Boundary integral equation based numerical solutions of helmholtz transmission problems for composite scatters

In this dissertation, an in-depth comparison between boundary integral equation solvers and Domain Decomposition Methods (DDM) for frequency domain Helmholtz transmission problems in composite two-dimensional media is presented. Composite media are characterized by piece-wise constant material properties (i.e., index of refraction) and thus, they exhibit interfaces of material discontinuity and multiple junctions. Whenever possible to use, boundary integral methods for solution of Helmholtz boundary value problems are computationally advantageous. Indeed, in addition to the dimensional reduction and straightforward enforcement of the radiation conditions that these methods enjoy, they do not suffer from the pollution effect present in volumetric discretization. The reformulation of Helmholtz transmission problems in composite media in terms of boundary integral equations via multi-traces constitutes one of the recent success stories in the boundary integral equation community. Multi-trace formulations (MTF) incorporate local Dirichlet and Neumann traces on subdomains within Green’s identities and use restriction and extension by zero operators to enforce the intradomain continuity of the fields and fluxes. Through usage of subdomain Calderon projectors, the transmission problem is cast into a linear system form whose unknowns are local Dirichlet and Neumann traces (two such traces per interface of material discontinuity) and whose operator matrix consists of diagonal block boundary integral operators associated with the subdomains and extension/projections off diagonal blocks. This particular form of the matrix operator associated with MTF is amenable to operator preconditioning via Calderon projectors. DDM rely on subdomain solutions that are matched via transmission conditions on the subdomain interfaces that are equivalent to the physical continuity of fields and traces. By choosing the appropriate transmission conditions, the convergence of DDM for frequency domain scattering problems can be accelerated. Traditionally, the intradomain transmission conditions were chosen to be the classical outgoing Robin/impedance boundary conditions. When the ensuing DDM linear system is solved via Krylov subspace methods, the convergence of DDM with classical Robin transmission conditions is slow and adversely affected by the number of subdomains. Heuristically, this behavior is explained by the fact that Robin boundary conditions are first order approximations of transparent boundary conditions, and thus there is significant information that is reflected back into a given subdomain from adjacent subdomains. Clearly, using more sophisticated transparent boundary conditions facilitates the information exchange between subdomains. For instance, Dirichlet-to-Neumann (DtN) operators of adjacent domains or suitable approximations of these can be used in the form of generalized Robin boundary conditions to increase the rate of the convergence of iterative solvers of DDM linear systems. The approximations of DtN operators that are expressed in terms of Helmholtz hypersingular operators (e.g., the normal derivative of the double layer operator) are used in this dissertation. The incorporation of these in a DDM framework is subtle, and an effective method is proposed to blend these transmission operators in the presence of multiple junctions. Conceptually, the information exchange between subdomains is realized through certain Robin-to-Robin (RtR) operators, which how to compute robustly via integral equation formulations is shown. All of the Helmholtz boundary integral operators that feature in Calderon’s calculus are discretized via Nystr¨om methods that rely on sigmoid transforms, trigonometric interpolation, and singular kernel splitting. Sigmoid transforms are means to polynomially accumulate discretization points toward corners without compromising the discretization density in smooth boundary portions. A wide variety of numerical results is presented in this dissertation that illustrate the merits of each of the two approaches (MTF and DDM) for the solution of transmission problems in composite domains

Time integrable weighted dispersive estimates for the fourth order Schr\"odinger equation in three dimensions

We consider the fourth order Schr\"odinger operator $H=\Delta^2+V$ and show that if there are no eigenvalues or resonances in the absolutely continuous spectrum of $H$ that the solution operator $e^{-itH}$ satisfies a large time integrable $|t|^{-\frac54}$ decay rate between weighted spaces. This bound improves what is possible for the free case in two directions; both better time decay and smaller spatial weights. In the case of a mild resonance at zero energy, we derive the operator-valued expansion $e^{-itH}P_{ac}(H)=t^{-\frac34} A_0+t^{-\frac54}A_1$ where $A_0:L^1\to L^\infty$ is an operator of rank at most four and $A_1$ maps between polynomially weighted spaces.Comment: 24 pages, submitted. Revised according to referee's comments. arXiv admin note: text overlap with arXiv:1905.0289

On the Regularizing Property of Stochastic Gradient Descent

Stochastic gradient descent is one of the most successful approaches for solving large-scale problems, especially in machine learning and statistics. At each iteration, it employs an unbiased estimator of the full gradient computed from one single randomly selected data point. Hence, it scales well with problem size and is very attractive for truly massive dataset, and holds significant potentials for solving large-scale inverse problems. In the recent literature of machine learning, it was empirically observed that when equipped with early stopping, it has regularizing property. In this work, we rigorously establish its regularizing property (under \textit{a priori} early stopping rule), and also prove convergence rates under the canonical sourcewise condition, for minimizing the quadratic functional for linear inverse problems. This is achieved by combining tools from classical regularization theory and stochastic analysis. Further, we analyze the preasymptotic weak and strong convergence behavior of the algorithm. The theoretical findings shed insights into the performance of the algorithm, and are complemented with illustrative numerical experiments.Comment: 22 pages, better presentatio