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1. The numerical unified transform method for initial-boundary value problems on the half-line

   https://doi.org/10.1093/imanum/drab007  

   Deconinck, Bernard ; Trogdon, Thomas ; Yang, Xin ( March 2021 , IMA Journal of Numerical Analysis)

   null (Ed.)

   Abstract We implement the unified transform method of Fokas as a numerical method to solve linear evolution partial differential equations on the half-line. The method computes the solution at any $x$ and $t$ without spatial discretization or time stepping. With the help of contour deformations and oscillatory integration techniques, the method’s complexity does not increase for large $x,t$ and the method is more accurate as $x,t$ increase (absolute errors are smaller, relative errors are bounded). Our goal is to make no assumptions on the functional form of the initial or boundary functions beyond some decay and smoothness, while maintaining high accuracy in a large region of the $(x,t)$ plane. 

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2. Embedding properties of network realizations of dissipative reduced order models

   Druskin, Guddati ( July 2022 , HOUSEHOLDER SYMPOSIUM XXI)

   Embedding properties of network realizations of dissipative reduced order models Jörn Zimmerling, Mikhail Zaslavsky,Rob Remis, Shasri Moskow, Alexander Mamonov, Murthy Guddati, Vladimir Druskin, and Liliana Borcea Mathematical Sciences Department, Worcester Polytechnic Institute https://www.wpi.edu/people/vdruskin Abstract Realizations of reduced order models of passive SISO or MIMO LTI problems can be transformed to tridiagonal and block-tridiagonal forms, respectively, via dierent modications of the Lanczos algorithm. Generally, such realizations can be interpreted as ladder resistor-capacitor-inductor (RCL) networks. They gave rise to network syntheses in the rst half of the 20th century that was at the base of modern electronics design and consecutively to MOR that tremendously impacted many areas of engineering (electrical, mechanical, aerospace, etc.) by enabling ecient compression of the underlining dynamical systems. In his seminal 1950s works Krein realized that in addition to their compressing properties, network realizations can be used to embed the data back into the state space of the underlying continuum problems. In more recent works of the authors Krein's ideas gave rise to so-called nite-dierence Gaussian quadrature rules (FDGQR), allowing to approximately map the ROM state-space representation to its full order continuum counterpart on a judicially chosen grid. Thus, the state variables can be accessed directly from the transfer function without solving the full problem and even explicit knowledge of the PDE coecients in the interior, i.e., the FDGQR directly learns" the problem from its transfer function. This embedding property found applications in PDE solvers, inverse problems and unsupervised machine learning. Here we show a generalization of this approach to dissipative PDE problems, e.g., electromagnetic and acoustic wave propagation in lossy dispersive media. Potential applications include solution of inverse scattering problems in dispersive media, such as seismic exploration, radars and sonars. To x the idea, we consider a passive irreducible SISO ROM fn(s) = Xn j=1 yi s + σj , (62) assuming that all complex terms in (62) come in conjugate pairs. We will seek ladder realization of (62) as rjuj + vj − vj−1 = −shˆjuj , uj+1 − uj + ˆrj vj = −shj vj , (63) for j = 0, . . . , n with boundary conditions un+1 = 0, v1 = −1, and 4n real parameters hi, hˆi, ri and rˆi, i = 1, . . . , n, that can be considered, respectively, as the equivalent discrete inductances, capacitors and also primary and dual conductors. Alternatively, they can be viewed as respectively masses, spring stiness, primary and dual dampers of a mechanical string. Reordering variables would bring (63) into tridiagonal form, so from the spectral measure given by (62 ) the coecients of (63) can be obtained via a non-symmetric Lanczos algorithm written in J-symmetric form and fn(s) can be equivalently computed as fn(s) = u1. The cases considered in the original FDGQR correspond to either (i) real y, θ or (ii) real y and imaginary θ. Both cases are covered by the Stieltjes theorem, that yields in case (i) real positive h, hˆ and trivial r, rˆ, and in case (ii) real positive h,r and trivial hˆ,rˆ. This result allowed us a simple interpretation of (62) as the staggered nite-dierence approximation of the underlying PDE problem [2]. For PDEs in more than one variables (including topologically rich data-manifolds), a nite-dierence interpretation is obtained via a MIMO extensions in block form, e.g., [4, 3]. The main diculty of extending this approach to general passive problems is that the Stieltjes theory is no longer applicable. Moreover, the tridiagonal realization of a passive ROM transfer function (62) via the ladder network (63) cannot always be obtained in port-Hamiltonian form, i.e., the equivalent primary and dual conductors may change sign [1]. 100 Embedding of the Stieltjes problems, e.g., the case (i) was done by mapping h and hˆ into values of acoustic (or electromagnetic) impedance at grid cells, that required a special coordinate stretching (known as travel time coordinate transform) for continuous problems. Likewise, to circumvent possible non-positivity of conductors for the non-Stieltjes case, we introduce an additional complex s-dependent coordinate stretching, vanishing as s → ∞ [1]. This stretching applied in the discrete setting induces a diagonal factorization, removes oscillating coecients, and leads to an accurate embedding for moderate variations of the coecients of the continuum problems, i.e., it maps discrete coecients onto the values of their continuum counterparts. Not only does this embedding yields an approximate linear algebraic algorithm for the solution of the inverse problems for dissipative PDEs, it also leads to new insight into the properties of their ROM realizations. We will also discuss another approach to embedding, based on Krein-Nudelman theory [5], that results in special data-driven adaptive grids. References [1] Borcea, Liliana and Druskin, Vladimir and Zimmerling, Jörn, A reduced order model approach to inverse scattering in lossy layered media, Journal of Scientic Computing, V. 89, N1, pp. 136,2021 [2] Druskin, Vladimir and Knizhnerman, Leonid, Gaussian spectral rules for the three-point second dierences: I. A two-point positive denite problem in a semi-innite domain, SIAM Journal on Numerical Analysis, V. 37, N 2, pp.403422, 1999 [3] Druskin, Vladimir and Mamonov, Alexander V and Zaslavsky, Mikhail, Distance preserving model order reduction of graph-Laplacians and cluster analysis, Druskin, Vladimir and Mamonov, Alexander V and Zaslavsky, Mikhail, Journal of Scientic Computing, V. 90, N 1, pp 130, 2022 [4] Druskin, Vladimir and Moskow, Shari and Zaslavsky, Mikhail LippmannSchwingerLanczos algorithm for inverse scattering problems, Inverse Problems, V. 37, N. 7, 2021, [5] Mark Adolfovich Nudelman The Krein String and Characteristic Functions of Maximal Dissipative Operators, Journal of Mathematical Sciences, 2004, V 124, pp 49184934 Go back to Plenary Speakers Go back to Speakers Go back 

