APL-UW

Daniel Shapero

Research Scientist/Engineer - Senior

Email

shapero@apl.washington.edu

Phone

206-543-1348

Department Affiliation

Polar Science Center

Education

B.S. Applied Mathematics, McGill University (Montreal, QC, Canada), 2010

Ph.D. Applied Mathematics, University of Washington - Seattle, 2017

Publications

2000-present and while at APL-UW

Consistent point data assimilation in Firedrake and Icepack

Nixon-Hill, R.W., D. Shapero, C.J. Cotter, and D.A. Ham, "Consistent point data assimilation in Firedrake and Icepack," Geosci. Model. Dev., 17, 5369-5386, doi:10.5194/gmd-17-5369-2024, 2024.

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12 Jul 2024

We present a high-level, differentiable, and composable abstraction for the point evaluation of the solution fields of partial differential equation models. The new functionality, embedded in the Firedrake automated finite element system, enables modellers to easily assimilate point data into their models at the point locations, rather than resorting to extrapolation to a computational mesh. We demonstrate the expressiveness and ease with which more mathematically defensible data assimilation can be performed with examples in the fields of groundwater hydrology and glaciology.

In various geoscience disciplines, modellers seek to estimate fields that are challenging to directly observe using measurements of other related fields. These measurements are often sparse, and it is common practice to first extrapolate these measurements to the grid or mesh used for computations. When this estimation procedure is viewed as a deterministic inverse problem, the extrapolation step is undesirable because the choice of extrapolation method introduces an arbitrary algorithmic degree of freedom that can alter the outcomes. When the estimation procedure is instead viewed through the lens of statistical inference, the extrapolation step is undesirable for the additional reason that it obscures the number of statistically independent measurements that are assimilated and thus makes it impossible to apply statistical goodness-of-fit tests or model selection criteria. The introduction of point evaluation into Firedrake, together with its integration into the automatic differentiation features of the system, greatly facilitates the direct assimilation of point data and thus improved methodology for solving both deterministic and statistical inverse problems.

Responses of the Pine Island and Thwaites glaciers to melt and sliding parameterizations

Joughin, I., D. Shapero, and P. Dutrieux, "Responses of the Pine Island and Thwaites glaciers to melt and sliding parameterizations," Cryosphere, 18, 2583-2601, doi:10.5194/tc-18-2583-2024, 2024.

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28 May 2024

The Pine Island and Thwaites glaciers are the two largest contributors to sea level rise from Antarctica. Here we examine the influence of basal friction and ice shelf basal melt in determining projected losses. We examine both Weertman and Coulomb friction laws with explicit weakening as the ice thins to flotation, which many friction laws include implicitly via the effective pressure. We find relatively small differences with the choice of friction law (Weertman or Coulomb) but find losses to be highly sensitive to the rate at which the basal traction is reduced as the area upstream of the grounding line thins. Consistent with earlier work on Pine Island Glacier, we find sea level contributions from both glaciers to vary linearly with the melt volume averaged over time and space, with little influence from the spatial or temporal distribution of melt. Based on recent estimates of melt from other studies, our simulations suggest that the combined melt-driven and sea level rise contribution from both glaciers may not exceed 10 cm by 2200, although the uncertainty in model parameters allows for larger increases. We do not include other factors, such as ice shelf breakup, that might increase loss, or factors such as increased accumulation and isostatic uplift that may mitigate loss.

High-order bounds-satisfying approximation of partial differential equations via finite element variational inequalities

Kirby, R.C., and D. Shapero, "High-order bounds-satisfying approximation of partial differential equations via finite element variational inequalities," Numer. Math., 156, 927-947, doi:10.1007/s00211-024-01405-y, 2024.

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30 Apr 2024

Solutions to many important partial differential equations satisfy bounds constraints, but approximations computed by finite element or finite difference methods typically fail to respect the same conditions. Chang and Nakshatrala (Comput Methods Appl Mech Eng 320:287–334, 2017) enforce such bounds in finite element methods through the solution of variational inequalities rather than linear variational problems. Here, we provide a theoretical justification for this method, including higher-order discretizations. We prove an abstract best approximation result for the linear variational inequality and estimates showing that bounds-constrained polynomials provide comparable approximation power to standard spaces. For any unconstrained approximation to a function, there exists a constrained approximation which is comparable in the W1,p norm. In practice, one cannot efficiently represent and manipulate the entire family of bounds-constrained polynomials, but applying bounds constraints to the coefficients of a polynomial in the Bernstein basis guarantees those constraints on the polynomial. Although our theoretical results do not guaruntee high accuracy for this subset of bounds-constrained polynomials, numerical results indicate optimal orders of accuracy for smooth solutions and sharp resolution of features in convection–diffusion problems, all subject to bounds constraints.

More Publications

In The News

Edge of Pine Island Glacier’s ice shelf is ripping apart, causing key Antarctic glacier to gain speed

UW News, Hannah Hickey

For decades, the ice shelf helping to hold back one of the fastest-moving glaciers in Antarctica has gradually thinned. Analysis of satellite images reveals a more dramatic process in recent years: From 2017 to 2020, large icebergs at the ice shelf’s edge broke off, and the glacier sped up.

11 Jun 2021

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