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1. 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 themore »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« less
2. The cohomology of semi-infinite Deligne–Lusztig varieties

   https://doi.org/10.1515/crelle-2019-0039  

   Chan, Charlotte ( January 2020 , Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite Deligne–Lusztig varieties attached to division algebras over local fields. We also prove the two conjectures of Boyarchenko on these varieties. It is known that in this setting, the semi-infinite Deligne–Lusztig varieties are ind-schemes comprised of limits of certain finite-type schemes X h {X_{h}} . Boyarchenko’s two conjectures are on the maximality of X h {X_{h}} and on the behavior of the torus-eigenspaces of their cohomology. Both of these conjectures were known in full generality only for division algebras with Hasse invariant 1 / n {1/n} in the case h = 2 {h=2} (the “lowest level”) by the work of Boyarchenko–Weinstein on the cohomology of a special affinoid in the Lubin–Tate tower. We prove that the number of rational points of X h {X_{h}} attains its Weil–Deligne bound, so that the cohomology of X h {X_{h}} is pure in a very strong sense. We prove that the torus-eigenspaces of the cohomology group H c i ⁢ ( X h ) {H_{c}^{i}(X_{h})} are irreducible representations and are supported in exactly one cohomological degree. Finally, we give a complete description of the homology groups of the semi-infinite Deligne–Lusztig varieties attachedmore »to any division algebra, thus giving a geometric realization of a large class of supercuspidal representations of these groups. Moreover, the correspondence θ ↦ H c i ⁢ ( X h ) ⁢ [ θ ] {\theta\mapsto H_{c}^{i}(X_{h})[\theta]} agrees with local Langlands and Jacquet–Langlands correspondences. The techniques developed in this paper should be useful in studying these constructions for p -adic groups in general.« less
3. Slope classicality in higher Coleman theory via highest weight vectors in completed cohomology

   https://doi.org/10.1073/pnas.2208249119  

   Howe, Sean ( November 2022 , Proceedings of the National Academy of Sciences)

   We give a proof of the slope classicality theorem in classical and higher Coleman theory for modular curves of arbitrary level using the completed cohomology classes attached to overconvergent modular forms. The latter give an embedding of the quotient of overconvergent modular forms by classical modular forms, which is the obstruction space for classicality in either cohomological degree, into a unitary representation of GL 2 ( ℚ p ) . The U p operator becomes a double coset, and unitarity yields slope vanishing.
4. Optimal Coding Theorems in Time-Bounded Kolmogorov Complexity

   Lu, Zhenjian ; Oliveira, Igor C. ; Zimand, Marius ( January 2022 , 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022))

   Mikołaj Bojańczyk and Emanuela Merelli and David P. Woodruff (Ed.)

   The classical coding theorem in Kolmogorov complexity states that if an n-bit string x is sampled with probability δ by an algorithm with prefix-free domain then K(x) ≤ log(1/δ) + O(1). In a recent work, Lu and Oliveira [31] established an unconditional time-bounded version of this result, by showing that if x can be efficiently sampled with probability δ then rKt(x) = O(log(1/δ)) + O(log n), where rKt denotes the randomized analogue of Levin’s Kt complexity. Unfortunately, this result is often insufficient when transferring applications of the classical coding theorem to the time-bounded setting, as it achieves a O(log(1/δ)) bound instead of the information-theoretic optimal log(1/δ). Motivated by this discrepancy, we investigate optimal coding theorems in the time-bounded setting. Our main contributions can be summarised as follows. • Efficient coding theorem for rKt with a factor of 2. Addressing a question from [31], we show that if x can be efficiently sampled with probability at least δ then rKt(x) ≤ (2 + o(1)) · log(1/δ) +O(log n). As in previous work, our coding theorem is efficient in the sense that it provides a polynomial-time probabilistic algorithm that, when given x, the code of the sampler, and δ, it outputs, withmore »probability ≥ 0.99, a probabilistic representation of x that certifies this rKt complexity bound. • Optimality under a cryptographic assumption. Under a hypothesis about the security of cryptographic pseudorandom generators, we show that no efficient coding theorem can achieve a bound of the form rKt(x) ≤ (2 − o(1)) · log(1/δ) + poly(log n). Under a weaker assumption, we exhibit a gap between efficient coding theorems and existential coding theorems with near-optimal parameters. • Optimal coding theorem for pKt and unconditional Antunes-Fortnow. We consider pKt complexity [17], a variant of rKt where the randomness is public and the time bound is fixed. We observe the existence of an optimal coding theorem for pKt, and employ this result to establish an unconditional version of a theorem of Antunes and Fortnow [5] which characterizes the worst-case running times of languages that are in average polynomial-time over all P-samplable distributions.« less
5. Boundary regularity and stability for spaces with Ricci bounded below

   https://doi.org/10.1007/s00222-021-01092-8  

   Bruè, Elia ; Naber, Aaron ; Semola, Daniele ( May 2022 , Inventiones mathematicae)

   Abstract This paper studies the structure and stability of boundaries in noncollapsed $${{\,\mathrm{RCD}\,}}(K,N)$$ RCD ( K , N ) spaces, that is, metric-measure spaces $$(X,{\mathsf {d}},{\mathscr {H}}^N)$$ ( X , d , H N ) with Ricci curvature bounded below. Our main structural result is that the boundary $$\partial X$$ ∂ X is homeomorphic to a manifold away from a set of codimension 2, and is $$N-1$$ N - 1 rectifiable. Along the way, we show effective measure bounds on the boundary and its tubular neighborhoods. These results are new even for Gromov–Hausdorff limits $$(M_i^N,{\mathsf {d}}_{g_i},p_i) \rightarrow (X,{\mathsf {d}},p)$$ ( M i N , d g i , p i ) → ( X , d , p ) of smooth manifolds with boundary, and require new techniques beyond those needed to prove the analogous statements for the regular set, in particular when it comes to the manifold structure of the boundary $$\partial X$$ ∂ X . The key local result is an $$\varepsilon $$ ε -regularity theorem, which tells us that if a ball $$B_{2}(p)\subset X$$ B 2 ( p ) ⊂ X is sufficiently close to a half space $$B_{2}(0)\subset {\mathbb {R}}^N_+$$ B 2 ( 0 ) ⊂more »R + N in the Gromov–Hausdorff sense, then $$B_1(p)$$ B 1 ( p ) is biHölder to an open set of $${\mathbb {R}}^N_+$$ R + N . In particular, $$\partial X$$ ∂ X is itself homeomorphic to $$B_1(0^{N-1})$$ B 1 ( 0 N - 1 ) near $$B_1(p)$$ B 1 ( p ) . Further, the boundary $$\partial X$$ ∂ X is $$N-1$$ N - 1 rectifiable and the boundary measure "Equation missing" is Ahlfors regular on $$B_1(p)$$ B 1 ( p ) with volume close to the Euclidean volume. Our second collection of results involve the stability of the boundary with respect to noncollapsed mGH convergence $$X_i\rightarrow X$$ X i → X . Specifically, we show a boundary volume convergence which tells us that the $$N-1$$ N - 1 Hausdorff measures on the boundaries converge "Equation missing" to the limit Hausdorff measure on $$\partial X$$ ∂ X . We will see that a consequence of this is that if the $$X_i$$ X i are boundary free then so is X .« less