mathematical and industrial applications' points of view.

The Gaussian Beam (GB) is an asymptotic solution of the elastodynamic equation in the paraxial vicinity of a central ray, which approaches better the wave field than the standard zero-order ray theory. The GB regularity in the description of the wave field, as well as its high accuracy in some singular regions of the propagation medium, provide us with a strong alternative to solve seismic modeling and imaging problems. In this paper, we use the concept of the projected Fresnel zone to limit the superposition integral of Gaussian beams, in order to obtain a more stable Gaussian beam propagation. This result is used to calculate synthetic seismograms with more effectiveness.