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3. Localized Model Reduction for Nonlinear Elliptic Partial Differential Equations: Localized Training, Partition of Unity, and Adaptive Enrichment

   https://doi.org/10.1137/22M148402X  

   Smetana, Kathrin ; Taddei, Tommaso ( June 2023 , SIAM Journal on Scientific Computing)

   We propose a component-based (CB) parametric model order reduction (pMOR) formulation for parameterized nonlinear elliptic partial differential equations. CB-pMOR is designed to deal with large-scale problems for which full-order solves are not affordable in a reasonable time frame or parameters' variations induce topology changes that prevent the application of monolithic pMOR techniques. We rely on the partition-of-unity method to devise global approximation spaces from local reduced spaces, and on Galerkin projection to compute the global state estimate. We propose a randomized data compression algorithm based on oversampling for the construction of the components' reduced spaces: the approach exploits random boundary conditions of controlled smoothness on the oversampling boundary. We further propose an adaptive residual-based enrichment algorithm that exploits global reduced-order solves on representative systems to update the local reduced spaces. We prove exponential convergence of the enrichment procedure for linear coercive problems; we further present numerical results for a two-dimensional nonlinear diffusion problem to illustrate the many features of our methodology and demonstrate its effectiveness. 

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4. Manakov system with parity symmetry on nonzero background and associated boundary value problems

   https://doi.org/10.1088/1751-8121/ac674a  

   Abeya, Asela ; Biondini, Gino ; Prinari, Barbara ( May 2022 , Journal of Physics A: Mathematical and Theoretical)

   Abstract We characterize initial value problems for the defocusing Manakov system (coupled two-component nonlinear Schrödinger equation) with nonzero background and well-defined spatial parity symmetry (i.e., when each of the components of the solution is either even or odd), corresponding to boundary value problems on the half line with Dirichlet or Neumann boundary conditions at the origin. We identify the symmetries of the eigenfunctions arising from the spatial parity of the solution, and we determine the corresponding symmetries of the scattering data (reflection coefficients, discrete spectrum and norming constants). All parity induced symmetries are found to be more complicated than in the scalar (i.e., one-component) case. In particular, we show that the discrete eigenvalues giving rise to dark solitons arise in symmetric quartets, and those giving rise to dark–bright solitons in symmetric octets. We also characterize the differences between the purely even or purely odd case (in which both components are either even or odd functions of x ) and the ‘mixed parity’ cases (in which one component is even while the other is odd). Finally, we show how, in each case, the spatial symmetry yields a constraint on the possible existence of self-symmetric eigenvalues, corresponding to stationary solitons, and we study the resulting behavior of solutions. 

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5. Riemann–Hilbert approach for an initial-boundary value problem of the two-component modified Korteweg-de Vries equation on the half-line

   https://doi.org/https://doi.org/10.1016/j.amc.2018.03.049  

   Hu, Bei-Bei ; Xia, Tie-Cheng ; Ma, Wen-Xiu ( September 2018 , Applied mathematics and computation)

   In this work, we investigate the two-component modified Korteweg-de Vries (mKdV) equation, which is a complete integrable system, and accepts a generalization of 4 × 4 matrix Ablowitz–Kaup–Newell-Segur (AKNS)-type Lax pair. By using of the unified transform approach, the initial-boundary value (IBV) problem of the two-component mKdV equation associated with a 4 × 4 matrix Lax pair on the half-line will be analyzed. Supposing that the solution {u1(x, t), u2(x, t)} of the two-component mKdV equation exists, we will show that it can be expressed in terms of the unique solution of a 4 × 4 matrix Riemann–Hilbert problem formulated in the complex λ-plane. Moreover, we will prove that some spectral functions s(λ) and S(λ) are not independent of each other but meet the global relationship. 

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